3 Presentation Markup

Overview: Mathematical Markup Language (MathML) Version 2.0
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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 Bidirectional Layout
   3.1.6 Summary of Presentation Elements
3.2 Token Elements
   3.2.1 MathML characters in token elements
   3.2.2 Mathematics style attributes common to token elements
   3.2.3 Identifier (mi)
   3.2.4 Number (mn)
   3.2.5 Operator, Fence, Separator or Accent (mo)
   3.2.6 Text (mtext)
   3.2.7 Space (mspace)
   3.2.8 String Literal (ms)
   3.2.9 Adding new character glyphs to MathML (mglyph)
3.3 General Layout Schemata
   3.3.1 Horizontally Group Sub-Expressions (mrow)
   3.3.2 Fractions (mfrac)
   3.3.3 Radicals (msqrt, mroot)
   3.3.4 Style Change (mstyle)
   3.3.5 Error Message (merror)
   3.3.6 Adjust Space Around Content (mpadded)
   3.3.7 Making Sub-Expressions Invisible (mphantom)
   3.3.8 Expression Inside Pair of Fences (mfenced)
   3.3.9 Enclose Expression Inside Notation (menclose)
3.4 Script and Limit Schemata
   3.4.1 Subscript (msub)
   3.4.2 Superscript (msup)
   3.4.3 Subscript-superscript Pair (msubsup)
   3.4.4 Underscript (munder)
   3.4.5 Overscript (mover)
   3.4.6 Underscript-overscript Pair (munderover)
   3.4.7 Prescripts and Tensor Indices (mmultiscripts)
3.5 Tables and Matrices
   3.5.1 Table or Matrix (mtable)
   3.5.2 Row in Table or Matrix (mtr)
   3.5.3 Labeled Row in Table or Matrix (mlabeledtr)
   3.5.4 Entry in Table or Matrix (mtd)
   3.5.5 Alignment Markers
3.6 Enlivening Expressions
   3.6.1 Bind Action to Sub-Expression (maction)

3.1 Introduction

This chapter specifies the `presentation' elements of MathML, which can be used to describe the layout structure of mathematical notation.

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. Because of the importance of traditional visual notation, the descriptions of the notational constructs the elements represent are usually given here in visual terms. However, the elements are medium-independent in the sense that they have been designed to contain enough information for good spoken renderings as well. 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).

MathML presentation elements only suggest (i.e. do not require) specific ways of rendering in order to allow for medium-dependent rendering and for individual preferences of style. 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 sub-expressions 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 MathML characters are used to name operators or identifiers that in traditional notation render the same as other symbols, such as ⅆ, ⅇ, or ⅈ, or operators that 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 [Characters, Entities and Fonts].

3.1.2 Terminology Used In This Chapter

It is strongly recommended that, before reading the present chapter, one read Section 2.4 [MathML Syntax and Grammar] on MathML syntax and grammar, which contains important information on MathML notations and conventions. In particular, in this chapter it is assumed that the reader has an understanding of basic XML terminology described in Section 2.4.2 [An XML Syntax Primer], and the attribute value notations and conventions described in Section 2.4.4 [MathML Attribute Values].

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, etc. In general, tokens can have only characters as content. The only exceptions are the vertical alignment element malignmark, mglyph, and entity references. 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 that 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, etc.), 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 sub-expressions are built into larger expressions. The terminology derives from the fact that each layout schema corresponds to a different way of `laying out' its sub-expressions 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 this chapter. These elements are listed in Section 3.1.6 [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 (described in Section 5.2.4 [Content Markup Contained in Presentation Markup]). A sub-expression of an expression E is any MathML expression that is part of the content of E, whether directly or indirectly, i.e. whether it is a `child' of E or not.

Since layout schemata attach special meaning to the number and/or positions of their children, a child of a layout schema is also called an argument of that element. As a consequence of the above definitions, the content of a layout schema consists exactly of a sequence of zero or more elements that are its arguments.

3.1.3 Required Arguments

Many of the elements described herein require a specific number of arguments (always 1, 2, or 3). 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. 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 in which they appear. See Section 3.2.7 [Space (mspace)] 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 1* argument (msqrt, mstyle, merror, menclose, mpadded, mphantom, mtd, and math) actually accept any number of arguments. However, 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. Although the math element is not a presentation element, it is listed below for completeness.

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 that 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 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. An argument count of 1* indicates an inferred mrow as described above.

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 3.2.7 [Space (mspace)]. This definition affects some of the suggested rendering rules for mo elements (Section 3.2.5 [Operator, Fence, Separator or Accent (mo)]).

