3. Presentation Markup

Mathematical Markup Language (MathML) Version 2.0
2 MathML Fundamentals
3 Presentation Markup
3.1 Introduction
   3.1.1 What Presentation Elements Represent
   3.1.2 Terminology Used In This Chapter
   3.1.3 Required Arguments
   3.1.4 Elements with Special Behaviors
   3.1.5 Summary of Presentation Elements
3.2 Token Elements
   3.2.1 Attributes common to token elements
   3.2.2 Identifiers
   3.2.3 Numbers
   3.2.4 Operator, Fence, Separator or Accent
   3.2.5 Text
   3.2.6 Space
   3.2.7 String Literal
3.3 General Layout Schemata
   3.3.1 Horizontally Group Any Number of Subexpressions
   3.3.2 Fractions
   3.3.3 Radicals
   3.3.4 Style Change
   3.3.5 Error Message
   3.3.6 Adjust Space Around Content
   3.3.7 Making Content Invisible
   3.3.8 Content Inside Pair of Fences
   3.3.9 Enclose Content Inside Notation
3.4 Script and Limit Schemata
   3.4.1 Subscript
   3.4.2 Superscript
   3.4.3 Subscript-superscript Pair
   3.4.4 Underscript
   3.4.5 Overscript
   3.4.6 Underscript-overscript Pair
   3.4.7 Prescripts and Tensor Indices
3.5 Tables and Matrices
   3.5.1 Table or Matrix
   3.5.2 Row in Table or Matrix
   3.5.3 Entry in Table or Matrix
   3.5.4 Alignment Markers
   3.5.5 Equation Numbering
3.6 Enlivening Expressions
   3.6.1 Bind Action to Subexpression
4 Content Markup

3.1. Introduction

This chapter specifies the `presentation' elements of MathML, which can be used to describe the layout structure of mathematical notation. It is strongly recommended that, before reading the present chapter, one read section 2.3 [MathML Syntax and Grammar] on MathML syntax and grammar, which contains important information on MathML notations and conventions necessary for understanding some of the material in this chapter.

3.1.1. What Presentation Elements Represent

Presentation elements correspond to the `constructors' of traditional mathematical notation - that is, to the basic kinds of symbols and expression-building structures out of which any particular piece of traditional mathematical notation is built. They are designed to be medium-independent, in the sense that there are sensible ways to render them in audio, as well as in traditional visual media for mathematics. Because of the importance of traditional visual notation, the descriptions of which notational constructs the elements represent, and how they are typically rendered, is often given here in visual terms. However, the elements have been designed to contain enough information for good spoken renderings as well, provided the conventions described herein for their proper use are followed. Some attributes of these elements may make sense only for visual media, but most attributes can be treated in an analogous way in audio as well (for example, by a correspondence between time duration and horizontal extent).

One major anticipated use of MathML is to describe mathematical expressions within HTML documents, using multiple MathML expressions embedded in some manner in an HTML document. Note that HTML in general describes logical structures such as headings, paragraphs, etcetera. but only suggests (i.e. does not require) specific ways of rendering various logical parts of the document, in order to allow for medium-dependent rendering and for individual preferences of style; MathML presentation elements are fully compatible with this philosophy. This specification describes suggested visual rendering rules in some detail, but a particular MathML renderer is free to use its own rules as long as its renderings are intelligible.

The presentation elements are meant to express the syntactic structure of mathematical notation in much the same way as titles, sections, and paragraphs capture the higher level syntactic structure of a textual document. Because of this, for example, a single row of identifiers and operators, such as `x + a / b', will often be represented not just by one mrow element (which renders as a horizontal row of its arguments), but by multiple nested mrow elements corresponding to the nested subexpressions of which one mathematical expression is composed - in this case,

<mrow>
  <mi> x </mi>
  <mo> + </mo>
  <mrow>
     <mi> a </mi>
     <mo> / </mo>
     <mi> b </mi>
  </mrow>
</mrow>

Similarly, superscripts are attached not just to the preceding character, but to the full expression constituting their base. This structure allows for better-quality rendering of mathematics, especially when details of the rendering environment such as display widths are not known to the document author; it also greatly eases automatic interpretation of the mathematical structures being represented.

Certain extended characters, represented by entity references, are used to name operators or identifiers which in traditional notation render the same as other symbols, such as ⅆ, ⅇ, or ⅈ, or operators which usually render invisibly, such as ⁢, ⁡, or ⁣. These are distinct notational symbols or objects, as evidenced by their distinct spoken renderings and in some cases by their effects on linebreaking and spacing in visual rendering, and as such should be represented by the appropriate specific entity references. For example, the expression represented visually as `f(x)' would usually be spoken in English as `f of x' rather than just `f x'; this is expressible in MathML by the use of the ⁡ operator after the `f', which (in this case) can be aurally rendered as `of'.

The complete list of MathML entities is described in chapter 6 [Entities, Characters and Fonts].

3.1.2. Terminology Used In This Chapter

The MathML specification uses a number of technical terms to describe MathML-specific rules and conventions. The most notable example is the attribute value notations and conventions described in section 2.3.3 [MathML Attribute Values]. (See also the brief description of XML terminology in section 2.3.1 [An XML Syntax Primer].)

The remainder of this section introduces MathML-specific terminology and conventions used in this chapter.

3.1.2.1. Types of presentation elements

The presentation elements are divided into two classes. Token elements represent individual symbols, names, numbers, labels, etcetera, and can have only characters and entity references (or the vertical alignment element malignmark) as content. Layout schemata build expressions out of parts, and can have only elements as content (except for whitespace, which they ignore). There are also a few empty elements used only in conjunction with certain layout schemata.

All individual `symbols' in a mathematical expression should be represented by MathML token elements. The primary MathML token element types are identifiers (e.g. variables or function names), numbers, and operators (including fences, such as parentheses, and separators, such as commas). There are also token elements for representing text or whitespace which has more aesthetic than mathematical significance, and for representing `string literals' for compatibility with computer algebra systems. Note that although a token element represents a single meaningful `symbol' (name, number, label, mathematical symbol, etcetera), such symbols may be comprised of more than one character. For example sin and 24 are represented by the single tokens <mi>sin</mi> and <mn>24</mn> respectively.

In traditional mathematical notation, expressions are recursively constructed out of smaller expressions, and ultimately out of single symbols, with the parts grouped and positioned using one of a small set of notational structures, which can be thought of as `expression constructors'. In MathML, expressions are constructed in the same way, with the layout schemata playing the role of the expression constructors. The layout schemata specify the way in which subexpressions are built into larger expressions. The terminology derives from the fact that each layout schema corresponds to a different way of `laying out' its subexpressions to form a larger expression in traditional mathematical typesetting.

3.1.2.2. Terminology for other classes of elements and their relationships

The terminology used in this chapter for special classes of elements, and for relationships between elements, is as follows: The presentation elements are the MathML elements defined in the chapter. These elements are listed in section 3.1.5 [Summary of Presentation Elements]. The content elements are the MathML elements defined in chapter 4 [Content Markup]. The content elements are listed in section 4.4 [The Content Markup Elements].

A MathML expression is a single instance of any of the presentation elements with the exception of the empty elements none or mprescripts, or is a single instance of any of the content elements which are allowed as content of presentation elements (listed in section 5.2.3 [Presentation Markup Contained in Content Markup]). A subexpression of an expression E is any MathML expression which is part of the content of E, whether directly or indirectly, i.e. whether it is a `child' of E or not.

A child of a layout schema is also called an argument of that element. Token elements have no arguments, by definition, even though they can contain the malignmark element; this means that a malignmark element in a token is not an argument, whereas in a layout schema it is one.

As a consequence of the above definitions, the content of a layout schema consists exactly of a sequence of zero or more nonoverlapping elements which are its arguments (possibly with intervening whitespace, which is ignored in MathML). Note that an argument is almost always a subexpression; the only exceptions are the empty elements none and mprescripts which are allowed only as special arguments of the mmultiscripts element, but are not subexpressions because they are not MathML expressions as defined above.

3.1.2.3. Descriptions of presentation elements

Each MathML presentation element is described below in detail. The description starts with the information needed by authors of MathML (or of programs which generate MathML). The intended use of each element is described, along with the argument syntax it accepts. (There is also a table of argument count requirements and argument roles in section 3.1.3 [Required Arguments].) The valid attributes, along with their permissible and default values, are listed, and the effect of each attribute is discussed.

For certain elements, further information of interest mainly to those implementing MathML renderers is given in a subsection. This includes many details of one suggested set of rendering rules which can be used to render MathML expressions in a manner reminiscent of traditional visual notation.

3.1.3. Required Arguments

Many of the elements described herein require a specific number of arguments (always 1, 2, or 3). Recall that MathML uses the term argument to describe a child element with additional MathML-specific requirements, usually related to which position it occupies in its parent.

In the detailed descriptions of element syntax given below, the number of required arguments is implicitly indicated by giving names for the arguments at various positions. The descriptions, interpreted according to the convention just stated, fully specify the allowed numbers of arguments for every element defined in this chapter. A few elements have additional requirements on the number or type of arguments, which are described with the individual element. For example, some elements accept sequences of zero or more arguments - that is, they are allowed to occur with no arguments at all.

Note that MathML elements encoding rendered space do count as arguments of the elements they appear in. See section 3.2.6 [Space] for a discussion of the proper use of such space-like elements.

3.1.3.1. Inferred mrows

The elements listed in the following table as requiring exactly 1 argument (msqrt, mstyle, merror, mpadded, mphantom, and mtd) actually accept any number of arguments, but if the number of arguments is 0, or is more than 1, they treat their contents as a single inferred mrow formed from all their arguments.

For example,

<mtd>
</mtd>

is treated as if it were

<mtd>
  <mrow>
  </mrow>
</mtd>

and

<msqrt>
   <mo> - </mo>
   <mn> 1 </mn>
</msqrt>

is treated as if it were

<msqrt>
   <mrow>
      <mo> - </mo>
      <mn> 1 </mn>
   </mrow>
</msqrt>

This feature allows MathML data not to contain (and its authors to leave out) many mrow elements which would otherwise be necessary.

In the descriptions in this chapter of the above-listed elements' rendering behaviors, their content can be assumed to consist of exactly one expression, which may be an mrow element formed from their arguments in this manner. However, their argument counts are shown in the following table as exactly 1, since they are most naturally understood as acting on a single expression.

3.1.3.2. Table of argument requirements

For convenience, here is a table of each element's argument count requirements, and the roles of individual arguments when these are distinguished. Recall that a required argument count of 1 may indicate an inferred mrow.

3.1.4. Elements with Special Behaviors

Certain MathML presentation elements exhibit special behaviors in certain contexts. Such special behaviors are discussed in the detailed element descriptions below. However, for convenience, some of the most important classes of special behavior are listed here.

Certain elements are considered space-like; these are defined in section 3.2.6 [Space]. This definition affects some of the suggested rendering rules for mo elements (section 3.2.4 [Operator, Fence, Separator or Accent]).

Certain elements, e.g. msup, are able to embellish operators which are their first argument. These elements are listed in section 3.2.4 [Operator, Fence, Separator or Accent], which precisely defines an `embellished operator' and explains how this affects the suggested rendering rules for stretchy operators.

Certain elements treat their arguments as the arguments of an `inferred mrow' if they are not given exactly one argument, as explained in section 3.1.3 [Required Arguments].

The mtable element can infer mtrs around its arguments, and the mtr element can infer mtds, as explained in the sections about those elements.

3.1.5. Summary of Presentation Elements

3.1.5.1. Token Elements

3.1.5.2. General Layout Schemata

3.1.5.3. Script and Limit Schemata

3.1.5.4. Tables and Matrices

3.1.5.5. Enlivening Expressions

3.2. Token Elements

Token elements can contain any sequence of zero or more characters, or extended characters represented by entity references. In particular, tokens with empty content are allowed, and should typically render invisibly, with no width except for the normal extra spacing for that kind of token element. The allowed set of entity references for extended characters is given in chapter 6 [Entities, Characters and Fonts].

In MathML, characters and MathML entity references are only allowed to occur as part of the content of a token element. The only exception is whitespace between elements, which is ignored.

The malignmark element (see section 3.5.4 [Alignment Markers]) is the only element allowed in the content of tokens. It marks a place which can be vertically aligned with other objects, as explained in that section.

3.2.1. Attributes common to token elements

Several attributes related to text formatting are provided on all token elements except mspace, but on no other elements except mstyle. These are:

(See section 2.3.3 [MathML Attribute Values] for terminology and notation used in attribute value descriptions.)

Token elements (other than mspace) should be rendered as their content (i.e. in the visual case, as a closely-spaced horizontal row of standard glyphs for the characters in their content) using the attributes listed above, with surrounding spacing modified by rules or attributes specific to each type of token element. Some of the individual attributes are further discussed below.

Recall that all MathML elements, including tokens, accept class, style, and id attributes for compatibility with style sheet mechanisms, as described in section 2.3.4 [Attributes Shared by all MathML Elements]. In principle, the font properties controlled by the attributes listed above might be better handled using style sheets. When style sheet support becomes available for XML, future revisions of MathML will likely revisit the issue of font control.

MathML expressions are often embedded in a textual data format such as HTML, and their renderings are likewise embedded in a rendering of the surrounding text. The renderer of the surrounding text (e.g. a browser) should provide the MathML renderer with information about the rendering environment, including attributes of the surrounding text such as its font size, so that the MathML can be rendered in a compatible style. For this reason, most attribute values affecting text rendering are inherited from the rendering environment, as shown in the `default' column in the table above. (Note that it is also important for the rendering environment to provide the renderer with additional information, such as the baseline position of surrounding text, which is not specified by any MathML attributes.)

The exception to the general pattern of inheritance is the fontstyle attribute, whose default value is normal (non-slanted) for most tokens, but for mi depends on the content in a way described in the section about mi, section 3.2.2 [Identifiers]. Note that fontstyle is not inherited in MathML, even though the corresponding CSS1 property `font-style' is inherited in CSS.

The fontsize attribute specifies the desired font size. v-unit represents a unit of vertical length (see section 2.3.3.3 [CSS-compatible attributes]). The most common unit for specifying font sizes in typesetting is pt (points).

If the requested size of the current font is not available, the renderer should approximate it in the manner likely to lead to the most intelligible, highest quality rendering.

Many MathML elements automatically change fontsize in some of their children; see the discussion of scriptlevel in the section on mstyle, section 3.3.4 [Style Change].

The value of the fontfamily attribute should be the name of a font which may be available to a MathML renderer, or information which permits the renderer to select a font in some manner; acceptable values and their meanings are dependent on the specific renderer and rendering environment in use, and are not specified by MathML (but see the note about css-fontfamily below). (Note that the renderer's mechanism for finding fonts by name may be case-sensitive.)

If the value of fontfamily is not recognized by a particular MathML renderer, this should never be interpreted as a MathML error; rather, the renderer should either use a font which it considers to be a suitable substitute for the requested font, or ignore the attribute and act as if no value had been given.

Note that any use of the fontfamily attribute is unlikely to be portable across all MathML renderers. In particular, it should never be used to try to achieve the effect of a reference to an extended character (for example, by using a reference to a character in some symbol font which maps ordinary characters to glyphs for extended characters). As a corollary to this principle, MathML renderers should attempt to always produce intelligible renderings for the extended characters listed in chapter 6 [Entities, Characters and Fonts], even when these characters are not available in the font family indicated. Such a rendering is always possible - as a last resort, a character can be rendered to appear as an XML-style entity reference using one of the entity names given for the same character in chapter 6 [Entities, Characters and Fonts].

The symbol css-fontfamily refers to a legal value for the font-family property in CSS1, which is a comma-separated list of alternative font family names or generic font types in order of preference, as documented in more detail in CSS1. MathML renderers are encouraged to make use of the CSS syntax for specifying fonts when this is practical in their rendering environment, even if they do not otherwise support CSS. (See also the subsection CSS-compatible attributes within section 2.3.3.3 [CSS-compatible attributes].

The syntax and meaning of the color attribute are as described for the same attribute of <mstyle> (section 3.3.4 [Style Change]).

3.2.2. Identifiers

3.2.2.1. Description

An mi element represents a symbolic name or arbitrary text which should be rendered as an identifier. Identifiers can include variables, function names, and symbolic constants.

Not all `mathematical identifiers' are represented by mi elements - for example, subscripted or primed variables should be represented using msub or msup respectively. Conversely, arbitrary text playing the role of a `term' (such as an ellipsis in a summed series) can be represented using an mi element, as shown in an example in section 3.2.5.4 [Mixing text and mathematics]@@.

It should be stressed that mi is a presentation element, and as such, it only indicates that its content should be rendered as an identifier. In the majority of cases, the contents of an mi will actually represent a mathematical identifier such as a variable or function name. However, as the preceding paragraph indicates, the correspondence between notations which should render like identifiers and notations which are actually intended to represent mathematical identifiers is not perfect. For an element whose semantics is guaranteed to be that of an identifier, see the description of ci in chapter 4 [Content Markup].

3.2.2.2. Attributes of mi

mi elements accept the attributes listed in section 3.2.1 [Attributes common to token elements], but in one case with a different default value:

A typical graphical renderer would render an mi element as the characters in its content, with no extra spacing around the characters (except spacing associated with neighboring elements). The default fontstyle would (typically) be normal (non-slanted) unless the content is a single character, in which case it would be italic. Note that this rule for fontstyle is specific to mi elements; the default value for the fontstyle attribute of other MathML token elements is normal.

3.2.2.3. Examples of mi

<mi> x </mi>
<mi> D </mi>
<mi> sin </mi>
<mi></mi>

An mi element with no content is allowed; <mi></mi> might, for example, be used by an `expression editor' to represent a location in a MathML expression which requires a `term' (according to conventional syntax for mathematics) but does not yet contain one.

Identifiers include function names such as `sin'. Expressions such as `sin x' should be written using the ⁡ operator (which also has the short name ⁡ as shown below; see also the discussion of invisible operators in section 3.2.4 [Operator, Fence, Separator or Accent].