Certain elements, e.g. msup, are able to embellish operators that are their first argument. These elements are listed in Section 3.2.5 [Operator, Fence, Separator or Accent (mo)], 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].

In MathML 1.x, the mtable element could infer mtr elements around its arguments, and the mtr element could infer mtd elements. In MathML 2.0, mtr and mtd elements must be explicit. However, for backward compatibility renderers may wish to continue supporting inferred mtr and mtd elements.

3.1.5 Bidirectional Layout

The term 'bidirectional layout' refers to the fact that letters from certain scripts, in particular Arabic and Hebrew, are written from right to left, and that mixing these with numbers or letters from scripts written left- to-right results in text runs of two differing directions within the same line or paragraph.

For ordinary text, Unicode defines a bidirectional algorithm [Bidi]. This algorithm assumes that the order of characters in a 'backing store' is in logical order (i.e. in the order it would be pronounced or typed in), and defines how the characters get reordered for display based on character properties and other directives. HTML, CSS, XSL, and SVG adopt this algorithm and provide ways to control it via markup or styling.

In mathematical expressions, bidirectional layout is more difficult than it is in text. In part, this is due to the 2-dimensional nature of mathematical layout, and the fact that spatial relationships are often used to convey meaning in mathematics notation. Another factor is the lack of established conventions for bidirectional mathematics layout, since this is relatively uncommon, even in right-to-left contexts.

For these reasons, MathML 2.0 only adopts a restricted version of the Unicode Bidirectional algorithm, as described in the remainder of this section.

3.1.5.1 Bidirectional Layout in Token Elements

For MathML token elements that can contain text (mtext, mo, mi, mn and ms), the implicit part of the Unicode bidirectional algorithm [Bidi] is applied when its content is rendered visually (i.e. characters are reordered based on character properties). The base directionality is left-to-right.

The implicit part of the Unicode bidirectional algorithm is identical to straightforward left-to-right layout if there is only one character, or if there are no strong right-to-left characters (i.e. no characters from the Arabic, Hebrew, or similar scripts).

Applications are not required to apply the Unicode bidirectional algorithm if they do not render strong right-to-left characters.

Please note that for the transfinite cardinals represented by Hebrew characters, the codepoints U+2135-U+2138 (ALEF SYMBOL, BET SYMBOL, GIMEL SYMBOL, DALET SYMBOL) should be used. These are strong left-to-right.

3.1.5.2 Bidirectional Layout of Mathematics Formulas

MathML 2.0 does not address right-to-left or bidirectional layout in mathematics formulas. Only left-to-right layout is supported. Right-to-left layout of mathematical formulas may be addressed in a future version of MathML.

3.1.6 Summary of Presentation Elements

3.1.6.1 Token Elements

3.1.6.2 General Layout Schemata

3.1.6.3 Script and Limit Schemata

3.1.6.4 Tables and Matrices

3.1.6.5 Enlivening Expressions

3.2 Token Elements

Token elements in presentation markup are broadly intended to represent the smallest units of mathematical notation which carry meaning. Tokens are roughly analogous to words in text. However, because of the precise, symbolic nature of mathematical notation, the various categories and properties of token elements figure prominently in MathML markup. By contrast, in textual data, individual words rarely need to be marked up or styled specially.

Frequently tokens consist of a single character denoting a mathematical symbol. Other cases, e.g. function names, involve multi-character tokens. Further, because traditional mathematical notation makes wide use of symbols distinguished by their typographical properties (e.g. a Fraktur 'g' for a Lie algebra, or a bold 'x' for a vector), care must be taken to insure that styling mechanisms respect typographical properties which carry meaning. Consequently, characters, tokens, and typographical properties of symbols are closely related to one another in MathML.

3.2.1 MathML characters in token elements

Character data in MathML markup is only allowed to occur as part of the content of token elements. The only exception is whitespace between elements, which is ignored. Token elements can contain any sequence of zero or more Unicode characters. 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 exceptions to this are the empty elements mspace and mglyph. The mspace element's width depends upon its attribute values. The mglyph element renders using the character described by its attributes.

While all Unicode character data is valid in token element content, MathML 2.0 distinguishes a special subset of named Unicode 3.2 characters, called MathML characters in this document. The complete list of MathML characters is defined in Chapter 6 [Characters, Entities and Fonts]. MathML characters can be either represented directly as Unicode character data, or indirectly via numeric or character entity references. See Chapter 6 [Characters, Entities and Fonts] for a discussion of the advantages and disadvantages of numeric character references versus entity references. New mathematics characters that arise, or non-standard glyphs for existing MathML characters, may be represented by means of the mglyph element.