<mrow>
   <mi> sin </mi>
   <mo> &ApplyFunction; </mo>
   <mi> x </mi>
</mrow>

Miscellaneous text that should be treated as a `term' can also be represented by an mi element, as in:

<mrow>
   <mn> 1 </mn>
   <mo> + </mo>
   <mi> ... </mi>
   <mo> + </mo>
   <mi> n </mi>
</mrow>

When an mi is used in such exceptional situations, explicitly setting the fontstyle attribute may give better results than the default behavior of some renderers.

The names of symbolic constants should be represented as mi elements:

<mi> &pi; </mi>
<mi> &ImaginaryI; </mi>
<mi> &ExponentialE; </mi>

Use of special entity references for such constants can simplify the interpretation of MathML presentation elements. See chapter 6 [Entities, Characters and Fonts] for a complete list of character entity references in MathML.

3.2.3. Numbers

3.2.3.1. Description

An mn element represents a `numeric literal' or other data which should be rendered as a numeric literal. Generally speaking, a numeric literal is a sequence of digits, perhaps including a decimal point, representing an unsigned integer or real number.

The concept of a mathematical `number' depends on the context, and is not well-defined in the abstract. As a consequence, not all mathematical numbers should be represented using mn; examples of mathematical numbers which should be represented differently are shown below, and include negative numbers, complex numbers, ratios of numbers shown as fractions, and names of numeric constants.

Conversely, since mn is a presentation element, there are a few situations where it may desirable to include arbitrary text in the content of an mn which should merely render as a numeric literal, even though that content may not be unambiguously interpretable as a number according to any particular standard encoding of numbers as character sequences. As a general rule, however, the mn element should be reserved for situations where its content is actually intended to represent a numeric quantity in some fashion. For an element whose semantics are guaranteed to be that of a particular kind of mathematical number, see the description of cn in chapter 4 [Content Markup].

3.2.3.2. Attributes of <mn>

mn elements accept the attributes listed in section 3.2.1 [Attributes common to token elements].

A typical graphical renderer would render an mn element as the characters of its content, with no extra spacing around them (except spacing from neighboring elements such as mo). Unlike mi, mn elements are (typically) rendered in an unslanted font by default, regardless of their content.

3.2.3.3. Examples of mn

<mn> 2 </mn>
<mn> 0.123 </mn>
<mn> 1,000,000 </mn>
<mn> 2.1e10 </mn>
<mn> 0xFFEF </mn>
<mn> MCMLXIX </mn>
<mn> twenty one </mn>

3.2.3.4. Examples of numbers which should not be written using mn alone

Many mathematical numbers should be represented using presentation elements other than mn alone; this includes negative numbers, complex numbers, ratios of numbers shown as fractions, and names of numeric constants. Examples of MathML representations of such numbers include:

<mrow> <mo> - </mo> <mn> 1 </mn> </mrow>
<mrow>
   <mn> 2 </mn>
   <mo> + </mo>
   <mrow>
      <mn> 3 </mn>
      <mo> &InvisibleTimes; </mo>
      <mi> &ImaginaryI; </mi>
   </mrow>
</mrow>
<mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac>
<mi> &pi; </mi>
<mi> &ExponentialE; </mi>

3.2.4. Operator, Fence, Separator or Accent

3.2.4.1. Description

An mo element represents an operator or anything which should be rendered as an operator. In general, the notational conventions for mathematical operators are quite complicated, and therefore MathML provides a relatively sophisticated mechanism for specifying the rendering behavior of an mo element. As a consequence, in MathML the list of things which should `render as an operator' includes a number of notations which are not mathematical operators in the ordinary sense. Besides ordinary operators with infix, prefix, or postfix forms, these include fence characters such as braces, parentheses, and `absolute value' bars, separators such as comma and semicolon, and mathematical accents such as a bar or tilde over a symbol.

The term `operator' as used in the present chapter means any symbol or notation which should render as an operator, and which is therefore representable by an mo element. That is, the term `operator' includes any ordinary operator, fence, separator, or accent unless otherwise specified or clear from the context.

All such symbols are represented in MathML with mo elements since they are subject to essentially the same rendering attributes and rules; subtle distinctions in the rendering of these classes of symbols, when they exist, are supported using the boolean attributes fence, separator and accent which can be used to distinguish these cases.

A key feature of the mo element is that its default attribute values are set on a case-by-case basis from an `operator dictionary' as explained below. In particular, default values for fence, separator and accent can usually be found in the operator dictionary and therefore need not be specified on each mo element.

Note that some mathematical operators are represented not by mo elements alone, but by mo elements `embellished' with (for example) surrounding superscripts; this is further described below. Conversely, as presentation elements, mo elements can contain arbitrary text, even when that text has no standard interpretation as an operator; for an example, see the discussion `Mixing text and mathematics' in section 3.2.5 [Text]. See also chapter 4 [Content Markup] for definitions of MathML content elements which are guaranteed to have the semantics of specific mathematical operators.

3.2.4.2. Attributes of mo

mo elements accept the attributes listed in section 3.2.1 [Attributes common to token elements], as described later in this section. When a dictionary entry is not found for a given mo element, the default value shown here in parentheses is used.

h-unit represents a unit of horizontal length (see section 2.3.3 [MathML Attribute Values]). v-unit represents a unit of vertical length. If no unit is given with maxsize or minsize, the number is a multiplier of the normal size of the operator in the direction (or directions) in which it stretches. These attributes are further explained below.

Typical graphical renderers show all mo elements as the characters of their content, with additional spacing around the element determined from the attributes listed above. Detailed rules for determining operator spacing in visual renderings are described in a subsection below. As always, MathML does not require a specific rendering, and these rules are provided as suggestions for the convenience of implementors.

Renderers without access to complete fonts for the MathML character set may choose not to render an mo element as precisely the characters in its content in some cases. For example, <mo> &le; </mo> might be rendered as <= to a terminal. However, as a general rule, renderers should attempt to render the content of an mo element as literally as possible. That is, <mo> &le </mo> and <mo> <= </mo> should render differently. (The first one should render as a single extended character representing a less-than-or-equal-to sign, and the second one as the two-character sequence <=.)

3.2.4.3. Examples of mo elements representing ordinary operators

<mo> + </mo>
<mo> &lt; </mo>
<mo> &le; </mo>
<mo> &lt;= </mo>
<mo> ++ </mo>
<mo> &sum; </mo>
<mo> .NOT. </mo>
<mo> and </mo>
<mo> &InvisibleTimes; </mo>

3.2.4.4. Examples of expressions using mo elements for fences and separators

Note that the mo elements in these examples don't need explicit fence or separator attributes, since these can be found using the operator dictionary as described below. Some of these examples could also be encoded using the mfenced element described in section 3.3.8 [Content Inside Pair of Fences].

(a+b)

<mrow>
   <mo> ( </mo>
   <mrow>
      <mi> a </mi>
      <mo> + </mo>
      <mi> b </mi>
   </mrow>
   <mo> ) </mo>
</mrow>

[0,1)

<mrow>
   <mo> [ </mo>
   <mrow>
      <mn> 0 </mn>
      <mo> , </mo>
      <mn> 1 </mn>
   </mrow>
   <mo> ) </mo>
</mrow>

f(x,y)

<mrow>
   <mi> f </mi>
   <mo> &ApplyFunction; </mo>
   <mrow>
      <mo> ( </mo>
      <mrow>
         <mi> x </mi>
         <mo> , </mo>
         <mi> y </mi>
      </mrow>
      <mo> ) </mo>
   </mrow>
</mrow>

3.2.4.5. Invisible operators

Certain operators which are `invisible' in traditional mathematical notation should be represented using specific entity references within mo elements, rather than simply by nothing. The entity references used for these `invisible operators' are:

The MathML representations of the examples in the above table are:

<mrow>
   <mi> x </mi>
   <mo> &InvisibleTimes; </mo>
   <mi> y </mi>
</mrow>
<mrow>
   <mi> f </mi>
   <mo> &ApplyFunction; </mo>
   <mrow>
      <mo> ( </mo>
      <mi> x </mi>
      <mo> ) </mo>
   </mrow>
</mrow>
<mrow>
   <mi> sin </mi>
   <mo> &ApplyFunction; </mo>
   <mi> x </mi>
</mrow>
<msub>
   <mi> m </mi>
   <mrow>
      <mn> 1 </mn>
      <mo> &InvisibleComma; </mo>
      <mn> 2 </mn>
   </mrow>
</msub>

The reasons for using specific mo elements for invisible operators include:

• such operators should often have specific effects on visual rendering (particularly spacing and linebreaking rules) which are not the same as either the lack of any operator, or spacing represented by <mspace/> or mtext elements;
• these operators should often have specific audio renderings different than that of the lack of any operator;
• automatic semantic interpretation of MathML presentation elements is made easier by the explicit specification of such operators.

For example, an audio renderer might render f(x) (represented as in the above examples) by speaking `f of x', but use the word `times' in its rendering of xy. Although its rendering must still be different depending on the structure of neighboring elements (sometimes leaving out `of' or `times' entirely), its task is made much easier by the use of a different mo element for each invisible operator.

3.2.4.6. Entity references for other special operators

For reasons like those for including special entities for invisible operators, MathML also includes ⅆ for use in an mo element representing the differential operator symbol usually denoted by `d'.

3.2.4.7. Detailed rendering rules for mo elements

Typical visual rendering behaviors for mo elements are more complex than for the other MathML token elements, so the rules for rendering them are described in this separate subsection.

Note that, like all rendering rules in MathML, these rules are suggestions rather than requirements. Furthermore, no attempt is made to specify the rendering completely; rather, enough information is given to make the intended effect of the various rendering attributes as clear as possible.

3.2.4.7.1. The operator dictionary

Many mathematical symbols, such as an integral sign, a plus sign, or a parenthesis, have a well-established, predictable, traditional notational usage. Typically, this usage amounts to certain default attribute values for mo elements with specific contents and a specific form attribute. Since these defaults vary from symbol to symbol, MathML anticipates that renderers will have an `operator dictionary' of default attributes for mo elements (see Appendix C) indexed by each mo element's content and form attribute. If an mo element is not listed in the dictionary, the default values shown in parentheses in the table of attributes for mo should be used, since these values are typically acceptable for a generic operator.

Some operators are `overloaded', in the sense that they can occur in more than one form (prefix, infix, or postfix), with possibly different rendering properties for each form. For example, `+' can be either a prefix or an infix operator. Typically, a visual renderer would add space around both sides of an infix operator, while only on the left of a prefix operator. The form attribute allows specification of which form to use, in case more than one form is possible according to the operator dictionary and the default value described below is not suitable.

3.2.4.7.2. Default value of form attribute

The form attribute does not usually have to be specified explicitly, since there are effective heuristic rules for inferring the value of the form attribute from the context. If it is not specified, and there is more than one possible form in the dictionary for an mo element with given content, the renderer should choose which form to use as follows (but see the exception for embellished operators, described later):

• If the operator is the first argument in an mrow of length (i.e. number of arguments) greater than one (ignoring all space-like arguments (see section 3.2.6 [Space]) in the determination of both the length and the first argument), the prefix form is used;
• if it is the last argument in an mrow of length greater than one (ignoring all space-like arguments), the postfix form is used;
• in all other cases, including when the operator is not part of an mrow, the infix form is used.

Note that these rules make reference to the mrow in which the mo element lies. In some situations, this mrow might be an inferred mrow implicitly present around the arguments of an element such as msqrt or mtd.

Opening (left) fences should have form="prefix", and closing (right) fences should have form="postfix"; separators are usually `infix', but not always, depending on their surroundings. As with ordinary operators, these values do not usually need to be specified explicitly.

If the operator does not occur in the dictionary with the specified form, the renderer should use one of the forms which is available there, in the order of preference: infix, postfix, prefix; if no forms are available for the given mo element content, the renderer should use the defaults given in parentheses in the table of attributes for mo.

3.2.4.7.3. Exception for embellished operators

There is one exception to the above rules for choosing an mo element's default form attribute. An mo element which is `embellished' by one or more nested subscripts, superscripts, surrounding text or whitespace, or style changes behaves differently. It is the embellished operator as a whole (this is defined precisely, below) whose position in an mrow is examined by the above rules and whose surrounding spacing is affected by its form, not the mo element at its core; however, the attributes influencing this surrounding spacing are taken from the mo element at the core (or from that element's dictionary entry).

For example, the `+<sub>4</sub>' in a+<sub>4</sub>b should be considered an infix operator as a whole, due to its position in the middle of an mrow, but its rendering attributes should be taken from the mo element representing the `+', or when those are not specified explicitly, from the operator dictionary entry for <mo form="infix"> + </mo>. The precise definition of an `embellished operator' is:

• an mo element;
• or one of the elements msub, msup, msubsup, munder, mover, munderover, mmultiscripts, mfrac, or semantics (section 4.2.7 [Syntax and Semantics]), whose first argument exists and is an embellished operator;
• or one of the elements mstyle, mphantom, or mpadded, such that an mrow containing the same arguments would be an embellished operator;
• or an maction element whose selected subexpression exists and is an embellished operator;
• or an mrow whose arguments consist (in any order) of one embellished operator and zero or more space-like elements.

Note that this definition permits nested embellishment only when there are no intervening enclosing elements not in the above list.

The above rules for choosing operator forms and defining embellished operators are chosen so that in all ordinary cases it will not be necessary for the author to specify a form attribute.

3.2.4.7.4. Rationale for definition of embellished operators

The following notes are included as a rationale for certain aspects of the above definitions, but should not be important for most users of MathML.

An mfrac is included as an `embellisher' because of the common notation for a differential operator:

<mfrac>
   <mo> &DifferentialD; </mo>
   <mrow>
      <mo> &DifferentialD; </mo>
      <mi> x </mi>
   </mrow>
</mfrac>

Since the definition of embellished operator affects the use of the attributes related to stretching, it is important that it includes embellished fences as well as ordinary operators; thus it applies to any mo element.

Note that an mrow containing a single argument is an embellished operator if and only if its argument is an embellished operator. This is because an mrow with a single argument must be equivalent in all respects to that argument alone (as discussed in section 3.3.1 [Horizontally Group Any Number of Subexpressions]). This means that an mo element which is the sole argument of an mrow will determine its default form attribute based on that mrow's position in a surrounding, perhaps inferred, mrow (if there is one), rather than based on its own position in the mrow it is the sole argument of.

Note that the above definition defines every mo element to be `embellished' - that is, `embellished operator' can be considered (and implemented in renderers) as a special class of MathML expressions, of which mo is a specific case.

3.2.4.7.5. Spacing around an operator

The amount of space added around an operator (or embellished operator), when it occurs in an mrow, can be directly specified by the lspace and rspace attributes. These values are in ems if no units are given. By convention, operators that tend to bind tightly to their arguments have smaller values for spacing than operators that tend to bind less tightly. This convention should be followed in the operator dictionary included with a MathML renderer. In T_EX, these values can only be one of three values; typically they are 3/18em, 4/18em, and 5/18em. MathML does not impose this limit.

Some renderers may choose to use no space around most operators appearing within subscripts or superscripts, as is done in T_EX.

Non-graphical renderers should treat spacing attributes, and other rendering attributes described here, in analogous ways for their rendering medium.

3.2.4.8. Stretching of operators, fences and accents

Four attributes govern whether and how an operator (perhaps embellished) stretches so that it matches the size of other elements: stretchy, symmetric, maxsize, and minsize. If an operator has the attribute stretchy=true, then it (that is, each character in its content) obeys the stretching rules listed below, given the constraints imposed by the fonts and font rendering system. In practice, typical renderers will only be able to stretch a small set of characters, and quite possibly will only be able to generate a discrete set of character sizes.

There is no provision in MathML for specifying in which direction (horizontal or vertical) to stretch a specific character or operator; rather, when stretchy=true it should be stretched in each direction for which stretching is possible. It is up to the renderer to know in which directions it is able to stretch each character. (Most characters can be stretched in at most one direction by typical renderers, but some renderers may be able to stretch certain characters, such as diagonal arrows, in both directions independently.)

The minsize and maxsize attributes limit the amount of stretching (in either direction). These two attributes are given as multipliers of the operator's normal size in the direction or directions of stretching, or as absolute sizes using units. For example, if a character has maxsize="3", then it can grow to be no more than three times its normal (unstretched) size.

The symmetric attribute governs whether the height and depth above and below the axis of the character are forced to be equal (by forcing both height and depth to become the maximum of the two). An example of a situation where one might set symmetric=false arises with parentheses around a matrix not aligned on the axis, which frequently occurs when multiplying non-square matrices. In this case, one wants the parentheses to stretch to cover the matrix, whereas stretching the parentheses symmetrically would cause them to protrude beyond one edge of the matrix. The symmetric attribute only applies to characters that stretch vertically (otherwise it is ignored).

If a stretchy mo element is embellished (as defined earlier in this section), the mo element at its core is stretched to a size based on the context of the embellished operator as a whole, i.e. to the same size as if the embellishments were not present. For example, the parentheses in the following example (which would typically be set to be stretchy by the operator dictionary) will be stretched to the same size as each other, and the same size they would have if they were not underlined and overlined, and furthermore will cover the same vertical interval:

<mrow>
   <munder>
      <mo> ( </mo>
      <mo> &UnderBar; </mo>
   </munder>
   <mfrac>
      <mi> a </mi>
      <mi> b </mi>
   </mfrac>
   <mover>
      <mo> ) </mo>
      <mo> &OverBar; </mo>
   </mover>
</mrow>

Note that this means that the stretching rules given below must refer to the context of the embellished operator as a whole, not just to the mo element itself.

3.2.4.8.1. Example of stretchy attributes

This shows one way to set the maximum size of a parenthesis so that it does not grow, even though its default value is stretchy=true.

<mrow>
   <mo maxsize="1"> ( </mo>
   <mfrac>
      <mi> a </mi> <mi> b </mi>
   </mfrac>
   <mo maxsize="1"> ) </mo>
</mrow>

The above should render as as opposed to the default rendering .