Apart from the mglyph element, the malignmark element is the only other element allowed in the content of tokens. See Section 3.5.5 [Alignment Markers] for details.

Token elements (other than mspace and mglyph) 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). Rendering algorithms should also take into account the mathematics style attributes as described below, and modify surrounding spacing by rules or attributes specific to each type of token element.

3.2.1.1 Alphanumeric symbol characters

A large class of mathematical symbols are single letter identifiers typically used as variable names in formulas. Different font variants of a letter are treated as separate symbols. For example, a Fraktur 'g' might denote a Lie algebra, while a Roman 'g' denotes the corresponding Lie group. These letter-like symbols are traditionally typeset differently than the same characters appearing in text, using different spacing and ligature conventions. These characters must also be treated specially by style mechanisms, since arbitrary style transformations can change meaning in an expression.

For these reasons, Unicode 3.1 will be adding more than nine hundred Math Alphanumeric Symbol characters corresponding to letter-like symbols. These characters are in the Secondary Multilingual Plane (SMP). See Chapter 6 [Characters, Entities and Fonts] for more information. As valid Unicode data, these characters are permitted in MathML 2.0, and as tools and fonts for them become widely available, we anticipate they will be the predominant way of denoting letter-like symbols.

Until support for SMP characters is widely available, however, it is still necessary to provide an alternative encoding using only Basic Multilingual Plane (BMP) characters together with markup. MathML 2.0 defines a correspondence between token elements with certain combinations of BMP character data and the mathvariant attribute and tokens containing SMP Math Alphanumeric Symbol characters. Processing applications that accept SMP characters are required to treat the corresponding BMP and attribute combinations identically. The next section discusses the mathvariant attribute in more detail, and a complete technical description of the corresponding characters is given in Section 6.2.3 [Mathematical Alphanumeric Symbols Characters.].

3.2.2 Mathematics style attributes common to token elements

MathML 2.0 introduces four new mathematics style attributes. These attributes are valid on all presentation token elements except mspace and mglyph, and on no other elements except mstyle. The attributes are:

(See Section 2.4.4 [MathML Attribute Values] for terminology and notation used in attribute value descriptions.)

The mathematics style attributes define logical classes of token elements. Each class is intended to correspond to a collection of typographically-related symbolic tokens that have a meaning within a given math expression, and therefore need to be visually distinguished and protected from inadvertent document-wide style changes which might change their meanings.

When MathML rendering takes place in an environment where CSS is available, the mathematics style attributes can be viewed as predefined selectors for CSS style rules. See Section 7.1.5 [Using CSS with MathML] and Appendix G [Sample CSS Style Sheet for MathML] for further discussion and a sample CSS style sheet. When CSS is not available, it is up to the internal style mechanism of the rendering application to visually distinguish the different logical classes.

At a theoretical level, renderers have complete freedom in mapping mathematics style attributes to specific rendering properties. However, in practice, the mathematics style attribute names and values suggest obvious typographical properties, and renderers should attempt to respect these natural interpretations as far as possible. For example, it is reasonable to render a token with the mathvariant attribute set to sans-serif in Helvetica or Arial. However, rendering the token in a Times Roman font could be seriously misleading and should be avoided.

A issue arises in that the natural interpretations of the mathvariant attribute values only make sense for certain characters. For example, there is no clear cut rendering for a 'fraktur' alpha, or a 'bold italic' Kanji character. In general, the only cases that have a clear interpretation are exactly the ones that correspond to SMP Math Alphanumeric Symbol characters.

Consequently, style sheet authors and application developers are encouraged in the strongest possible terms to respect the obvious typographical interpretation of the mathvariant attribute when applied to characters that have SMP Math Alphanumeric Symbol counterparts. In all other cases, it is up to the renderer to determine what effect, if any, the mathvariant attribute will have. For example, a renderer might sensibly choose to display a token with the contents ∑ (a character with no SMP counterpart) in bold face font if it has the mathvariant attribute set to bold or to bold-fraktur, and to display it in a default Roman font if the mathvariant attribute is set to fraktur. As this example indicates, authors should refrain from using the mathvariant attribute with characters that do not have SMP counterparts, since renderings may not be useful or predictable.