Note that each parenthesis is sized independently; if only one of them had maxsize="1", they would render with different sizes.

3.2.4.8.2. Vertical Stretching Rules

• If a stretchy operator is a direct subexpression of an mrow element, or is the sole direct subexpression of an mtd element in some row of a table, then it should stretch to cover the height and depth (above and below the axis) of the non-stretchy direct subexpressions in the mrow element or table row, unless stretching is constrained by minsize or maxsize attributes.
• In the case of an embellished stretchy operator, the preceding rule applies to the stretchy operator at its core.
• If symmetric=true, then the maximum of the height and depth is used to determine the size, before application of the minsize or maxsize attributes.
• The preceding rules also apply in situations where the mrow or mtd element is inferred (see section 3.5.1 [Table or Matrix] for a discussion of inferred mtd elements).

Most common opening and closing fences are defined in the operator dictionary to stretch by default; and they stretch vertically. Also, operators such as ∑, ∫, /, and vertical arrows stretch vertically by default.

In the case of a stretchy operator in a table cell (i.e. within an mtd element), the above rules assume each cell of the table row containing the stretchy operator covers exactly one row. (Equivalently, the value of the rowspan attribute is assumed to be 1 for all the table cells in the table row, including the cell containing the operator.) When this is not the case, the operator should only be stretched vertically to cover those table cells which are entirely within the set of table rows that the operator's cell covers. Table cells which extend into rows not covered by the stretchy operator's table cell should be ignored.

3.2.4.8.3. Horizontal Stretching Rules

• If a stretchy operator, or an embellished stretchy operator, is a direct subexpression of an munder, mover, or munderover element, or if it is the sole direct subexpression of an mtd element (perhaps an inferred one) in some column of a table (see mtable), then it, or the mo element at its core, should stretch to cover the width of the other direct subexpressions in the given element (or in the same table column), given the constraints mentioned above.
• If a stretchy operator is a direct subexpression of an munder, mover, or munderover element, or if it is the sole direct subexpression of an mtd element in some column of a table, then it should stretch to cover the width of the other direct subexpressions in the given element (or in the same table column), given the constraints mentioned above.
• In the case of an embellished stretchy operator, the preceding rule applies to the stretchy operator at its core.
• The preceding rules also apply in situations where the mtd element is inferred (see section 3.5.1 [Table or Matrix] for a discussion of inferred mtd elements).

By default, most horizontal arrows and some accents stretch horizontally.

In the case of a stretchy operator in a table cell (i.e. within an mtd element), the above rules assume each cell of the table column containing the stretchy operator covers exactly one column. (Equivalently, the value of the columnspan attribute is assumed to be 1 for all the table cells in the table row, including the cell containing the operator.) When this is not the case, the operator should only be stretched horizontally to cover those table cells which are entirely within the set of table columns that the operator's cell covers. Table cells which extend into columns not covered by the stretchy operator's table cell should be ignored.

The rules for horizontal stretching include mtd elements to allow arrows to stretch for use in commutative diagrams laid out using mtable. The rules for the horizontal stretchiness include scripts to make examples such as the following work:

<mrow>
   <mi> x </mi>
   <munder>
      <mo> &RightArrow; </mo>
      <mtext> maps to </mtext>
   </munder>
   <mi> y </mi>
</mrow>

This displays as .

3.2.4.8.4. Rules Common to both Vertical and Horizontal Stretching

If a stretchy operator is not required to stretch (i.e. if it is not in one of the locations mentioned above, or if there are no other expressions whose size it should stretch to match), then it has the standard (unstretched) size determined by the font and current fontsize.

If a stretchy operator is required to stretch, but all other expressions in the containing element or object (as described above) are also stretchy, all elements that can stretch should grow to the maximum of the normal unstretched sizes of all elements in the containing object, if they can grow that large. If the value of minsize or maxsize prevents this then that (min or max) size is used.

For example, in an mrow containing nothing but vertically stretchy operators, each of the operators should stretch to the maximum of all of their normal unstretched sizes, provided no other attributes are set which override this behavior. Of course, limitations in fonts or font rendering may result in the final, stretched sizes being only approximately the same.

3.2.4.9. Other attributes of mo

The largeop attribute specifies whether the operator should be drawn larger than normal if displaystyle=true in the current rendering environment. This roughly corresponds to T_EX's \displaystyle style setting. MathML uses two attributes, displaystyle and scriptlevel, to control orthogonal presentation features that T_EX encodes into one `style' attribute with values \displaystyle, \textstyle, \scriptstyle, and \scriptscriptstyle. These attributes are discussed further in section 3.3.4 [Style Change] describing the mstyle element. Note that these attributes can be specified directly on an mstyle element's begin tag, but not on most other elements. Examples of large operators include ∫ and ∏.

The movablelimits attribute specifies whether underscripts and overscripts attached to this mo element should be drawn as subscripts and superscripts when displaystyle=false. movablelimits=false means that underscripts and overscripts should never be drawn as subscripts and superscripts. In general, displaystyle is true for displayed mathematics and false for inline mathematics. Also, displaystyle is false by default within tables, scripts and fractions, and a few other exceptional situations detailed in section 3.3.4 [Style Change]. Thus, operators with movablelimits=true will display with limits (i.e. underscripts and overscripts) in displayed mathematics, and with subscripts and superscripts in inline mathematics, tables, scripts and so on. Examples of operators that typically have movablelimits=true are sum, prod, and lim.

The accent attribute determines whether this operator should be treated by default as an accent (diacritical mark) when used as an underscript or overscript; see munder, mover, and munderover (sections 3.4.4-3.4.6).

The separator attribute may affect automatic linebreaking in renderers which position ordinary infix operators at the beginnings of broken lines rather than at the ends (that is, which avoid linebreaking just after such operators), since linebreaking should be avoided just before separators, but is acceptable just after them.

The fence attribute has no effect in the suggested visual rendering rules given here; it is not needed for properly rendering traditional notation using these rules. It is provided so that specific MathML renderers, especially non-visual renderers, have the option of using this information.

3.2.5. Text

3.2.5.1. Description

An mtext element is used to represent arbitrary text which should be rendered as itself. In general, the mtext element is intended to denote commentary text which is not central to the mathematical meaning or notational structure of the expression it is contained in.

Note that some text with a clearly defined notational role might be more appropriately marked up using mi or mo; this is discussed further below.

An mtext element can be used to contain `renderable whitespace', i.e. invisible characters which are intended to alter the positioning of surrounding elements. In non-graphical media, such characters are intended to have an analogous effect, such as introducing positive or negative time delays or affecting rhythm in an audio renderer. This is not related to any whitespace in the source MathML consisting of blanks, newlines, tabs, or carriage returns; whitespace present directly in the source is trimmed and collapsed, as described in section 2.3.5 [Collapsing Whitespace in Input]. Whitespace which is intended to be rendered as part of an element's content must be represented by entity references (unless it consists only of single blanks between non-whitespace characters).

Renderable whitespace can have a positive or negative width, as in   and ​, or zero width, as in ​. The complete list of such characters is given in chapter 6 [Entities, Characters and Fonts]. Note that there is no formal distinction in MathML between renderable whitespace characters and any other class of characters, in mtext or in any other element.

Renderable whitespace can also include characters that affect alignment or linebreaking. Some of these characters are:

For the complete list of MathML entities, consult chapter 6 [Entities, Characters and Fonts].

3.2.5.2. Attributes of mtext

mtext elements accept the attributes listed in section 3.2.1 [Attributes common to token elements].

See also the warnings about the legal grouping of `space-like elements' in section 3.2.6 [Space], and about the use of such elements for `tweaking' or conveying meaning in section 3.3.6 [Adjust Space Around Content].

3.2.5.3. Examples of mtext

<mtext> Theorem 1: </mtext>
<mtext> &ThinSpace; </mtext>
<mtext> &ThickSpace;&ThickSpace; </mtext>
<mtext> /* a comment */ </mtext>

3.2.5.4. Mixing text and mathematics

In some cases, text embedded in mathematics could be more appropriately represented using mo or mi elements. For example, the expression `there exists such that f(x) <1' is equivalent to and could be represented as:

<mrow>
   <mo> there exists </mo>
   <mrow>
      <mrow>
         <mi> &delta; </mi>
         <mo> &gt; </mo>
         <mn> 0 </mn>
      </mrow>
      <mo> such that </mo>
      <mrow>
         <mrow>
            <mi> f </mi>
            <mo> &ApplyFunction; </mo>
            <mrow>
               <mo> ( </mo>
               <mi> x </mi>
               <mo> ) </mo>
            </mrow>
         </mrow>
         <mo> &lt; </mo>
         <mn> 1 </mn>
      </mrow>
   </mrow>
</mrow>

An example involving an mi element is: x+x²+···+xⁿ. In this example, ellipsis should be represented using an mi element, since it takes the place of a term in the sum (see section 3.2.2 [Identifiers], mi).

On the other hand, expository text within MathML is best represented with an mtext element. An example of this is:

Theorem 1: if x > 1, then x² > x.

However, when MathML is embedded in HTML, the example is probably best rendered with only the two inequalities represented as MathML at all, letting the text be part of the surrounding HTML.

Another factor to consider in deciding how to mark up text is the effect on rendering. Text enclosed in an mo element is unlikely to be found in a renderer's operator dictionary, so it will be rendered with the format and spacing appropriate for an `unrecognized operator', which may or may not be better than the format and spacing for `text' obtained by using an mtext element. An ellipsis entity in an mi element is apt to be spaced more appropriately for taking the place of a term within a series than if it appeared in an mtext element.

3.2.6. Space

3.2.6.1. Description

An mspace empty element represents a blank space of any desired size, as set by its attributes. The default value for each attribute is 0em or 0ex, so it will not be useful without some attributes specified.

3.2.6.2. Attributes of mspace

h-unit and v-unit represent units of horizontal or vertical length, respectively (see section 2.3.3 [MathML Attribute Values]).

Note the warning about the legal grouping of `space-like elements' given below, and the warning about the use of such elements for `tweaking' or conveying meaning in section 3.3.6 [Adjust Space Around Content]. See also the other elements which can render as whitespace, namely mtext, mphantom, and maligngroup.

3.2.6.3. Definition of space-like elements

A number of MathML presentation elements are `space-like' in the sense that they typically render as whitespace, and do not affect the mathematical meaning of the expressions in which they appear. As a consequence, these elements often function in somewhat exceptional ways in other MathML expressions. For example, space-like elements are handled specially in the suggested rendering rules for mo given in section 3.2.4 [Operator, Fence, Separator or Accent]. The following MathML elements are defined to be `space-like':

• an mtext, mspace, maligngroup, or malignmark element;
• an mstyle, mphantom, or mpadded element, all of whose direct subexpressions are space-like;
• an maction element whose selected subexpression exists and is space-like;
• an mrow all of whose direct subexpressions are space-like.

Note that an mphantom is not automatically defined to be space-like, unless its content is space-like. This is because operator spacing is affected by whether adjacent elements are space-like. Since the mphantom element is primarily intended as an aid in aligning expressions, operators adjacent to an mphantom should behave as if they were adjacent to the contents of the mphantom, rather than to an equivalently sized area of whitespace.

3.2.6.4. Legal grouping of space-like elements

Authors who insert space-like elements or mphantom elements into an existing MathML expression should note that such elements are counted as arguments, in elements which require a specific number of arguments, or which interpret different argument positions differently.

Therefore, space-like elements inserted into such a MathML element should be grouped with a neighboring argument of that element by introducing an mrow for that purpose. For example, to allow for vertical alignment on the right edge of the base of a superscript, the expression

<msup> <mi> x </mi> <malignmark edge="right"/> <mn> 2 </mn> </msup>

is illegal, because msup must have exactly 2 arguments; the correct expression would be:

<msup>
   <mrow>
      <mi> x </mi>
      <malignmark edge="right"/>
   </mrow>
   <mn> 2 </mn>
</msup>

See also the warning about `tweaking' in section 3.3.6 [Adjust Space Around Content].

3.2.7. String Literal

3.2.7.1. Description

The ms element is used to represent `string literals' in expressions meant to be interpreted by computer algebra systems or other systems containing `programming languages'. By default, string literals are displayed surrounded by double quotes. As explained in section 3.2.5 [Text], ordinary text embedded in a mathematical expression should be marked up with mtext, or in some cases mo or mi, but never with ms.

Note that the string literals encoded by ms are `Unicode strings' rather than `ASCII strings'. In practice, non-ASCII characters will typically be represented by entity references. For example, <ms>&</ms> represents a string literal containing a single character, &, and <ms>&</ms> represents a string literal containing 5 characters, the first one of which is &. (In fact, MathML string literals are even more general than Unicode string literals, since not all MathML entity references necessarily refer to existing Unicode characters, as discussed in chapter 6 [Entities, Characters and Fonts].)

Like all token elements, ms does trim and collapse whitespace in its content according to the rules of section 2.3.5 [Collapsing Whitespace in Input], so whitespace intended to remain in the content should be encoded as described in that section.

3.2.7.2. Attributes of ms

ms elements accept the attributes listed in section 3.2.1 [Attributes common to token elements], and additionally:

In visual renderers, the content of an ms element is typically rendered with no extra spacing added around the string, and a quote character at the beginning and the end of the string. By default, the left and right quote characters are both the standard double quote character ". However, these characters can be changed with the lquote and rquote attributes respectively.

The content of ms elements should be rendered with visible `escaping' of certain characters in the content, including at least `double quote' itself, and preferably whitespace other than individual blanks. The intent is for the viewer to see that the expression is a string literal, and to see exactly which characters form its content. For example, <ms>double quote is "</ms> might be rendered as `double quote is \"'.

3.3. General Layout Schemata

Besides tokens there are several families of MathML presentation elements. One family of elements deals with various `scripting' notations, such as subscript and superscript. Another family is concerned with matrices and tables. The remainder of the elements, discussed in this section, describe other basic notations such as fractions and radicals, or deal with general functions such as setting style properties and error handling.

3.3.1. Horizontally Group Any Number of Subexpressions

3.3.1.1. Description

An mrow element is used to group together any number of subexpressions, usually consisting of one or more mo elements acting as `operators' on one or more other expressions which are their `operands'.

Several elements automatically treat their arguments as if they were contained in an mrow element. See the discussion of inferred mrows in section 3.1.3 [Required Arguments]. See also mfenced (section 3.3.8 [Content Inside Pair of Fences]), which can effectively form an mrow containing its arguments separated by commas.

3.3.1.2. Attributes of mrow

None (except the attributes allowed for all MathML elements, listed in section 2.3.4 [Attributes Shared by all MathML Elements]).

mrow elements are typically rendered visually as a horizontal row of their arguments, left to right in the order in which the arguments occur, or audibly as a sequence of renderings of the arguments. The description in section 3.2.4 [Operator, Fence, Separator or Accent] of suggested rendering rules for mo elements assumes that all horizontal spacing between operators and their operands is added by the rendering of mo elements (or, more generally, embellished operators), not by the rendering of the mrows they are contained in.

MathML is designed to allow renderers to automatically linebreak expressions (that is, to break excessively long expressions into several lines), without requiring authors to specify explicitly how this should be done. This is because linebreaking positions can't be chosen well without knowing the width of the display device and the current font size, which for many uses of MathML will not be known except by the renderer at the time of each rendering.

Determining good positions for linebreaks is complex, and rules for this are not described here; whether and how it is done is up to each MathML renderer. Typically, linebreaking will involve selection of `good' points for insertion of linebreaks between successive arguments of mrow elements.

Although MathML does not require linebreaking or specify a particular linebreaking algorithm, it has several features designed to allow such algorithms to produce good results. These include the use of special entities for certain operators, including invisible operators (see section 3.2.4 [Operator, Fence, Separator or Accent]), or for providing hints related to linebreaking when necessary (see section 3.2.5 [Text]), and the ability to use nested mrows to describe subexpression structure (see below).

3.3.1.2.1. mrow of one argument

MathML renderers are required to treat an mrow element containing exactly one argument as equivalent in all ways to the single argument occurring alone, provided there are no attributes on the mrow element's begin tag. If there are attributes on the mrow element's begin tag, no requirement of equivalence is imposed. This equivalence condition is intended to simplify the implementation of MathML-generating software such as template-based authoring tools. It directly affects the definitions of embellished operator and space-like element and the rules for determining the default value of the form attribute of an mo element; see sections 3.2.4 and 3.2.6. See also the discussion of equivalence of MathML expressions in chapter 7 [The MathML Interface].

3.3.1.3. Proper grouping of subexpressions using mrow

Subexpressions should be grouped by the document author in the same way as they are grouped in the mathematical interpretation of the expression; that is, according to the underlying `syntax tree' of the expression. Specifically, operators and their mathematical arguments should occur in a single mrow; more than one operator should occur directly in one mrow only when they can be considered (in a syntactic sense) to act together on the interleaved arguments, e.g. for a single parenthesized term and its parentheses, for chains of relational operators, or for sequences of terms separated by + and -. A precise rule is given below.

Proper grouping has several purposes: it improves display by possibly affecting spacing; it allows for more intelligent linebreaking and indentation; and it simplifies possible semantic interpretation of presentation elements by computer algebra systems, and audio renderers.

Although improper grouping will sometimes result in suboptimal renderings, and will often make interpretation other than pure visual rendering difficult or impossible, any grouping of expressions using mrow is allowed in MathML syntax; that is, renderers should not assume the rules for proper grouping will be followed.

3.3.1.3.1. Precise rule for proper grouping

A precise rule for when and how to nest subexpressions using mrow is especially desirable when generating MathML automatically by conversion from other formats for displayed mathematics, such as T_EX, which don't always specify how subexpressions nest. When a precise rule for grouping is desired, the following rule should be used:

Two adjacent operators (i.e. mo elements, possibly embellished), possibly separated by operands (i.e. anything other than operators), should occur in the same mrow only when the left operator has an infix or prefix form (perhaps inferred), the right operator has an infix or postfix form, and the operators are listed in the same group of entries in the operator dictionary provided in Appendix C. In all other cases, nested mrows should be used.