Finally, there is a redundancy problem with the mathvariant attribute that must be dealt with as a special case. When the mathvariant attribute is used on an mi element containing a single character from the specific ranges of BMP character data detailed in Section 6.2.3 [Mathematical Alphanumeric Symbols Characters.], the resulting rendering will be visually indistinguishable from an mi element with no attributes containing the corresponding SMP character. Therefore MathML 2.0 mandates that processing applications treat these two representations as equivalent. This is primarily an issue for applications that support searching and/or equality testing.

Token elements also permit id, xref, class and style attributes for compatibility with style sheet mechanisms, as described in Section 2.4.5 [Attributes Shared by all MathML Elements]. However, some care must be taken when using CSS generally. Using CSS to produce visual effects that alter the meaning of an equation should be especially avoided, since MathML is used in many non-CSS environments. Similarly, care should be taken to insure arbitrary document-wide style transformations do not affect mathematics expressions in such a way that meaning is altered.

Since MathML expressions are often embedded in a textual data format such as XHTML, the surrounding text and the MathML must share rendering attributes such as font size, so that the renderings will be compatible in 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. (In cases where the surrounding text and the MathML are being rendered by separate software, e.g. a browser and a plug-in, it is also important for the rendering environment to provide the MathML renderer with additional information, such as the baseline position of surrounding text, which is not specified by any MathML attributes.) Note, however, that MathML 2.0 doesn't specify the mechanism by which style information is inherited from the rendering environment. For example, one browser plug-in might choose to rely completely on the CSS inheritance mechanism and use the fully resolved CSS properties for rendering, while another application might only consult a style environment at the root node, and then use its own internal style inheritance rules.

Most MathML renderers will probably want to rely on some degree to additional, internal style processing algorithms. In particular, inheritance of the mathvariant attribute does not follow the CSS model. The default value for this attribute is normal (non-slanted) for all tokens except mi. For mi tokens, the default depends on the number of characters in tokens' content. (The deprecated fontslant attribute also behaves this way.) See Section 3.2.3 [Identifier (mi)] for details.

3.2.2.1 Deprecated style attributes on token elements

The MathML 1.01 style attributes listed below have been deprecated in MathML 2.0. In rendering environments that support CSS, it is preferable to use CSS to control the rendering properties corresponding to these attributes. However as explained above, direct manipulation of these rendering properties by whatever means should usually be avoided.

If both a new mathematics style attribute and conflicting deprecated attributes are given, the new math style attribute value should be used. For example

<mi fontweight='bold' mathvariant='normal'> a </mi>

should render in a normal weight font, and

<mi fontweight='bold' mathvariant='sans-serif'> a </mi>

should render in a normal weight sans serif font. In the example

<mi fontweight='bold' mathvariant='fraktur'> a1 </mi>

the mathvariant attribute still overrides fontweight attribute, even though fraktur generally shouldn't be applied to a '1' since there is no corresponding SMP Math Alphanumeric Symbol character. In the absence of fonts containing Fraktur digits, this would probably render as a Fraktur 'a' followed by a Roman '1' in most renderers.

The new mathematics style attributes also override deprecated 1.01 style attribute values that are inherited. Thus

<mstyle fontstyle='italic'>
  <mi mathvariant='bold'> a </mi>
</mstyle>

renders in a bold upright font, not a bold italic font.

At the same time, the MathML 1.01 attributes still serve a purpose. Since they correspond directly to rendering properties needed for mathematics layout, they are very useful for describing MathML layout rules and algorithms. For this reason, and for backward compatibility, the MathML rendering rules suggested in this chapter continue to be described in terms of the rendering properties described by these MathML 1.01 style attributes.

The deprecated attributes are:

The fontsize attribute specifies the desired font size. v-unit represents a unit of vertical length (see Section 2.4.4.3 [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 (mstyle)].

The value of the fontfamily attribute should be the name of a font that may be available to a MathML renderer, or information that 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 that 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 a non-ASCII MathML character (for example, by using a reference to a character in some symbol font that maps ordinary characters to glyphs for non-ASCII characters). As a corollary to this principle, MathML renderers should attempt to always produce intelligible renderings for the MathML characters listed in Chapter 6 [Characters, Entities 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 [Characters, Entities and Fonts].

The symbol css-fontfamily refers to a legal value for the font-family property in CSS, 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 CSS[CSS2]. 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 2.4.4.3 [CSS-compatible attributes]).

3.2.2.2 Color-related attributes

The mathcolor (and deprecated 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.

The values of mathcolor, color, mathbackground, 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 below.