When forming a nested mrow (during generation of MathML) which includes just one of two successive operators with the forms mentioned above (which mean that either operator could in principle act on the intervening operand or operands), it is necessary to decide which operator acts on those operands directly (or would do so, if they were present). Ideally, this should be determined from the original expression; for example, in conversion from an operator-precedence-based format, it would be the operator with the higher precedence. If this cannot be determined directly from the original expression, the operator which occurs later in the suggested operator dictionary (Appendix C) can be assumed to have a higher precedence for this purpose.

Note that the above rule has no effect on whether any MathML expression is valid, only on the recommended way of generating MathML from other formats for displayed mathematics or directly from written notation.

(Some of the terminology used in stating the above rule in defined in section 3.2.4 [Operator, Fence, Separator or Accent].)

3.3.1.4. Examples of mrow

As an example, 2x+y+x should be written as:

<mrow>
  <mrow>
     <mn> 2 </mn>
     <mo> &InvisibleTimes; </mo>
     <mi> x </mi>
  </mrow>
  <mo> + </mo>
  <mi> y </mi>
  <mo> - </mo>
  <mi> z </mi>
</mrow>

The proper encoding of (x, y) furnishes a less obvious example of nesting mrows:

<mrow>
   <mo> ( </mo>
   <mrow>
      <mi> x </mi>
      <mo> , </mo>
      <mi> y </mi>
   </mrow>
   <mo> ) </mo>
</mrow>

In this case, a nested mrow is required inside the parentheses, since parentheses and commas, thought of as fence and separator `operators', do not act together on their arguments.

3.3.2. Fractions

3.3.2.1. Description

The mfrac element is used for fractions. It can also be used to mark up fraction-like objects such as binomial coefficients and Legendre symbols. The syntax for mfrac is <mfrac> numerator denominator </mfrac>

3.3.2.2. Attributes of mfrac

The linethickness attribute indicates the thickness of the horizontal `fraction bar', or `rule', typically used to render fractions. A fraction with linethickness="0" renders without the bar, and might be used within binomial coefficients. A linethickness greater than one might be used with nested fractions. These cases are shown below:

The numalign and denomalign attributes control the horizontal alignment of the numerator and denominator respectively. Typically, numerators and denominators are centered, but a very long numerator or denominator might be displayed on several lines and a left alignment might be more appropriate for displaying them.

The beveled attribute determines whether the fraction is displayed with the numerator above the denominator separated by a horizontal line or whether a diagonal line is used to separate a slightly raised numerator from a slightly lowered denominator. The later form corresponds to the attribute value being true and provides for a more compact form for simple numerator and denominators

In general, the value of linethickness can be a number, as a multiplier of the default thickness of the fraction bar (the default thickness is not specified by MathML), or a number with a unit of vertical length (see section 2.3.3 [MathML Attribute Values]), or one of the keywords medium (same as 1), thin (thinner than 1, otherwise up to the renderer), or thick (thicker than 1, otherwise up to the renderer).

The mfrac element sets displaystyle to false, or if it was already false increments scriptlevel by 1, within numerator and denominator. These attributes are inherited by every element from its rendering environment, but can be set explicitly only on the mstyle element. (See section 3.3.4 [Style Change].)

3.3.2.3. Examples of mfrac

The examples shown above can be represented in MathML as:

<mrow>
   <mo> ( </mo>
   <mfrac linethickness="0">
      <mi> a </mi>
      <mi> b </mi>
   </mfrac>
   <mo> ) </mo>
</mrow>
<mfrac linethickness="2">
   <mfrac>
      <mi> a </mi>
      <mi> b </mi>
   </mfrac>
   <mfrac>
      <mi> c </mi>
      <mi> d </mi>
   </mfrac>
</mfrac>

A more generic example is:

<mfrac>
   <mrow>
      <mn> 1 </mn>
      <mo> + </mo>
      <msqrt>
         <mn> 5 </mn>
      </msqrt>
   </mrow>
   <mn> 2 </mn>
</mfrac>

3.3.3. Radicals

3.3.3.1. Description

These elements construct radicals. The msqrt element is used for square roots, while the mroot element is used to draw radicals with indices, e.g. a cube root. The syntax for these elements is

<msqrt> base </msqrt>
<mroot> base index </mroot>

The mroot element requires exactly 2 arguments. However, msqrt accepts any number of arguments; if this number is not 1, its contents are treated as a single `inferred mrow' containing its arguments, as described in section 3.1.3 [Required Arguments].

3.3.3.2. Attributes of msqrt and mroot

None (except the attributes allowed for all MathML elements, listed in section 2.3.4 [Attributes Shared by all MathML Elements]).

The mroot element increments scriptlevel by 2, and sets displaystyle to false, within index, but leaves both attributes unchanged within base. The msqrt element leaves both attributes unchanged within all its arguments. These attributes are inherited by every element from its rendering environment, but can be set explicitly only on mstyle. (See section 3.3.4 [Style Change].)

3.3.4. Style Change

3.3.4.1. Description

The mstyle element is used to make style changes which affect the rendering of its contents. mstyle can be given any attribute accepted by any MathML presentation element; it can also be given certain special attributes listed below.

The mstyle element accepts any number of arguments. If this number is not 1, its contents are treated as a single `inferred mrow' formed from all its arguments, as described in section 3.1.3 [Required Arguments].

Loosely speaking, the effect of the mstyle element is to change the default value of an attribute for the elements it contains. Style changes work in one of two ways, depending on whether an attribute's default value is documented as inherited. The two cases are:

• Some attributes, such as displaystyle or scriptlevel (explained below), are inherited from the surrounding context when they are not explicitly set. Specifying such an attribute on an mstyle element sets the value which will be inherited by its child elements. Unless a child element overrides this inherited value, it will pass it on to its children, and they will pass it to their children, and so on. But if a child element does override it, either by an explicit attribute setting or automatically (as is common for scriptlevel), the new (overriding) value will be passed on to that element's children, and then to their children, etc, until it is again overridden.
• Other attributes, such as linethickness on mfrac, have default values which are not normally inherited. That is, if the linethickness attribute is not set on the begin tag of an mfrac element, it will normally use the default value of 1, even if it was contained in a larger mfrac element which set this attribute to a different value. For attributes like this, specifying a value with an mstyle element has the effect of changing the default value for all elements within its scope. The net effect is that setting the attribute value with mstyle propagates the change to all the elements it contains directly or indirectly, except for the individual elements on which the value is overridden. Unlike in the case of inherited attributes, elements which explicitly override this attribute have no effect on this attribute's value in their children.

Note that attribute values inherited from an mstyle, in either manner, affect a given element in the mstyle's content only if that attribute is not given a value in that element's begin tag. On any element for which the attribute is set explicitly, the value specified on the begin tag overrides the inherited value. The only exception to this rule is when the value given on the begin tag is documented as specifying an incremental change to the value inherited from that element's context or rendering environment.

Note also that the difference between inherited and non-inherited attributes set by mstyle, explained above, only matters when the attribute is set on some element within the mstyle's contents which has children which also set it. Thus it never matters for attributes, such as color, which can only be set on token elements (or on mstyle itself).

There is one exceptional element, mpadded, whose attributes cannot be set with mstyle. When the attributes width, height and depth are specified on an mstyle element, they apply only to the mspace element. Similarly, when lspace is set with mstyle, it applies only to the mo element.

3.3.4.2. Attributes accepted by mstyle

As stated above, mstyle accepts all attributes of all MathML presentation elements. Additionally, mstyle can be given the following special attributes which are implicitly inherited by every MathML element as part of its rendering environment:

3.3.4.2.1. scriptlevel and displaystyle

MathML uses two attributes, displaystyle and scriptlevel, to control orthogonal presentation features that T_EX encodes into one style attribute with values \displaystyle, \textstyle, \scriptstyle, and \scriptscriptstyle. The corresponding values of displaystyle and scriptlevel for those T_EX styles would be true and 0, false and 0, false and 1, and false and 2, respectively.

The main effect of the displaystyle attribute is that it determines the effect of other attributes such as the largeop and movablescripts attributes of mo. The main effect of the scriptlevel attribute is to control the font size. Typically, the higher the scriptlevel, the smaller the font size. (Non-visual renderers can respond to the font size in an analogous way for their medium.) More sophisticated renderers may also choose to use these attributes in other ways, such as rendering expressions with displaystyle=false in a more vertically compressed manner.

These attributes are given initial values for the outermost expression of an instance of MathML based on its rendering environment. A short list of layout schemata described below modify these values for some of their subexpressions. Otherwise, values are determined by inheritance whenever they are not directly specified on a given element's start tag.

For an instance of MathML embedded in a textual data format (such as HTML) in `display' mode, i.e. in place of a paragraph, displaystyle = true and scriptlevel = 0 for the outermost expression of the embedded MathML; if the MathML is embedded in `inline' mode, i.e. in place of a character, displaystyle = false and scriptlevel = 0 for the outermost expression. See chapter 7 [The MathML Interface] for further discussion of the distinction between `display' and `inline' embedding of MathML and how this can be specified in particular instances. In general, a MathML renderer may determine these initial values in whatever manner is appropriate for the location and context of the specific instance of MathML it is rendering, or if it has no way to determine this, based on the way it is most likely to be used; as a last resort it is suggested that it use the most generic values displaystyle = "true" and scriptlevel = "0".

The MathML layout schemata which typically display some of their arguments in smaller type or with less vertical spacing, namely the elements for scripts, fractions, radicals, and tables or matrices, set displaystyle to false, and in some cases increase scriptlevel, for those arguments. The new values are inherited by all subexpressions within those arguments, unless they are overridden.

The specific rules by which each element modifies displaystyle and/or scriptlevel are given in the specification for each element which does so; the complete list of elements which modify either attribute are: the `scripting' elements msub, msup, msubsup, munder, mover, munderover, and mmultiscripts; and the elements mfrac, mroot, and mtable.

When mstyle is given a scriptlevel attribute with no sign, it sets the value of scriptlevel within its contents to the value given, which must be a nonnegative integer. When the attribute value consists of a sign followed by an integer, the value of scriptlevel is incremented (for '+') or decremented (for '-') by the amount given. The incremental syntax for this attribute is an exception to the general rules for setting inherited attributes using mstyle, and is not allowed by any other attribute on mstyle.

Whenever the scriptlevel is changed, either automatically or by being explicitly incremented, decremented, or set, the current font size is multiplied by the value of scriptsizemultiplier to the power of the change in scriptlevel. For example, if scriptlevel is increased by 2, the font size is multiplied by scriptsizemultiplier twice in succession; if scriptlevel is explicitly set to 2 when it had been 3, the font size is divided by scriptsizemultiplier.

The default value of scriptsizemultiplier is less than one (in fact, it is approximately the square root of 1/2), resulting in a smaller font size with increasing scriptlevel. To prevent scripts from becoming unreadably small, the font size is never allowed to go below the value of scriptminsize as a result of a change to scriptlevel, though it can be set to a lower value using the fontsize attribute (section 3.2.1 [Attributes common to token elements]) on mstyle or on token elements. If a change to scriptlevel would cause the font size to become lower than scriptminsize using the above formula, the font size is instead set equal to scriptminsize within the subexpression for which scriptlevel was changed.

In the syntax for scriptminsize, v-unit represents a unit of vertical length (as described in section 2.3.3 [MathML Attribute Values]). The most common unit for specifying font sizes in typesetting is pt (points).

Explicit changes to the fontsize attribute have no effect on the value of scriptlevel.

3.3.4.2.2. Further details on scriptlevel for renderers

For MathML renderers which support CSS1 style sheets, or some other analogous style sheet mechanism, absolute or relative changes to fontsize (or other attributes) may occur implicitly on any element in response to a style sheet. Changes to fontsize of this kind also have no effect on scriptlevel. A style sheet-induced change to fontsize overrides scriptminsize in the same way as for an explicit change to fontsize in the element's begin tag (discussed above), whether it is specified in the style sheet as an absolute or a relative change. (However, any subsequent scriptlevel-induced change to fontsize will still be affected by it.) As is required for inherited attributes in CSS1, the style sheet-modified fontsize is inherited by child elements.

If the same element is subject to both a style sheet-induced and an automatic (scriptlevel-related) change to its own fontsize, the scriptlevel-related change is done first - in fact, in the simplest implementation of the element-specific rules for scriptlevel, this change would be done by the element's parent as part of producing the rendering properties it passes to the given element, since it is the parent element which knows whether scriptlevel should be changed for each of its child elements.

If the element's own fontsize is changed by a style sheet and it also changes scriptlevel (and thus fontsize) for one of its children, the style sheet-induced change is done first, followed by the change inherited by that child. If more than one child's scriptlevel is changed, the change inherited by each child has no effect on the other children. (As a mnemonic rule which applies to a `parse tree' of elements and their children, style sheet-induced changes to fontsize can be associated to nodes of the tree, i.e. to MathML elements, and scriptlevel-related changes can be associated to the edges between parent and child elements; then the order of the associated changes corresponds to the order of nodes and edges in each path down the tree.) For general information on the relative order of processing of properties set by style sheets vs. by attributes, see the appropriate subsection of CSS-compatible attributes in section 2.3.3.3 [CSS-compatible attributes].

If scriptlevel is changed incrementally by an mstyle element which also sets certain other attributes, the overall effect of the changes may depend on the order in which they are processed. In such cases, the attributes in the following list should be processed in the following order, regardless of the order in which they occur in the XML-format attribute list of the mstyle begin tag: scriptsizemultiplier, scriptminsize, scriptlevel, fontsize.

Note that scriptlevel can, in principle, attain any integral value by being decremented sufficiently, even though it can only be explicitly set to nonnegative values. Negative values of scriptlevel generated in this way are legal and should work as described, generating font sizes larger than those of the surrounding expression. Since scriptlevel is initially 0 and never decreases automatically, it will always be nonnegative unless it is decremented past 0 using mstyle.

Explicit decrements of scriptlevel after the font size has been limited by scriptminsize as described above would produce undesirable results. This might occur, for example, in a representation of a continued fraction, in which the scriptlevel was decremented for part of the denominator back to its value for the fraction as a whole, if the continued fraction itself was located in a place which had a high scriptlevel. To prevent this problem, MathML renderers should, when decrementing scriptlevel, use as the initial font size the value the font size would have had if it had never been limited by scriptminsize. They should not, however, ignore the effects of explicit settings of fontsize, even to values below scriptminsize.

Since MathML renderers may be unable to make use of arbitrary font sizes with good results, they may wish to modify the mapping from scriptlevel to fontsize to produce better renderings in their judgement. In particular, if fontsizes have to be rounded to available values, or limited to values within a range, the details of how this is done are up to the renderer. Renderers should, however, ensure that a series of incremental changes to scriptlevel resulting in its return to the same value for some subexpression that it had in a surrounding expression results in the same fontsize for that subexpression as for the surrounding expression.

3.3.4.2.3. Color and background attributes

The color attribute controls the color in which the content of tokens is rendered. Additionally, when inherited from mstyle or from a MathML expression's rendering environment, it controls the color of all other drawing by MathML elements, including the lines or radical signs that can be drawn by mfrac, mtable, or msqrt.

Note that the background attribute, though not inherited, has the default value `transparent' (as in CSS1), which effectively allows an element's parent to control its background.

The values of color and background can be specified as a string consisting of '#' followed without intervening whitespace by either 1-digit or 2-digit hexadecimal values for the red, green, and blue components, respectively, of the desired color, with the same number of digits used for each component (or as the keyword `transparent' for background). The hexadecimal digits are not case-sensitive. The possible 1-digit values range from 0 (component not present) to F (component fully present), and the possible 2-digit values range from 00 (component not present) to FF (component fully present), with the 1-digit value x being equivalent to the 2-digit value xx (rather than x0). % x0 would be a more strictly correct notation, but renders terribly in some browsers.

These attributes can also be specified as an html-color-name, which is defined in the following subsection.

3.3.4.2.4. CSS compatibility of color attributes

The color syntax described above is a subset of the syntax of the color and background-color properties of CSS1. (The background-color syntax is in turn a subset of the full CSS1 background property syntax, which also permits specification of (for example) background images with optional repeats. The more general attribute name background is used in MathML to facilitate possible extensions to the attribute's scope in future versions of MathML.)

Color values on either attribute can also be specified as an html-color-name, that is, as one of the color-name keywords defined in [HTML40]. The list of allowed color names includes most of the commonest English color words, though not orange, brown, or pink, and also includes a number of less-common color words; see the reference for the complete list and the equivalent RGB values. Note that the color name keywords are not case-sensitive, unlike most keywords in MathML attribute values. (The same color name keywords are defined for the CSS1 color property, but with unspecified RGB values. See also section 2.3.3.3 [CSS-compatible attributes].)

3.3.4.2.5. Precise background region not specified

The suggested MathML visual rendering rules do not define the precise extent of the region whose background is affected by using the background attribute on mstyle, except that, when mstyle's content does not have negative dimensions and its drawing region is not overlapped by other drawing due to surrounding negative spacing, this region should lie behind all the drawing done to render the content of the mstyle, but should not lie behind any of the drawing done to render surrounding expressions. The effect of overlap of drawing regions caused by negative spacing on the extent of the region affected by the background attribute is not defined by these rules.

3.3.4.3. Example of mstyle

The example of limiting the stretchiness of a parenthesis shown in the section on <mo>,

<mrow>
   <mo maxsize="1"> ( </mo>
   <mfrac> <mi> a </mi> <mi> b </mi> </mfrac>
   <mo maxsize="1"> ) </mo>
</mrow>

can be rewritten using mstyle as:

<mstyle maxsize="1">
   <mrow>
      <mo> ( </mo>
      <mfrac> <mi> a </mi> <mi> b </mi> </mfrac>
      <mo> ) </mo>
   </mrow>
</mstyle>

3.3.5. Error Message

3.3.5.1. Description

The merror element displays its contents as an `error message'. This might be done, for example, by displaying the contents in red, flashing the contents, or changing the background color. The contents can be any expression or expression sequence.

merror accepts any number of arguments; if this number is not 1, its contents are treated as a single `inferred mrow' as described in section 3.1.3 [Required Arguments].

The intent of this element is to provide a standard way for programs that generate MathML from other input to report syntax errors in their input. Since it is anticipated that preprocessors that parse input syntaxes designed for easy hand entry will be developed to generate MathML, it is important that they have the ability to indicate that a syntax error occurred at a certain point. See section 7.2.2 [Handling of Errors].

The suggested use of merror for reporting syntax errors is for a preprocessor to replace the erroneous part of its input with an merror element containing a description of the error, while processing the surrounding expressions normally as far as possible. By this means, the error message will be rendered where the erroneous input would have appeared, had it been correct; this makes it easier for an author to determine from the rendered output what portion of the input was in error.

No specific error message format is suggested here, but as with error messages from any program, the format should be designed to make as clear as possible (to a human viewer of the rendered error message) what was wrong with the input and how it can be fixed. If the erroneous input contains correctly formatted subsections, it may be useful for these to be preprocessed normally and included in the error message (within the contents of the merror element), taking advantage of the ability of merror to contain arbitrary MathML expressions rather than only text.

3.3.5.2. Attributes of merror

None (except the attributes allowed for all MathML elements, listed in section 2.3.4 [Attributes Shared by all MathML Elements]).

3.3.5.3. Example of merror

If a MathML syntax-checking preprocessor received the input

<mfraction>
   <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow>
   <mn> 2 </mn>
</mfraction>

which contains the non-MathML element mfraction (presumably in place of the MathML element mfrac), it might generate the error message

<merror>
   <mtext> Unrecognized element: <mfraction>;
           arguments were:  </mtext>
   <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow>
   <mtext>  and  </mtext>
   <mn> 2 </mn>
</merror>

Note that the preprocessor's input is not, in this case, valid MathML, but the error message it outputs is valid MathML.

3.3.6. Adjust Space Around Content

3.3.6.1. Description

An mpadded element renders the same as its content, but with its overall size and other dimensions (such as baseline position) modified according to its attributes. The mpadded element does not rescale (stretch or shrink) its content; its only effect is to modify the apparent size and position of the `bounding box' around its content, so as to affect the relative position of the content with respect to the surrounding elements. The name of the element reflects the use of mpadded to effectively add `padding', or extra space, around its content. If the `padding' is negative, it is possible for the content of mpadded to be rendered outside the mpadded element's bounding box; see below for warnings about several potential pitfalls of this effect.

The mpadded element accepts any number of arguments; if this number is not 1, its contents are treated as a single `inferred mrow' as described in section 3.1.3 [Required Arguments].

It is suggested that audio renderers add (or shorten) time delays based on the attributes representing horizontal space (width and lspace).

3.3.6.2. Attributes of mpadded

(The pseudo-unit syntax symbol is described below.)

These attributes modify the dimensions of the `bounding box' of the mpadded element. The dimensions (which have the same names as the attributes) are defined in the next subsection. Depending on the format of the attribute value, a dimension may be set to a new value, or to an incremented or decremented version of the content's corresponding dimension. Values may be specified as multiples or percentages of any of the dimensions of the normal rendering of the element's content (using so-called `pseudo-units'), or as multiples of standard section 2.3.3.2 [Attributes with units].

If an attribute value begins with a + or - sign, it specifies an increment or decrement of the corresponding dimension by the following length value (interpreted as explained below). Otherwise, the corresponding dimension is set directly to the following length value. Note that the + and - do not mean that the following value is positive or negative, even when an explicit length unit (h-unit or v-unit) is given. In particular, these attributes cannot directly set a dimension to a negative value.

Length values (after the optional sign, which is not part of the length value) can be specified in several formats. Each format begins with an unsigned-number, which may be followed by a % sign and an optional `pseudo-unit' (denoted by pseudo-unit in the attribute syntaxes above), by a pseudo-unit alone, or by one of the length units (denoted by h-unit or v-unit) specified in section 2.3.3.3 [CSS-compatible attributes], not including %. The possible pseudo-units are the keywords width, lspace, height, and depth; they each represent the length of the same-named dimension of the mpadded element's content (not of the mpadded element itself). The lengths represented by h-unit or v-unit are described in section 2.3.3.3 [CSS-compatible attributes].

In any of these formats, the length value specified is the product of: the specified number; 0.01 if % is given; and the length represented by the unit or pseudo-unit. If no pseudo-unit is given after %, the one with the same name as the attribute being specified is assumed.

Some examples of attribute formats using pseudo-units (explicit or default) are as follows: depth="100% height" and depth="1.0 height" both set the depth of the mpadded element to the height of its content. depth="105%" sets the depth to 1.05 times the content's depth, and either depth="+100%" or depth="200%" sets the depth to twice the content's depth.

Dimensions that would be positive if the content was rendered normally cannot be made negative using mpadded; a positive dimension is set to 0 if it would otherwise become negative. Dimensions which are initially 0 can be made negative, but this should generally be avoided. See the warnings below on the use of negative spacing for `tweaking' or conveying meaning.

The rules given above imply that all of the following attribute settings have the same effect, which is to leave the content's dimensions unchanged:

<mpadded width="+0em"> ... </mpadded>
<mpadded width="+0%"> ... </mpadded>
<mpadded width="-0em"> ... </mpadded>
<mpadded width="- 0 height"> ... </mpadded>
<mpadded width="100%"> ... </mpadded>
<mpadded width="100% width"> ... </mpadded>
<mpadded width="1 width"> ... </mpadded>
<mpadded width="1.0 width"> ... </mpadded>
<mpadded> ... </mpadded>

3.3.6.3. Meanings of dimension attributes

See appendix F [Glossary] for further information about some of the typesetting terms used here.

The width attribute refers to the overall horizontal width of a bounding box. By default (i.e. when lspace is not modified), the bounding box of the content of an mpadded element should be rendered flush with the left edge of the mpadded element's bounding box. Thus, increasing width alone effectively adds space on the right edge of the box.

The lspace attribute refers to the amount of space between the left edge of a bounding box and where the rendering of its contents' bounding box actually begins. Unlike the other dimensions, lspace does not correspond to a real property of a bounding box, but exists only transiently during the computations done by each instance of mpadded. It is provided so that there is a way to add space on the left edge of a bounding box.

The rationale behind using width and lspace to control horizontal padding instead of more symmetric attributes, such as a hypothetical rspace and lspace, is that it is desirable to have a `width' pseudo unit, in part because `width' is an actual property of a bounding box.

The height attribute refers to the amount of vertical space between the baseline (the line along the bottom of most letter glyphs in normal text rendering) and the top of the bounding box.

The depth attribute refers to the amount of vertical space between the bottom of the bounding box and the baseline.

MathML renderers should ensure that, except for the effects of the attributes, relative spacing between the contents of mpadded and surrounding MathML elements is not modified by replacing an mpadded element with an mrow element with the same content. This holds even if linebreaking occurs within the mpadded element. However, if an mpadded element with non-default attribute values is subjected to linebreaking, MathML does not define how its attributes or rendering interact with the linebreaking algorithm.

3.3.6.4. Warning: nonportability of `tweaking'

A likely temptation for the use of the mpadded and mspace elements (and perhaps also mphantom and mtext) will be for an author to improve the spacing generated by a specific renderer by slightly modifying it in specific expressions, i.e. to `tweak' the rendering.

Authors are strongly warned that different MathML renderers may use different spacing rules for computing the relative positions of rendered symbols in expressions which have no explicit modifications to their spacing; if renderer B improves upon renderer A's spacing rules, explicit spacing added to improve the output quality of renderer A may produce very poor results in renderer B, very likely worse than without any `tweaking' at all.

Even when a specific choice of renderer can be assumed, its spacing rules may be improved in successive versions, so that the effect of tweaking in a given MathML document may grow worse with time. Also, when style sheet mechanisms are extended to MathML, even one version of a renderer may use different spacing rules for users with different style sheets.

Therefore, it is suggested that MathML markup never use mpadded or mspace elements to tweak the rendering of specific expressions, unless the MathML is generated solely to be viewed using one specific version of one MathML renderer, using one specific style sheet (if style sheets are available in that renderer).

In cases where the temptation to improve spacing proves too strong, careful use of mpadded, mphantom, or the alignment elements (section 3.5.4 [Alignment Markers]) may give more portable results than the direct insertion of extra space using mspace or mtext. Advice given to the implementors of MathML renderers might be still more productive, in the long run.

3.3.6.5. Warning: spacing should not be used to convey meaning

MathML elements which permit `negative spacing', namely mspace, mpadded, and mtext, could in theory be used to simulate new notations or `overstruck' characters by the visual overlap of the renderings of more than one MathML subexpression.

This practice is strongly discouraged in all situations, for the following reasons:

• it will give different results in different MathML renderers (so the warning about `tweaking' applies);
• it is likely to appear much worse than a more standard construct supported by good renderers;
• such expressions are almost certain to be uninterpretable by audio renderers, computer algebra systems, text searches for standard symbols, or other processors of MathML input.

More generally, any construct which uses spacing to convey mathematical meaning, rather than simply as an aid to viewing expression structure, is discouraged. That is, the constructs which are discouraged are those which would be interpreted differently by a human viewer of rendered MathML if all explicit spacing was removed.

If such constructs are used in spite of this warning, they should be enclosed in a semantics element which also provides an additional MathML expression which can be interpreted in a standard way.

For example, the MathML expression

<mrow>
   <mpadded width="0"> <mi> C </mi> </mpadded>
   <mspace width="0.3em"/>
   <mtext> | </mtext>
</mrow>

forms an overstruck symbol in violation of the policy stated above; it might be intended to represent the set of complex numbers for a MathML renderer which lacks support for the standard symbol used for this purpose. This kind of construct should always be avoided in MathML, for the reasons stated above; indeed, it should never be necessary for standard symbols, since a MathML renderer with no better method of rendering them is free to use overstriking internally, so that it can still support general MathML input.

However, if for whatever reason such a construct is used in MathML, it should always be enclosed in a semantics element such as

<semantics>
   <mrow>
      <mpadded width="0"> <mi> C </mi> </mpadded>
      <mspace width="0.3em"/>
      <mtext> | </mtext>
   </mrow>
   <annotation-xml encoding="mathml">
      <mi> &DoubleStruckCapitalC; </mi>
   </annotation-xml>
</semantics>

which provides an alternative, standard encoding for the desired symbol, which is much more easily interpreted than the construct using negative spacing. (The alternative encoding in this example uses MathML presentation elements; the content elements described in chapter 4 [Content Markup] should also be considered.)

(The above warning also applies to most uses of rendering attributes to alter the meaning conveyed by an expression, with the exception of attributes on mi (such as fontweight) used to distinguish one variable from another.)

3.3.7. Making Content Invisible

3.3.7.1. Description

The mphantom element renders invisibly, but with the same size and other dimensions, including baseline position, that its contents would have if they were rendered normally. mphantom can be used to align parts of an expression by invisibly duplicating subexpressions.

The mphantom element accepts any number of arguments; if this number is not 1, its contents are treated as a single `inferred mrow' formed from all its arguments, as described in section 3.1.3 [Required Arguments].

It is suggested that audio renderers render mphantom elements in an analogous way for their medium, by rendering them as silence of the same duration as the normal rendering of their contents.

3.3.7.2. Attributes of mphantom

None (except the attributes allowed for all MathML elements, listed in section 2.3.4 [Attributes Shared by all MathML Elements]).

Note that it is possible to wrap both an mphantom and an mpadded element around one MathML expression, as in <mphantom><mpadded attribute-settings> ... </mpadded></mphantom>, to change its size and make it invisible at the same time.

MathML renderers should ensure that the relative spacing between the contents of an mphantom element and the surrounding MathML elements is the same as it would be if the mphantom element were replaced by an mrow element with the same content. This holds even if linebreaking occurs within the mphantom element.

For the above reason, mphantom is not considered space-like (section 3.2.6 [Space]) unless its content is space-like, since the suggested rendering rules for operators are affected by whether nearby elements are space-like. Even so, the warning about the legal grouping of space-like elements may apply to uses of mphantom.

There is one situation where the preceding rule for rendering an mphantom may not give the desired effect. When an mphantom is wrapped around a subsequence of the arguments of an mrow, the default determination of the form attribute for an mo element within the subsequence can change. (See the default value of the form attribute described in section 3.2.4 [Operator, Fence, Separator or Accent].) It may be necessary to add an explicit form attribute to such an mo in these cases. This is illustrated in the following example.

3.3.7.3. Example of mphantom

In this example, mphantom is used to ensure alignment of corresponding parts of the numerator and denominator of a fraction:

<mfrac>
   <mrow>
      <mi> x </mi>
      <mo> + </mo>
      <mi> y </mi>
      <mo> + </mo>
      <mi> z </mi>
   </mrow>
   <mrow>
      <mi> x </mi>
      <mphantom>
         <mo form="infix"> + </mo>
         <mi> y </mi>
      </mphantom>
      <mo> + </mo>
      <mi> z </mi>
   </mrow>
</mfrac>

This would render as something like

rather than as

The explicit attribute setting form="infix" on the mo element inside the mphantom sets the form attribute to what it would have been in the absence of the surrounding mphantom. This is necessary since otherwise, the + sign would be interpreted as a prefix operator, which might have slightly different spacing.

Alternatively, this problem could be avoided without any explicit attribute settings, by wrapping each of the arguments <mo>+</mo> and <mi>y</mi> in its own mphantom element, i.e.

<mfrac>
   <mrow>
      <mi> x </mi>
      <mo> + </mo>
      <mi> y </mi>
      <mo> + </mo>
      <mi> z </mi>
   </mrow>
   <mrow>
      <mi> x </mi>
      <mphantom>
         <mo> + </mo>
      </mphantom>
      <mphantom>
         <mi> y </mi>
      </mphantom>
      <mo> + </mo>
      <mi> z </mi>
   </mrow>
</mfrac>

3.3.8. Content Inside Pair of Fences

3.3.8.1. Description

The mfenced element provides a convenient form in which to express common constructs involving fences (i.e. braces, brackets, and parentheses), possibly including separators (such as comma) between the arguments.

For example, <mfenced> <mi>x</mi> </mfenced> renders as `(x)'and is equivalent to

<mrow> <mo> ( </mo> <mi>x</mi> <mo> ) </mo> </mrow>

and <mfenced> <mi>x</mi> <mi>y</mi> </mfenced> renders as `(x, y)' and is equivalent to

<mrow>
   <mo> ( </mo>
   <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow>
   <mo> ) </mo>
</mrow>

Individual fences or separators are represented using mo elements, as described in section 3.2.4 [Operator, Fence, Separator or Accent]. Thus, any mfenced element is completely equivalent to an expanded form described below; either form can be used in MathML, at the convenience of an author or of a MathML-generating program. A MathML renderer is required to render either of these forms in exactly the same way.

In general, an mfenced element can contain zero or more arguments, and will enclose them between fences in an mrow; if there is more than one argument, it will insert separators between adjacent arguments, using an additional nested mrow around the arguments and separators for proper grouping (section 3.3.1 [Horizontally Group Any Number of Subexpressions]). The general expanded form is shown below. The fences and separators will be parentheses and comma by default, but can be changed using attributes, as shown in the following table.

3.3.8.2. Attributes of mfenced

A generic mfenced element, with all attributes explicit, looks as follows:

<mfenced open="opening-fence"
         close="closing-fence"
         separators="sep#1 sep#2 ... sep#(n-1)" >
   arg#1
   ...
   arg#n
</mfenced>

The opening-fence and closing-fence are arbitrary strings. (Since they are used as the content of mo elements, any whitespace they contain will be trimmed and collapsed as described in section 2.3.5 [Collapsing Whitespace in Input].)

The value of separators is a sequence of zero or more separator characters (or entity references), optionally separated by whitespace. Each sep#i consists of exactly one character or entity reference. Thus, separators=",;" is equivalent to separators=" , ; ".

The general mfenced element shown above is equivalent to the following expanded form:

<mrow>
   <mo fence="true"> opening-fence </mo>
   <mrow>
      arg#1
      <mo separator="true"> sep#1 </mo>
      ...
      <mo separator="true"> sep#(n-1) </mo>
      arg#n
   </mrow>
   <mo fence="true"> closing-fence </mo>
</mrow>

Each argument except the last is followed by a separator. The inner mrow is added for proper grouping, as described in section 3.3.1 [Horizontally Group Any Number of Subexpressions].

When there is only one argument, the above form has no separators; since <mrow> arg#1 </mrow> is equivalent to arg#1 (as described in section 3.3.1 [Horizontally Group Any Number of Subexpressions]), this case is also equivalent to:

<mrow>
   <mo fence="true"> opening-fence </mo>
   arg#1
   <mo fence="true"> closing-fence </mo>
</mrow>

If there are too many separator characters, the extra ones are ignored. If separator characters are given, but there are too few, the last one is repeated as necessary. Thus, the default value of separators="," is equivalent to separators=",,", separators=",,,", etcetera. If there are no separator characters provided but some are needed, for example if separators=" " or "" and there is more than one argument, then no separator elements are inserted at all - that is, the elements <mo separator="true"> sep#i </mo> are left out entirely. Note that this is different from inserting separators consisting of mo elements with empty content.

Finally, for the case with no arguments, i.e.

<mfenced open="opening-fence"
         close="closing-fence"
         separators="anything" >
</mfenced>

the equivalent expanded form is defined to include just the fences within an mrow:

<mrow>
   <mo fence="true"> opening-fence </mo>
   <mo fence="true"> closing-fence </mo>
</mrow>

Note that not all `fenced expressions' can be encoded by an mfenced element. Such exceptional expressions include those with an `embellished' separator or fence or one enclosed in an mstyle element, a missing or extra separator or fence, or a separator with multiple content characters. In these cases, it is necessary to encode the expression using an appropriately modified version of an expanded form. As discussed above, it is always permissible to use the expanded form directly, even when it is not necessary. In particular, authors cannot be guaranteed that MathML preprocessors won't replace occurrences of mfenced with equivalent expanded forms.