The color syntax described above is a subset of the syntax of the color and background-color properties of CSS. The background-color syntax is in turn a subset of the full CSS 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 [HTML4] (aqua, black, blue, fuchsia, gray, green, lime, maroon, navy, olive, purple, red, silver, teal, white, and yellow). Note that the color name keywords are not case-sensitive, unlike most keywords in MathML attribute values for compatibility with CSS and HTML.

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.

3.2.3 Identifier (mi)

3.2.3.1 Description

An mi element represents a symbolic name or arbitrary text that 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 3.2.6.4 [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 that should render like identifiers and notations that 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].

3.2.3.2 Attributes

mi elements accept the attributes listed in Section 3.2.2 [Mathematics style 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 mathvariant and 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 mathvariant and fontstyle attributes is specific to mi elements; the default value for the mathvariant and fontstyle attributes on other MathML token elements is normal.

Note that for purposes of determining equivalences of Math Alphanumeric Symbol characters (See Section 6.2.3 [Mathematical Alphanumeric Symbols Characters.] and Section 3.2.1.1 [Alphanumeric symbol characters]) the value of the mathvariant attribute should be resolved first, including the special defaulting behavior described above.

3.2.3.3 Examples

<mi> x </mi>
<mi> D </mi>
<mi> sin </mi>
<mi mathvariant='script'> L </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 3.2.5 [Operator, Fence, Separator or Accent (mo)].

<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 [Characters, Entities and Fonts] for a complete list of character entity references in MathML.

3.2.4 Number (mn)

3.2.4.1 Description

An mn element represents a `numeric literal' or other data that 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 mathematical concept of a `number' can be quite subtle and involved, depending on the context. As a consequence, not all mathematical numbers should be represented using mn; examples of mathematical numbers that should be represented differently are shown below, and include 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 that 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].

3.2.4.2 Attributes

mn elements accept the attributes listed in Section 3.2.2 [Mathematics style 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.

3.2.4.3 Examples

<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>

3.2.4.4 Numbers that should not be written using mn alone

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

<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>

3.2.5 Operator, Fence, Separator or Accent (mo)

3.2.5.1 Description

An mo element represents an operator or anything that 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 that should `render as an operator' includes a number of notations that 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 that should render as an operator, and that 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 3.2.6 [Text (mtext)]. See also Chapter 4 [Content Markup] for definitions of MathML content elements that are guaranteed to have the semantics of specific mathematical operators.

3.2.5.2 Attributes

mo elements accept the attributes listed in Section 3.2.2 [Mathematics style attributes common to token elements], and the additional attributes listed here. Most attributes get their default values from the Section 3.2.5.7.1 [The operator dictionary], 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, and v-unit represents a unit of vertical length (see Section 2.4.4.2 [Attributes with units]). namedspace is one of veryverythinmathspace, verythinmathspace, thinmathspace, mediummathspace, thickmathspace, verythickmathspace, or veryverythickmathspace. These values can be set by using the mstyle element as is further discussed in Section 3.3.4 [Style Change (mstyle)].

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> &lt;= </mo> should render differently. (The first one should render as a single character representing a less-than-or-equal-to sign, and the second one as the two-character sequence <=.)

3.2.5.3 Examples with 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>
<mo mathvariant='bold'> + </mo>

3.2.5.4 Examples with 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 [Expression Inside Pair of Fences (mfenced)].

(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>

3.2.5.5 Invisible operators

Certain operators that 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) that 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.

3.2.5.6 Names for other special operators

MathML also includes ⅆ for use in an mo element representing the differential operator symbol usually denoted by `d'. The reasons for explicitly using this special entity are similar to those for using the special entities for invisible operators described in the preceding section.

3.2.5.7 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.

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 F [Operator Dictionary]) 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.

Default value of the 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 3.2.7 [Space (mspace)]) 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 that 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.

Exception for embellished operators

There is one exception to the above rules for choosing an mo element's default form attribute. An mo element that 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 4.2.6 [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 sub-expression 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.

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 Sub-Expressions (mrow)]). This means that an mo element that 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 in which it is the sole argument.

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.

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. For example, more space might translate into a longer pause in an audio rendering.

3.2.5.8 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.

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.

Vertical Stretching Rules

• If a stretchy operator is a direct sub-expression of an mrow element, or is the sole direct sub-expression 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 sub-expressions 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 element is inferred.

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 that are entirely within the set of table rows that the operator's cell covers. Table cells that extend into rows not covered by the stretchy operator's table cell should be ignored. See Section 3.5.4.2 [Attributes] for details about the rowspan attribute.