Note that the equivalent expanded forms shown above include attributes on the mo elements which identify them as fences or separators. Since the most common choices of fences and separators already occur in the operator dictionary with those attributes, authors would not normally need to specify those attributes explicitly when using the expanded form directly. Also, the rules for the default form attribute (section 3.2.4 [Operator, Fence, Separator or Accent]) cause the opening and closing fences to be effectively given the values form="prefix" and form="postfix" respectively, and the separators to be given the value form="infix".

Note that it would be incorrect to use mfenced with a separator of, for instance, `+', as an abbreviation for an expression using `+' as an ordinary operator, e.g.

<mrow>
  <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi>
</mrow>

This is because the + signs would be treated as separators, not infix operators. That is, it would render as if they were marked up as <mo separator="true">+</mo>, which might therefore render inappropriately.

3.3.8.3. Examples of mfenced

(a+b)

<mfenced>
   <mrow>
      <mi> a </mi>
      <mo> + </mo>
      <mi> b </mi>
   </mrow>
</mfenced>

Note that the above mrow is necessary so that the mfenced has just one argument. Without it, this would render incorrectly as `(a, +, b)'.

[0,1)

<mfenced open="[">
   <mn> 0 </mn>
   <mn> 1 </mn>
</mfenced>

f(x,y)

<mrow>
   <mi> f </mi>
   <mo> &ApplyFunction; </mo>
   <mfenced>
      <mi> x </mi>
      <mi> y </mi>
   </mfenced>
</mrow>

3.3.9. Enclose Content Inside Notation

3.3.9.1. Description

The menclose element renders its content inside the enclosing notation specified by its notation attribute. menclose accepts any number of arguments; if this number is not 1, its contents are treated as a single `inferred mrow' containing its arguments, as described in section 3.1.3 [Required Arguments].

3.3.9.2. Attributes of menclose

With notation has the value longdiv, the contents are drawn enclosed by a long division symbol. A complete example of long division is accomplished by also using mtable and malign. When notation is specified as actuarial, the contents are drawn enclosed by an actuarial symbol. The case of notation=radical is equivalent to the msqrt schema.

3.3.9.3. Examples

The following markup might be used to encode an elementary US-style long division problem.

<mtable columnspacing='0' rowspacing='0'>
<mtr>
  <mtd></mtd>
  <mtd columnalign='right'><mn>10</mn></mtd>
</mtr>
<mtr>
  <mtd columnalign='right'><mn>131</mn></mtd>
  <mtd columnalign='right'>
    <menclose notation='longdiv'><mn>1413</mn></menclose>
  </mtd>
</mtr>
<mtr>
  <mtd></mtd>
  <mtd columnalign='right'>
    <mrow>
    <munder>
      <mn>131</mn>
      <mo>&bar;</mo>
    </munder>
    <mphantom><mn>3</mn></mphantom>
    </mrow>
  </mtd>
</mtr>
<mtr>
  <mtd></mtd>
  <mtd columnalign='right'><mn>103</mn></mtd>
</mtr>
</mtable>

This might render as:

.

An example of using menclose for actuarial notation is

<msub>
  <mi>a</mi>
  <mrow>
    <menclose notation='actuarial'>
      <mi>n</mi>
    </menclose>
    <mo>&it;</mo>
    <mi>i</mi>
  </mrow>
</msub>

which renders as

a_
 n|i

3.4. Script and Limit Schemata

The elements described in this section position one or more scripts around a base. Attaching various kinds of scripts and embellishments to symbols is a very common notational device in mathematics. For purely visual layout, a single general-purpose element could suffice for positioning scripts and embellishments in any of the traditional script locations around a given base. However, in order to capture the abstract structure of common notation better, MathML provides several more specialized scripting elements.

In addition to sub/superscript elements, MathML has over/underscript elements which place scripts above and below the base. These elements can be used to place limits on large operators, or for placing accents and lines above or below the base. The rules for rendering accents differ from those for overscripts and underscripts, and this difference can be controlled with the accent and accentunder attributes, as described in the appropriate sections below.

Rendering of scripts is affected by the scriptlevel and displaystyle attributes, which are part of the environment inherited by the rendering process of every MathML expression, and are described under mstyle (section 3.3.4 [Style Change]). These attributes cannot be given explicitly on a scripting element, but can be specified on the start tag of a surrounding mstyle element if desired.

MathML also provides an element for attachment of tensor indices. Tensor indices are distinct from ordinary subscripts and superscripts in that they must align in vertical columns. Tensor indices can also occur in prescript positions.

Because presentation elements should be used to describe the abstract notational structure of expressions, it is important that the base expression in all `scripting' elements (i.e. the first argument expression) should be the entire expression that is being scripted, not just the rightmost character. For example, (x+y)² should be written as:

<msup>
   <mrow>
      <mo> ( </mo>
      <mrow>
         <mi> x </mi>
         <mo> + </mo>
         <mi> y </mi>
      </mrow>
      <mo> ) </mo>
   </mrow>
   <mn> 2 </mn>
</msup>

3.4.1. Subscript

3.4.1.1. Description

The syntax for the msub element is <msub> base subscript</msub>.

3.4.1.2. Attributes of msub

The subscriptshift attribute specifies the minimum amount to shift the baseline of subscript down.

v-unit represents a unit of vertical length (see section 2.3.3 [MathML Attribute Values]).

The msub element increments scriptlevel by 1, and sets displaystyle to false, within subscript, but leaves both attributes unchanged within base. (These attributes are inherited by every element through its rendering environment, but can be set explicitly only on mstyle; see section 3.3.4 [Style Change].)

3.4.2. Superscript

3.4.2.1. Description

The syntax for the msup element is <msup> base superscript </msup>.

3.4.2.2. Attributes of msup

The superscriptshift attribute specifies the minimum amount to shift the baseline of superscript up.

v-unit represents a unit of vertical length (see section 2.3.3 [MathML Attribute Values]).

The msup element increments scriptlevel by 1, and sets displaystyle to false, within superscript, but leaves both attributes unchanged within base. (These attributes are inherited by every element through its rendering environment, but can be set explicitly only on mstyle; see section 3.3.4 [Style Change].)

3.4.3. Subscript-superscript Pair

3.4.3.1. Description

The msubsup element is used so that the subscript and superscript are both tight against the base, i.e. vertically aligned as in the second case shown here: versus .

The syntax for the msubsup element is <msubsup>base subscript superscript</msubsup>

3.4.3.2. Attributes of msubsup

The subscriptshift attribute specifies the minimum amount to shift the baseline of subscript down. The superscriptshift attribute specifies the minimum amount to shift the baseline of superscript up.

v-unit represents a unit of vertical length (see section 2.3.3 [MathML Attribute Values]).

The msubsup element increments scriptlevel by 1, and sets displaystyle to false, within subscript and superscript, but leaves both attributes unchanged within base. (These attributes are inherited by every element through its rendering environment, but can be set explicitly only on mstyle; see section 3.3.4 [Style Change].)

3.4.3.3. Examples of msubsup

The msubsup is most commonly used for adding sub/superscript pairs to identifiers as illustrated above. However, another important use is placing limits on certain large operators whose limits are traditionally displayed in the script positions even when rendered in display style. The most common of these is the integral. For example,

would be represented as

<mrow>
   <msubsup>
      <mo> &int; </mo>
      <mn> 0 </mn>
      <mn> 1 </mn>
   </msubsup>
   <mrow>
      <msup>
         <mi> &ExponentialE; </mi>
         <mi> x </mi>
      </msup>
      <mo> &InvisibleTimes; </mo>
      <mrow>
         <mo> &DifferentialD; </mo>
         <mi> x </mi>
      </mrow>
   </mrow>
</mrow>

3.4.4. Underscript

3.4.4.1. Description

The syntax for the munder element is

<munder> base underscript </munder>

3.4.4.2. Attributes of munder

The accentunder attribute controls whether underscript is drawn as an `accent' or as a limit. The main difference between an accent and a limit is that the limit is reduced in size whereas an accent is the same size as the base. A second difference is that the accent is drawn closer to the base.

The default value of accentunder is false, unless underscript is an mo element or an embellished operator (see section 3.2.4 [Operator, Fence, Separator or Accent]). If underscript is an mo element, the value of its accent attribute is used as the default value of accentunder. If underscript is an embellished operator, the accent attribute of the mo element at its core is used as the default value. As with all attributes, an explicitly given value overrides the default.

Here is an example (accent versus underscript): versus . The MathML representation for this example is shown below.

If the base is an operator with movablelimits=true (or an embellished operator whose mo element core has movablelimits=true), and displaystyle=false, then underscript is drawn in a subscript position. In this case, the accentunder attribute is ignored. This is often used for limits on symbols such as ∑.

Within underscript, munder always sets displaystyle to false, but increments scriptlevel by 1 only when accentunder is false. Within base, it always leaves both attributes unchanged. (These attributes are inherited by every element through its rendering environment, but can be set explicitly only on mstyle; see section 3.3.4 [Style Change].)

3.4.4.3. Examples of munder

The MathML representation for the example shown above is:

<mrow>
   <munder accentunder="true">
      <mrow>
         <mi> x </mi>
         <mo> + </mo>
         <mi> y </mi>
         <mo> + </mo>
         <mi> z </mi>
      </mrow>
      <mo> &UnderBrace; </mo>
   </munder>
   <mtext>   vs   </mtext>
   <munder accentunder="false">
      <mrow>
         <mi> x </mi>
         <mo> + </mo>
         <mi> y </mi>
         <mo> + </mo>
         <mi> z </mi>
      </mrow>
      <mo> &UnderBrace; </mo>
   </munder>
</mrow>

3.4.5. Overscript

3.4.5.1. Description

The syntax for the mover element is

<mover> base overscript> </mover>

.

3.4.5.2. Attributes of mover

The accent attribute controls whether overscript is drawn as an `accent' (diacritical mark) or as a limit. The main difference between an accent and a limit is that the limit is reduced in size whereas an accent is the same size as the base. A second difference is that the accent is drawn closer to the base. This is shown below (accent versus limit): versus .

These differences also apply to `mathematical accents' such as bars over expressions: versus. . The MathML representation for each of these examples is shown below.

The default value of accent is false, unless overscript is an mo element or an embellished operator (see section 3.2.4 [Operator, Fence, Separator or Accent]). If overscript is an mo element, the value of its accent attribute is used as the default value of accent for mover. If overscript is an embellished operator, the accent attribute of the mo element at its core is used as the default value.

If the base is an operator with movablelimits=true (or an embellished operator whose mo element core has movablelimits=true), and displaystyle=false, then overscript is drawn in a superscript position. In this case, the accent attribute is ignored. This is often used for limits on symbols such as ∑.

Within overscript, mover always sets displaystyle to false, but increments scriptlevel by 1 only when accent is false. Within base, it always leaves both attributes unchanged. (These attributes are inherited by every element through its rendering environment, but can be set explicitly only on mstyle; see section 3.3.4 [Style Change].)

3.4.5.3. Examples of mover

The MathML representation for the examples shown above is:

<mrow>
   <mover accent="true">
      <mi> x </mi>
      <mo> &hat; </mo>
   </mover>
   <mtext>   vs   </mtext>
   <mover accent="false">
      <mi> x </mi>
      <mo> &hat; </mo>
   </mover>
</mrow>

<mrow>
   <mover accent="true">
      <mrow>
         <mi> x </mi>
         <mo> + </mo>
         <mi> y </mi>
         <mo> + </mo>
         <mi> z </mi>
      </mrow>
      <mo> &OverBar; </mo>
   </mover>
   <mtext>   vs   </mtext>
   <mover accent="false">
      <mrow>
         <mi> x </mi>
         <mo> + </mo>
         <mi> y </mi>
         <mo> + </mo>
         <mi> z </mi>
      </mrow>
      <mo> &OverBar; </mo>
   </mover>
</mrow>

3.4.6. Underscript-overscript Pair

3.4.6.1. Description

The syntax for the munderover element is

<munderover> base underscript overscript</munderover>

3.4.6.2. Attributes of munderover

The munderover element is used so that the underscript and overscript are vertically spaced equally in relation to the base and so that they follow the slant of the base as in the second expression shown below:

versus

The MathML representation for this example is shown below.

The difference in the vertical spacing is too small to be noticed on a low resolution display at a normal font size, but is noticeable on a higher resolution device such as a printer and when using large font sizes. In addition to the visual differences, attaching both the underscript and overscript to the same base more accurately reflects the semantics of the expression.

The accent and accentunder attributes have the same effect as the attributes with the same names on mover (section 3.4.5 [Overscript]) and munder (section 3.4.4 [Underscript]), respectively. Their default values are also computed in the same manner as described for those elements, with the default value of accent depending on overscript and the default value of accentunder depending on underscript.

If the base is an operator with movablelimits=true (or an embellished operator whose mo element core has movablelimits=true), and displaystyle=false, then underscript and overscript are drawn in a subscript and superscript position, respectively. In this case, the accent and accentunder attributes are ignored. This is often used for limits on symbols such as ∑.

Within underscript, munderover always sets displaystyle to false, but increments scriptlevel by 1 only when accentunder is false. Within overscript, munderover always sets displaystyle to false, but increments scriptlevel by 1 only when accent is false. Within base, it always leaves both attributes unchanged. (These attributes are inherited by every element through its rendering environment, but can be set explicitly only on mstyle; see section 3.3.4 [Style Change]).

3.4.6.3. Example of munderover

The MathML representation for the example shown above with the left expression made using separate munder and mover elements, and the right one using an munderover element, is:

<mrow>
   <mover>
      <munder>
         <mo> &int; </mo>
         <mn> 0 </mn>
      </munder>
      <mi> &infinity; </mi>
   </mover>
   <mtext>   vs   </mtext>
   <munderover>
      <mo> &int; </mo>
      <mn> 0 </mn>
      <mi> &infinity; </mi>
   </munderover>
</mrow>

3.4.7. Prescripts and Tensor Indices

3.4.7.1. Description

The syntax for the mmultiscripts element is

<mmultiscripts>
    base
     (subscript superscript)*
     [ <mprescripts/> (presubscript presuperscript)* ]
</mmultiscripts>

Presubscripts and tensor notations are represented by a single element, mmultiscripts. This element allows the representation of any number of vertically-aligned pairs of subscripts and superscripts, attached to one base expression. It supports both postscripts (to the right of the base in visual notation) and prescripts (to the left of the base in visual notation). Missing scripts can be represented by the empty element none.

The prescripts are optional, and when present are given after the postscripts, because prescripts are relatively rare compared to tensor notation.

The argument sequence consists of the base followed by zero or more pairs of vertically-aligned subscripts and superscripts (in that order) that represent all of the postscripts. This list is optionally followed by an empty element mprescripts and a list of zero or more pairs of vertically-aligned presubscripts and presuperscripts that represent all of the prescripts. The pair lists for postscripts and prescripts are given in a left-to-right order. If no subscript or superscript should be rendered in a given position, then the empty element none should be used in that position.

The base, subscripts, superscripts, the optional separator element mprescripts, the presubscripts, and the presuperscripts, are all direct subexpressions of the mmultiscripts element, i.e. they are all at the same level of the expression tree. Whether a script argument is a subscript or a superscript, or whether it is a presubscript or a presuperscript is determined by whether it occurs in an even-numbered or odd-numbered argument position, respectively, ignoring the empty element mprescripts itself when determining the position. The first argument, the base, is considered to be in position 1. The total number of arguments must be odd, if mprescripts is not given, or even, if it is.

The empty elements mprescripts and none are only allowed as direct subexpressions of mmultiscripts.

3.4.7.2. Attributes of mmultiscripts

Same as attributes of msubsup.

The mmultiscripts element increments scriptlevel by 1, and sets displaystyle to false, within each of its arguments except base, but leaves both attributes unchanged within base. (These attributes are inherited by every element through its rendering environment, but can be set explicitly only on mstyle; see section 3.3.4 [Style Change].)

3.4.7.3. Examples of mmultiscripts

Two examples of the use of mmultiscripts are:

₀F₁(;a;z).

<mrow>
   <mmultiscripts>
      <mi> F </mi>
      <mn> 1 </mn>
      <none/>
      <mprescripts/>
      <mn> 0 </mn>
      <none/>
   </mmultiscripts>
   <mo> &ApplyFunction; </mo>
   <mrow>
      <mo> ( </mo>
      <mrow> 
         <mo> ; </mo>
         <mi> a </mi>
         <mo> ; </mo>
         <mi> z </mi>
      </mrow> 
      <mo> ) </mo>
   </mrow>
</mrow>

(where k and l are different indices)

<mmultiscripts>
   <mi> R </mi>
   <mi> i </mi>
   <none/>
   <none/>
   <mi> j </mi>
   <mi> k </mi>
   <none/>
   <mi> l </mi>
   <none/>
</mmultiscripts>

3.5. Tables and Matrices

Matrices, arrays and other table-like mathematical notation are marked up using mtable, mtr and mtd elements. These elements are similar to the TABLE, TR and TD elements of HTML, except that they provide specialized attributes in order to provide the fine layout control necessary for commutative diagrams, block matrices and so on.

3.5.1. Table or Matrix

3.5.1.1. Description

A matrix or table is specified using the mtable element. Inside of the mtable element, mtr and mtd elements can be given. If some argument to mtable is not an mtr element, renderers should assume a one-column row (i.e. the argument is effectively wrapped with an inferred mtr .../mtr begin-end tag pair). Similarly, if some argument to a (possibly inferred) mtr element is not an mtd element, that argument is treated as a table entry by effectively wrapping it with an inferred mtd .../mtd begin-end tag pair. The mtr or mtd elements must be given explicitly if they have attributes different from those they would inherit from the enclosing mtable or mtr.

Note that the above rules imply that an mtable whose arguments are expressions other than mtr or mtd elements forms a single column of those expressions.

Table rows that have fewer columns than other rows of the same table (whether the other rows precede or follow them) are effectively padded on the right with empty mtd elements so that the number of columns in each row equals the maximum number of columns in any row of the table. Note that the use of mtd elements with non-default values of the rowspan or columnspan attributes may affect the number of mtd elements which should be given in subsequent mtr elements to cover a given number of columns.

Note that the usual representation of a matrix consists of an mtable element surrounded by parentheses; see the example below. This can be constructed using either mrow or mfenced. The parentheses are not rendered by the mtable element itself.

3.5.1.2. Attributes of mtable

Note that the default value for each of rowlines, columnlines and frame is the literal string `none', meaning that the default is to render no lines, rather than that there is no default.

As described in section 2.3.3 [MathML Attribute Values], the notation (x | y) + means one or more occurrences of either x or y, separated by whitespace. For example, possible values for columnalign are left, left left, and left right center center. If there are more entries than are necessary (e.g. more entries than columns for columnalign), then only the first entries will be used. If there are fewer entries, then the last entry is repeated as often as necessary. For example, if columnalign="right center" and the table has three columns, the first column will be right aligned and the second and third columns will be centered.

The align attribute specifies where to align the table with respect to its environment. `axis' means to align the center of the table on the environment's axis. (The axis of an equation is an alignment line used by typesetters. It is the line on which a minus sign typically lies.) `center' and `baseline' both mean to align the center of the table on the environment's baseline. `top' or `bottom' aligns the top or bottom of the table on the environment's baseline.

If the align attribute value ends with a rownumber between 1 and n (for a table with n rows), the specified row is aligned in the way described above, rather than the table as a whole; the top (first) row is numbered 1, and the bottom (last) row is numbered n. The same is true if the rownumber is negative, between -1 and -n, except that the bottom row is referred to as -1 and the top row as -n. Other values of rownumber are illegal.

The rowalign attribute specifies how the entries in each row should be aligned. For example, `top' means that the tops of each entry in each row should be aligned with the tops of the other entries in that row. The columnalign attribute specifies how the entries in each column should be aligned.

The groupalign and alignmentscope attributes are described with the alignment elements, maligngroup and malignmark, in section 3.5.4 [Alignment Markers].

The columnwidth attribute specifies how wide a column should be. The auto value means that the column should be as wide as needed, which is the default. If an explicit value is given, then the column is exactly that wide and the contents of that column are made to fit in that width. The contents are linewrapped or clipped at the discretion of the renderer. If fit is given as a value, the remaining pagewidth after subtracting the widths for columns specified as auto and/or specific widths is divided equally among the fit columns and this value is used for the column width. If insufficient room remains to hold the contents of the fit columns, renderers may linewrap or clip the contents of the fit columns.

The rowspacing and columnspacing attributes specify how much space should be added between each row and column. However, spacing before the first row and after the last row (i.e. at the top and bottom of the table) is given by the second number in the value of the framespacing attribute, and spacing before the first column and after the last column (i.e. on the left and on the right of the table) is given by the first number in the value of the framespacing attribute.

In those attributes' syntaxes, h-unit or v-unit represents a unit of horizontal or vertical length, respectively (see section 2.3.3.3 [CSS-compatible attributes]). The units shown in the attributes' default values (em or ex) are typically used.

The rowlines and columnlines attributes specify whether and what kind of lines should be added between each row and column. Lines before the first row or column and after the last row or column are given using the frame attribute.

If a frame is desired around the table, the frame attribute is used. If the attribute value is not `none', then framespacing is used to add spacing between the lines of the frame and the first and last rows and columns of the table. If frame="none", then the framespacing attribute is ignored. The frame and framespacing attributes are not part of the rowlines/columnlines, rowspacing/columnspacing options because having them be so would often require that rowlines and columnlines would need to be fully specified instead of just giving a single value. For example, if a table had five columns and we wanted lines between the columns, but no frame, then we would have to write columnlines="none solid solid solid solid none". By separating the frame from the internal lines, we only need to write columnlines="solid".

The equalrows attribute forces the rows all to be the same total height when set to true. The equalcolumns attribute forces the columns all to be the same width when set to true.

The displaystyle attribute specifies the value of displaystyle (described under mstyle in section 3.3.4 [Style Change]) within each cell (mtd element, perhaps inferred) of the table. Setting displaystyle=true can be useful for tables whose elements are whole mathematical expressions; the default value of false is appropriate when the table is part of an expression, for example, when it represents a matrix. In either case, scriptlevel (section 3.3.4 [Style Change]) is not changed for the table cells.

3.5.1.3. Example of mtable

A 3 by 3 identity matrix could be represented as follows:

<mrow>
   <mo> ( </mo>
   <mtable>
      <mtr> <mn>1</mn> <mn>0</mn> <mn>0</mn> </mtr>
      <mtr> <mn>0</mn> <mn>1</mn> <mn>0</mn> </mtr>
      <mtr> <mn>0</mn> <mn>0</mn> <mn>1</mn> </mtr>
   </mtable>
   <mo> ) </mo>
</mrow>

This might be rendered as:

Note that the parentheses must be represented explicitly; they are not part of the mtable element's rendering. This allows use of other surrounding fences, such as brackets, or none at all.

3.5.2. Row in Table or Matrix

3.5.2.1. Description

An mtr element represents one row in a table or matrix. An mtr element is only allowed as a direct subexpression of an mtable element, and specifies that its contents should form one row of the table. Each argument of mtr is placed in a different column of the table, starting at the leftmost column.

As described under mtable, mtr elements can be inferred. Inferred mtr elements necessarily have exactly one argument.

If some argument to a (possibly inferred) mtr element is not an mtd element, that argument is treated as a table entry by effectively wrapping it with an inferred mtd .../mtd begin-end tag pair. Note that mtd elements must be given explicitly if they have attributes different from those they would inherit from the enclosing mtr or mtable.

All mtr elements which cover fewer columns than other rows of the same table (whether the other rows precede or follow them) are effectively padded on the right with empty mtd elements, as described under mtable. The required number of arguments in some mtr elements may be affected by the use of the rowspan and columnspan attributes on mtd elements.

3.5.2.2. Attributes of mtr

The rowalign and columnalign attributes allow a specific row to override the alignment specified by the same attributes in the surrounding mtable element.

As with mtable, if there are more entries than necessary in the value of columnalign (i.e. more entries than columns in the row), then the extra entries will be ignored. If there are fewer entries than columns, then the last entry will be repeated as many times as needed.

The groupalign attribute is described with the alignment elements, maligngroup and malignmark, in section 3.5.4 [Alignment Markers].

3.5.3. Entry in Table or Matrix

3.5.3.1. Description

An mtd element represents one entry in a table or matrix. An mtd element is only allowed as a direct subexpression of an mtr element (perhaps an inferred one).

As described under mtr and mtable, mtd elements can be inferred. They must be given explicitly for table elements which have attributes different than those of the enclosing mtr or mtable.

The mtd element accepts any number of arguments; if this number is not 1, its contents are treated as a single `inferred mrow' formed from all its arguments, as described in section 3.1.3 [Required Arguments].

3.5.3.2. Attributes of mtd

The rowspan and columnspan attributes allow a specific matrix element to be treated as if it occupied the number of rows or columns specified. The interpretation of how this larger element affects specifying subsequent rows and columns is meant to correspond with the similar attributes for HTML 3.2 tables.

The rowalign and columnalign attributes allow a specific matrix element to override the alignment specified by a surrounding mtable or mtr element.

The groupalign attribute is described with the alignment elements, maligngroup and malignmark, in section 3.5.4 [Alignment Markers].

3.5.4. Alignment Markers

3.5.4.1. Description

These are space-like elements (see section 3.2.6 [Space]) which can be used to vertically align specified points within a column of MathML expressions, by the automatic insertion of the necessary amount of horizontal space between specified subexpressions.

The discussion that follows will use the example of a set of simultaneous equations which should be rendered with vertical alignment of the coefficients and variables of each term, by inserting spacing somewhat like that shown here:

    8.44x + 55  y =  0
    3.1 x -  0.7y = -1.1

If the example expressions shown above were arranged in a column but not aligned, they would appear as:

    8.44x + 55y = 0
    3.1x - 0.7y = -1.1

(For audio renderers, it is suggested that the alignment elements produce the analogous behavior of altering the rhythm of pronunciation so that it is the same for several subexpressions in a column, by the insertion of the appropriate time delays in place of the extra horizontal spacing described here.)

The expressions whose parts are to be aligned (each equation, in the example above) must be given as the table elements (i.e. as the mtd elements, perhaps inferred) of one column of an mtable. To avoid confusion, the term `table cell' rather than `table element' will be used in the remainder of this section.

All interactions between alignment elements are limited to the mtable column they arise in. That is, every column of a table specified by an mtable element acts as an `alignment scope' which contains within it all alignment effects arising from its contents. It also excludes any interaction between its own alignment elements and the alignment elements inside any nested alignment scopes it might contain.

The reason mtable columns are used as alignment scopes is that they are the only general way in MathML to arrange expressions into vertical columns. Future versions of MathML may provide an malignscope element which allows an alignment scope to be created around any MathML element, but even then, table columns would still sometimes need to act as alignment scopes, and since they are not elements themselves, but rather are made from corresponding parts of the content of several mtr elements, they could not individually be the content of an alignment scope element.

An mtable element can be given the attribute alignmentscope=false to cause its columns not to act as alignment scopes. This is discussed further at the end of this section. Otherwise, the discussion in this section assumes that this attribute has its default value of true.

3.5.4.2. Specifying alignment groups

To cause alignment, it is necessary to specify, within each expression to be aligned, the points to be aligned with corresponding points in other expressions, and the beginning of each alignment group of subexpressions which can be horizontally shifted as a unit to effect the alignment. Each alignment group must contain one alignment point. It is also necessary to specify which expressions in the column have no alignment groups at all, but are affected only by the ordinary column alignment for that column of the table, i.e. by the columnalign attribute, described elsewhere.

The alignment groups start at the locations of invisible maligngroup elements, which are rendered with zero width when they occur outside of an alignment scope, but within an alignment scope are rendered with just enough horizontal space to cause the desired alignment of the alignment group which follows them. A simple algorithm by which a renderer can achieve this is given later. In the example above, each equation would have one maligngroup element before each coefficient, variable, and operator on the left-hand side, one before the = sign, and one before the constant on the right-hand side.

In general, a table cell containing n maligngroup elements contains n alignment groups, with the ith group consisting of the elements entirely after the ith maligngroup element and before the (i+1)-th; no element within the table cell's content should occur entirely before its first maligngroup element.

Note that the division into alignment groups does not necessarily fit the nested expression structure of the MathML expression containing the groups - that is, it is permissible for one alignment group to consist of the end of one mrow, all of another one, and the beginning of a third one, for example. This can be seen in the MathML markup for the present example, given at the end of this section.

The nested expression structure formed by mrows and other layout schemata should reflect the mathematical structure of the expression, not the alignment-group structure, to make possible optimal renderings and better automatic interpretations; see the discussion of proper grouping in section 3.3.1. Insertion of alignment elements (or other space-like elements) should not alter the correspondence between the structure of a MathML expression and the structure of the mathematical expression it represents.

Although alignment groups need to coincide with the nested expression structure of layout schemata, there are nonetheless restrictions on where an maligngroup element is allowed within a table cell. The maligngroup element may only be contained within elements of the following types (which are themselves contained in the table cell):

• an mrow element, including an inferred mrow such as the one formed by a multi-argument mtd element;
• an mstyle element;
• an mphantom element;
• an mfenced element;
• an maction element, though only its selected subexpression is checked;
• a semantics element.

These restrictions are intended to ensure that alignment can be unambiguously specifed, while avoiding complexities involving things like overscripts, radical signs and fraction bars. They also ensure that a simple algorithm suffices to accomplish the desired alignment.

Note that some positions for an maligngroup element, although legal, are not useful, such as for an maligngroup element to be an argument of an mfenced element. When inserting an maligngroup element before a given element in pre-existing MathML, it will often be necessary, and always acceptable, to form a new mrow element to contain just the maligngroup element and the element it is inserted before. In general, this will be necessary except when the maligngroup element is inserted directly into an mrow or into an element which can form an inferred mrow from its contents. See the warning about the legal grouping of `space-like elements' in section 3.2.6 [Space].

For the table cells which are divided into alignment groups, every element in their content must be part of exactly one alignment group, except the elements from the above list which contain maligngroup elements inside them, and the maligngroup elements themselves. This means that, within any table cell containing alignment groups, the first complete element must be an maligngroup element, though this may be preceded by the begin tags of other elements.

This requirement removes a potential confusion about how to align elements before the first maligngroup element, and makes it easy to identify table cells which are left out of their column's alignment process entirely.

Note that it is not required that the table cells in a column which are divided into alignment groups each contain the same number of groups. If they don't, zero-width alignment groups are effectively added on the right side of each table cell which has fewer groups than other table cells in the same column.

3.5.4.3. Table cells which are not divided into alignment groups

Expressions in a column which are to have no alignment groups should contain no maligngroup elements. Expressions with no alignment groups are aligned using only the columnalign attribute which applies to the table column as a whole, and are not affected by the groupalign attribute described below. If such an expression is wider than the column width needed for the table cells containing alignment groups, all the table cells containing alignment groups will be shifted as a unit within the column as described by the columnalign attribute for that column. For example, a column heading with no internal alignment could be added to the column of two equations given above by preceding them with another table row containing an mtext element for the heading, and using the default columnalign="center" for the table, to produce:

equations with aligned variables
      8.44x + 55  y =  0
      3.1 x -  0.7y = -1.1

or, with a shorter heading,

   some equations
8.44x + 55  y =  0
3.1 x -  0.7y = -1.1

3.5.4.4. Specifying alignment points using malignmark

Each alignment group's alignment point can either be specified by an malignmark element anywhere within the alignment group (except within another alignment scope wholly contained inside it), or it is determined automatically from the groupalign attribute. The groupalign attribute can be specified on the group's preceding maligngroup element or on its surrounding mtd, mtr, or mtable elements. In typical cases, using the groupalign attribute is sufficient to describe the desired alignment points, so no malignmark elements need to be provided.

The malignmark element indicates that the alignment point should occur on the right edge of the preceding element, or the left edge of the following element or character, depending on the edge attribute of malignmark. Note that it may be necessary to introduce an mrow to group an malignmark element with a neighboring element, in order not to alter the argument count of the containing element. (See the warning about the legal grouping of `space-like elements' in section 3.2.6 [Space]).

When an malignmark element is provided within an alignment group, it can occur in an arbitrarily deeply nested element within the group, as long as it is not within a nested alignment scope. It is not subject to the same restrictions on location as maligngroup elements. However, its immediate surroundings need to be such that the element to its immediate right or left (depending on its edge attribute) can be unambiguously identified. If no such element is present, renderers should behave as if a zero-width element had been inserted there.

For the purposes of alignment, an element X is considered to be to the immediate left of an element Y, and Y to the immediate right of X, whenever X and Y are successive arguments of one (possibly inferred) mrow element, with X coming before Y. In the case of mfenced elements, renderers should evaluate this relation as if the mfenced element had been replaced by the equivalent expanded form involving mrow. Similarly, an maction element should be treated as if it were replaced by its currently selected subexpression. In all other cases, no relation of `to the immediate left or right' is defined for two elements X and Y. However, in the case of content elements interspersed in presentation markup, renderers should attempt to evaluate this relation in a sensible way. For example, if a renderer maintains an internal presentation structure for rendering content elements, the relation could be evaluated with respect to that. (See chapter 4 [Content Markup] and chapter 5 [Combining Presentation and Content Markup] for further details about mixing presentation and content markup.)

Unlike all other elements in MathML, malignmark elements are allowed to occur within the content of token elements, such as mn, mi, or mtext. When this occurs, the character immediately before or after the malignmark element will carry the alignment point; in all other cases, the element to its immediate left or right will carry the alignment point. The rationale for this is that it is sometimes desirable to align on the edges of specific characters within multi-character token elements.

If there is more than one malignmark element in an alignment group, all but the first one will be ignored. MathML renderers may wish to provide a mode in which they will warn about this situation, but it is not an error, and should trigger no warnings by default. (Rationale: it would be inconvenient to have to remove all unnecessary malignmark elements from automatically generated data, in certain cases, such as when they are used to specify alignment on `decimal points' other than the '.' character.)

3.5.4.5. Attributes of malignmark

malignmark has one attribute, edge, which specifies whether the alignment point will be found on the left or right edge of some element or character. The precise location meant by `left edge' or `right edge' is discussed below. If edge="right", the alignment point is the right edge of the element or character to the immediate left of the malignmark element. If edge="left", the alignment point is the left edge of the element or character to the immediate right of the malignmark element. Note that the attribute refers to the choice of edge rather than to the direction in which to look for the element whose edge will be used.

For malignmark elements which occur within the content of MathML token elements, the preceding or following character in the token element's content is used; if there is no such character, a zero-width character is effectively inserted for the purpose of carrying the alignment point on its edge. For all other malignmark elements, the preceding or following element is used; if there is no such element, a zero-width element is effectively inserted to carry the alignment point.

The precise definition of the `left edge' or `right edge' of a character or glyph (e.g. whether it should coincide with an edge of the character's bounding box) is not specified by MathML, but is at the discretion of the renderer; the renderer is allowed to let the edge position depend on the character's context as well as on the character itself.

For proper alignment of columns of numbers (using groupalign values of left, right, or decimalpoint), it is likely to be desirable for the effective width (i.e. the distance between the left and right edges) of decimal digits to be constant, even if their bounding box widths are not constant (e.g. if `1' is narrower than other digits). For other characters, such as letters and operators, it may be desirable for the aligned edges to coincide with the bounding box.

The `left edge' of a MathML element or alignment group refers to the left edge of the leftmost glyph drawn to render the element or group, except that explicit space represented by mspace or mtext elements should also count as `glyphs' in this context, as should glyphs which would be drawn if not for mphantom elements around them. The `right edge' of an element or alignment group is defined similarly.

3.5.4.6. Attributes of maligngroup

maligngroup has one attribute, groupalign, which is used to determine the position of its group's alignment point when no malignmark element is present. The following discussion assumes that no malignmark element is found within a group.

In the example given at the beginning of this section, there is one column of 2 table cells, with 7 alignment groups in each table cell; thus there are 7 columns of alignment groups, with 2 groups, one above the other, in each column. These columns of alignment groups should be given the 7 groupalign values `decimalpoint left left decimalpoint left left decimalpoint', in that order. How to specify this list of values for a table cell or table column as a whole, using attributes on elements surrounding the maligngroup element is described later.

If groupalign is `left', `right', or `center', the alignment point is defined to be at the group's left edge, at its right edge, or halfway between these edges, respectively. The meanings of `left edge' and `right edge' are as discussed above in relation to malignmark.

If groupalign is `decimalpoint', the alignment point is the right edge of the last character before the decimal point. The decimal point is the first `.' character (ASCII 0x2e) in the first mn element found along the alignment group's baseline. More precisely, the alignment group is scanned recursively, depth-first, for the first mn element, descending into all arguments of each element of the types mrow (including inferred mrows), mstyle, mpadded, mphantom, mfenced, or msqrt, descending into only the first argument of each `scripting' element (msub, msup, msubsup, munder, mover, munderover, mmultiscripts) or of each mroot or semantics element, descending into only the selected subexpression of each maction element, and skipping the content of all other elements. The first mn so found always contains the alignment point, which is the right edge of the last character before the first decimal point in the content of the mn element. If there is no decimal point in the mn element, the alignment point is the right edge of the last character in the content. If the decimal point is the first character of the mn element's content, the right edge of a zero-width character inserted before the decimal point is used. If no mn element is found, the right edge of the entire alignment group is used (as for groupalign="right").

In order to permit alignment on decimal points in cn elements, a MathML renderer can convert a content expression into a presentation expression which renders the same way before searching for decimal points as described above.

If characters other than `.' should be used as `decimal points' for alignment, they should be preceded by malignmark elements within the mn token's content itself.

For any of the groupalign values, if an explicit malignmark element is present anywhere within the group, the position it specifies (described earlier) overrides the automatic determination of alignment point from the groupalign value.

3.5.4.7. Inheritance of groupalign values

It is not usually necessary to put a groupalign attribute on every maligngroup element. Since this attribute is usually the same for every group in a column of alignment groups to be aligned, it can be inherited from an attribute on the mtable which was used to set up the alignment scope as a whole, or from the mtr or mtd elements surrounding the alignment group. It is inherited via an `inheritance path' which proceeds from mtable through successively contained mtr, mtd, and maligngroup elements. There is exactly one element of each of these kinds in this path from an mtable to any alignment group inside it, though the mtr and/or mtd elements might be inferred ones, as described in the sections on those elements. In general, the value of groupalign will be inherited by any given alignment group from the innermost element which surrounds the alignment group and which provides an explicit setting for this attribute.

Note, however, that each mtd element needs, in general, a list of groupalign values, one for each maligngroup element inside it, rather than just a single value. Furthermore, an mtr or mtable element needs, in general, a list of lists of groupalign values, since it spans multiple mtable columns, each potentially acting as an alignment scope. Such lists of group-alignment values are specified using the following syntax rules:

group-alignment            := left | right | center | decimalpoint
group-alignment-list       := group-alignment +
>group-alignment-list-list := ( '{' group-alignment-list '}' ) +

As described in section 2.3.3 [MathML Attribute Values], | separates alternatives; + represents optional repetition (i.e. 1 or more copies of what precedes it), with extra values ignored and the last value repeated if necessary to cover additional table columns or alignment group columns; '{' and '}' represent literal braces; and ( and ) are used for grouping, but do not literally appear in the attribute value.

The permissible values of the groupalign attribute of the elements that have this attribute are specified using the above syntax definitions as follows:

In the example near the beginning of this section, the group alignment values could be specified on every mtd element using groupalign = `decimalpoint left left decimalpoint left left decimalpoint', or on every mtr element using groupalign = `{decimalpoint left left decimalpoint left left decimalpoint}', or (most conveniently) on the mtable as a whole using groupalign = `{decimalpoint left left decimalpoint left left decimalpoint}', which provides a single braced list of group-alignment values for the single column of expressions to be aligned.

3.5.4.8. MathML representation of an alignment example

The above rules are sufficient to explain the MathML representation of the example given near the start of this section. To repeat the example, the desired rendering is:

    8.44x + 55  y =  0
    3.1 x -  0.7y = -1.1

One way to represent that in MathML is:

<mtable groupalign="decimalpoint left left decimalpoint left left decimalpoint">
   <mtd>
      <mrow>
         <mrow>
            <maligngroup/>
            <mn> 8.44 </mn>
            <mo> &InvisibleTimes; </mo>
            <maligngroup/>
            <mi> x </mi>
         </mrow>
         <maligngroup/>
         <mo> + </mo>
         <mrow>
            <maligngroup/>
            <mn> 55 </mn>
            <mo> &InvisibleTimes; </mo>
            <maligngroup/>
            <mi> y </mi>
         </mrow>
      </mrow>
      <maligngroup/>
      <mo> = </mo>
      <maligngroup/>
      <mn> 0 </mn>
   </mtd>
   <mtd>
      <mrow>
         <mrow>
            <maligngroup/>
            <mn> 3.1 </mn>
            <mo> &InvisibleTimes; </mo>
            <maligngroup/>
            <mi> x </mi>
         </mrow>
         <maligngroup/>
         <mo> - </mo>
         <mrow>
            <maligngroup/>
            <mn> 0.7 </mn>
            <mo> &InvisibleTimes; </mo>
            <maligngroup/>
            <mi> y </mi>
         </mrow>
      </mrow>
      <maligngroup/>
      <mo> = </mo>
      <maligngroup/>
      <mrow>
         <mo> - </mo>
         <mn> 1.1 </mn>
      </mrow>
   </mtd>
</mtable>

3.5.4.9. Further details of alignment elements

The alignment elements maligngroup and malignmark can occur outside of alignment scopes, where they are ignored. The rationale behind this is that in situations in which MathML is generated, or copied from another document, without knowing whether it will be placed inside an alignment scope, it would be inconvenient for this to be an error.

An mtable element can be given the attribute alignmentscope=false to cause its columns not to act as alignment scopes. In general, this attribute has the syntax (true | false) +; if its value is a list of boolean values, each boolean value applies to one column, with the last value repeated if necessary to cover additional columns, or with extra values ignored. Columns which are not alignment scopes are part of the alignment scope surrounding the mtable element, if there is one. Use of alignmentscope=false allows nested tables to contain malignmark elements for aligning the inner table in the surrounding alignment scope.

As discussed above, processing of alignment for content elements is not well-defined, since MathML does not specify how content elements should be rendered. However, many renderers are likely to find it convenient to internally convert content elements to presentation elements which render the same way. Thus, as a general rule, even if a renderer does not perform such conversions internally, it is recommended that the alignment elements should be processed as if it did perform them.

A particularly important case for renderers to handle gracefully is the interaction of alignment elements with the matrix content element, since this element may or may not be internally converted to an expression containing an mtable element for rendering. To partially resolve this ambiguity, it is suggested, but not required, that if the matrix element is converted to an expression involving an mtable element, that the mtable element be given the attribute alignmentscope=false, which will make the interaction of the matrix element with the alignment elements no different than that of a generic presentation element (in particular, it will allow it to contain malignmark elements which operate within the alignment scopes created by the columns of an mtable which contains the matrix element in one of its table cells).

The effect of alignment elements within table cells which have non-default values of the columnspan or rowspan attributes is not specified, except that such use of alignment elements is not an error. Future versions of MathML may specify the behavior of alignment elements in such table cells.

The effect of possible linebreaking of an mtable element on the alignment elements is not specified.

3.5.4.10. A simple alignment algorithm

A simple algorithm by which a MathML renderer can perform the alignment specified in this section is given here. Since the alignment specification is deterministic (except for the definition of the left and right edges of a character), any correct MathML alignment algorithm will have the same behavior as this one. Each mtable column (alignment scope) can be treated independently; the algorithm given here applies to one mtable column, and takes into account the alignment elements, the groupalign attribute described in this section, and the columnalign attribute described under mtable (section 3.5.1 [Table or Matrix]).

First, a rendering is computed for the contents of each table cell in the column, using zero width for all maligngroup and malignmark elements. The final rendering will be identical except for horizontal shifts applied to each alignment group and/or table cell. The positions of alignment points specified by any malignmark elements are noted, and the remaining alignment points are determined using groupalign values.

For each alignment group, the horizontal positions of the left edge, alignment point, and right edge are noted, allowing the width of the group on each side of the alignment point (left and right) to be determined. The sum of these two `side-widths', i.e. the sum of the widths to the left and right of the alignment point, will equal the width of the alignment group.

Second, each column of alignment groups, from left to right, is scanned. The ith scan covers the ith alignment group in each table cell containing any alignment groups. Table cells with no alignment groups, or with fewer than i alignment groups, are ignored. Each scan computes two maximums over the alignment groups scanned: the maximum width to the left of the alignment point, and the maximum width to the right of the alignment point, of any alignment group scanned.

The sum of all the maximum widths computed (two for each column of alignment groups) gives one total width, which will be the width of each table cell containing alignment groups. Call the maximum number of alignment groups in one cell n; each such cell's width is divided into 2n adjacent sections, called L(i) and R(i) for i from 1 to n, using the 2n maximum side-widths computed above; for each i, the width of all sections called L(i) is the maximum width of any cell's ith alignment group to the left of its alignment point, and the width of all sections called R(i) is the maximum width of any cell's ith alignment group to the right of its alignment point.

The alignment groups are then positioned in the unique way which places the part of each ith group to the left of its alignment point in a section called L(i), and places the part of each ith group to the right of its alignment point in a section called R(i). This results in the alignment point of each ith group being on the boundary between adjacent sections L(i) and R(i), so that all alignment points of ith groups have the same horizontal position.

The widths of the table cells which contain no alignment groups were computed as part of the initial rendering, and may be different for each cell, and different from the single width used for cells containing alignment groups. The maximum of all the cell widths (for both kinds of cells) gives the width of the table column as a whole.

The position of each cell in the column is determined by the applicable part of the value of the columnalign attribute of the innermost surrounding mtable, mtr, or mtd element which has an explicit value for it, as described in the sections on those elements. This may mean that the cells containing alignment groups will be shifted within their column, in addition to their alignment groups having been shifted within the cells as described above, but since each such cell has the same width, it will be shifted the same amount within the column, thus maintaining the vertical alignment of the alignment points of the corresponding alignment groups in each cell.

3.5.5. Equation Numbering

3.5.5.1. Description

MathML uses tables to allow for different styles of equation numbering. Equation numbering is accomplished by using the columnwidth attribute of mtable, and the meqno element.

The columnwidth attribute value fit allows equations to use up as much of a page as possible, after accounting for any equation number. This allows for styles of equation numbers that are left or right justified on a page.

In cases where only a few equations in a document are numbered, using the columnwidth attribute alone is probably sufficient.

<mtable columnalign='left right' columnwidth='fit auto'>
   <mtr>
      <mrow>
         <mi>E</mi>
         <mo>=</mo>
         <mrow>
            <mi>m</mi>
            <mo>&it;</mo>
            <msup>
               <mi>c</mi>
               <mn>2</mn>
            </msup>
         </mrow>
      </mrow>
      <mrow> <mo> ( </mo> <mn> 2.1 </mn> <mo> ) </mo> </mrow>
   </mtr>
</mtable>

This should be rendered as:

In larger documents with many numbered equations, automatic numbering becomes important. While automatic equation numbering and automatically resolving references to equation numbers is outside the scope of MathML, these problems can be addressed by the use of XSL style sheet or other means. The MathML meqno construction provides support for both of these functions in a way that is intended to facilitate XSL processing.

An equation number is indicated by the meqno element. As noted above, the use of meqno to indicate equation numbers is not required to do equation numbering, though it is strongly recommended. The meqno element simply renders its contents, which can be empty. The meqno element accepts any number of arguments; if this number is not 1, its contents are treated as a single `inferred mrow' formed from all its arguments, as described in section 3.1.3 [Required Arguments].

meqno is used to explicitly mark what is an equation number so that transformation languages can match and change the equation number or its style, such as whether the number is surrounded by parentheses or whether the equation number appears on the left or right side. In the later case, the transformation rule would need to rewrite the row, not just the contents of the meqno element.

3.5.5.2. Attributes of meqno

The eqno attribute allows a numbered equation to have a name that can be referenced elsewhere in the document without refering to its number which may not be known at the time the document is authored. For example, an XML application containing embedded MathML might define a cite element, with a eqno attribute. by looking for matching eqno attributes, a tranformation engine such as XSL could then replace all appropriate cite elements and the content of the corresponding meqno element with an automatically generated equation number.

3.5.5.3. Example

Using the meqno construction, the example given above might be coded as follows:

<mtable columnalign='left right' columnwidth='fit auto'>
   <mtr>
      <mrow>
         <mi>E</mi>
         <mo>=</mo>
         <mrow>
            <mi>m</mi>
            <mo>&it;</mo>
            <msup>
               <mi>c</mi>
               <mn>2</mn>
            </msup>
         </mrow>
      </mrow>
      <meqno eqno='famous formula'>
         <mrow> <mo> ( </mo> <mn> 2.1 </mn> <mo> ) </mo> </mrow>
      </meqno>
   </mtr>
</mtable>

3.6. Enlivening Expressions

3.6.1. Bind Action to Subexpression

Authors can make links from MathML subexpressions by using the standard simple link construction of XML, <name ... href="...">, as described in section 7.1.5 [Mixing and Linking MathML and HTML]. However, links are only one of many ways which specific renderers might provide authors for making mathematical notation active. For example, in lengthy mathematical expressions, the ability to `fold' expressions might be provided, i.e. a renderer might allow a reader to toggle between an ellipsis and a much longer expression which it represents.

To provide a mechanism for binding actions to expressions, MathML provides the maction element. This element accepts any number of subexpressions as arguments, and the following attributes:

By default, renderers which do not recognize the specified actiontype should render the selected subexpression as defined below. If no selected subexpression exists, it is a MathML error; the appropriate rendering in that case is as described in section 7.2.2 [Handling of Errors] on the treatment of MathML errors.

Since a MathML-compliant renderer is not required to recognize any particular actiontypes, a renderer can be fully MathML compliant just by implementing the above-described default behavior.

The selection attribute is provided for those actiontypes which permit someone viewing a document to select one of several subexpressions for viewing. Its value should be a positive integer which indicates one of the subexpressions of the maction element, numbered from 1 to the number of children of the element. When this is the case, the subexpression so indicated is defined to be the `selected subexpression' of the maction element; otherwise the `selected subexpression' does not exist, which is an error. When the selection attribute is not specified (including for actiontypes for which it makes no sense), its default value is 1, so the selected subexpression will be the first subexpression.

Note that the selection attribute of an maction element affects not only the rendering of the element itself, but potentially the rendering of elements containing it; this can be due either to the reference to an maction's selected subexpression in the definition of embellished operator in section 3.2.4 [Operator, Fence, Separator or Accent], or to alignment elements contained in the selected subexpression (section 3.5.4 [Alignment Markers]). This means that a change of the selection by the viewer may require a rerendering of the surroundings, not just their repositioning due to a change in size of the maction element itself.

Furthermore, as described in chapter 7 [The MathML Interface], if a MathML renderer responds to a user command to copy the currently selected MathML subexpression to the environment's `clipboard', any maction elements present in what is copied should be given selection attributes which correspond to their selection state in the MathML rendering at the time of the copy command.

A suggested list of actiontypes and their associated actions is given below. Keep in mind, however, that this list is mainly for illustration, and recognized values and behaviors will vary from renderer to renderer. (Note that hyperlinks, which are part of the actual MathML specification, are not included here, but are described separately in chapter 7 [The MathML Interface].)

<maction actiontype="toggle" selection="positive-integer" > (first expression) (second expression)... </maction>

For this action type, a renderer would alternately display the given expressions, cycling through them when a reader clicked on the active expression, starting with the selected expression and updating the selection attribute value as described above. Typical uses would be for exercises in education, ellipses in long computer algebra output, or to illustrate alternate notations. Note that the expressions may be of significantly different size, so that size negotiation with the browser may be desirable. If size negotiation is not available, scrolling, elision, panning, or some other method may be necessary to allow full viewing.

<maction actiontype="statusline"> (expression) (message) </maction>

In this case, the renderer would display the expression in context on the screen. When a reader clicked on the expression or moved the mouse over it, the renderer would send a rendering of the message to the browser statusline. Since most browsers in the forseeable future are likely to be limited to displaying text on their statusline, authors would presumably use plain text in an mtext element for the message in most circumstances. For non-mtext messages, renderers might provide a natural language translation of the markup, but this is not required.

<maction actiontype="tooltip"> (expression) (message) </maction>

Here the renderer would also display the expression in context on the screen. When the mouse pauses over the expression for a long enough delay time, the renderer displays a rendering of the message in a pop-up `tooltip' box near the expression. These message boxes are also sometimes called `balloon help' boxes. Presumably authors would use plain text in an mtext element for the message in most circumstances. For non-mtext messages, renderers may provide a natural language translation of the markup if full MathML rendering is not practical, but this is not required.

<maction actiontype="highlight" other="color='#ff0000'"> expression </maction> <maction actiontype="highlight" other="background='#ff0000'"> expression </maction>

In this case, a renderer might highlight the enclosed expression on a `mouse-over' event. In the example given above, use is being made of the `other' attribute to pass non-standard attributes to renderers which support them, without violating the MathML DTD (see 7.2.3). The `color' attribute changes the color of the characters in the presentation, while the `background' attribute changes the color of the background behind the characters.

<maction actiontype="menu" selection="positive-integer" > (menu item 1) (menu item 2) ... </maction>

This action type instructs a renderer to provide a pop up menu. This allows a one-to-many linking capability. Note that the menu items may be other <maction actiontype="menu">...</maction> expressions, thereby allowing nested menus.