Horizontal Stretching Rules

• If a stretchy operator, or an embellished stretchy operator, is a direct sub-expression of an munder, mover, or munderover element, or if it is the sole direct sub-expression of an mtd element 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 sub-expressions in the given element (or in the same table column), given the constraints mentioned above.
• If a stretchy operator is a direct sub-expression of an munder, mover, or munderover element, or if it is the sole direct sub-expression of an mtd element in some column of a table, then it should stretch to cover the width of the other direct sub-expressions 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.

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 that are entirely within the set of table columns that the operator's cell covers. Table cells that extend into columns not covered by the stretchy operator's table cell should be ignored. See Section 3.5.4.2 [Attributes] for details about the rowspan attribute.

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 .

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 (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 that override this behavior. Of course, limitations in fonts or font rendering may result in the final, stretched sizes being only approximately the same.

3.2.5.9 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 (mstyle)] describing the mstyle element. Note that these attributes can be specified directly on an mstyle element's start 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 (mstyle)]. 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 (Section 3.4.4 [Underscript (munder)], Section 3.4.5 [Overscript (mover)] and Section 3.4.6 [Underscript-overscript Pair (munderover)]).

The separator attribute may affect automatic linebreaking in renderers that 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.

3.2.6 Text (mtext)

3.2.6.1 Description

An mtext element is used to represent arbitrary text that should be rendered as itself. In general, the mtext element is intended to denote commentary text.

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 that 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 2.4.6 [Collapsing Whitespace in Input]. Whitespace that is intended to be rendered as part of an element's content must be represented by entity references or mspace elements (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 [Characters, Entities 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 [Characters, Entities and Fonts].

3.2.6.2 Attributes

mtext elements accept the attributes listed in Section 3.2.2 [Mathematics style attributes common to token elements].

See also the warnings about the legal grouping of `space-like elements' in Section 3.2.7 [Space (mspace)], and about the use of such elements for `tweaking' or conveying meaning in Section 3.3.6 [Adjust Space Around Content (mpadded)].

3.2.6.3 Examples

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

3.2.6.4 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 3.2.3 [Identifier (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, or another document markup language, 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.

3.2.7 Space (mspace)

3.2.7.1 Description

An mspace empty element represents a blank space of any desired size, as set by its attributes. It can also be used to make linebreaking suggestions to a visual renderer. Note that the default values for attributes have been chosen so that they typically will have no effect on rendering. Thus, the mspace element is generally used with one or more attribute values explicitly specified.

3.2.7.2 Attributes

In addition to the attributes listed below, mspace permits id, xref, class and style attributes, as described in Section 2.4.5 [Attributes Shared by all MathML Elements].

h-unit and v-unit represent units of horizontal or vertical length, respectively (see Section 2.4.4.2 [Attributes with units]).

The linebreak attribute is used to give a linebreaking hint to a visual renderer. The default value is auto, which indicates that a renderer should use whatever default linebreaking algorithm it would normally use. The meaning of the other possible values for the linebreak attribute are described above in the discussion on renderable whitespace in the mtext element. See Section 3.2.6 [Text (mtext)] for details.

In the case when both dimensional attributes and a linebreaking attribute are set, the linebreaking attribute is ignored.

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 (mpadded)]. See also the other elements that can render as whitespace, namely mtext, mphantom, and maligngroup.

3.2.7.3 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 3.2.5 [Operator, Fence, Separator or Accent (mo)]. 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 sub-expressions are space-like;
• an maction element whose selected sub-expression exists and is space-like;
• an mrow all of whose direct sub-expressions 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.

3.2.7.4 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 that require a specific number of arguments, or that 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 (mpadded)].

3.2.8 String Literal (ms)

3.2.8.1 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 3.2.6 [Text (mtext)], 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>&amp</ms> represents a string literal containing a single character, &, and <ms>&amp;amp;</ms> represents a string literal containing 5 characters, the first one of which is &.

Like all token elements, ms does trim and collapse whitespace in its content according to the rules of Section 2.4.6 [Collapsing Whitespace in Input], so whitespace intended to remain in the content should be encoded as described in that section.

3.2.8.2 Attributes

ms elements accept the attributes listed in Section 3.2.2 [Mathematics style 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 space characters. 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 \"".

3.2.9 Adding new character glyphs to MathML (mglyph)

3.2.9.1 Description

Unicode defines a large number of characters used in mathematics, and in most cases, glyphs representing these characters are widely available in a variety of fonts. Although these characters should meet almost all users needs, MathML recognizes that mathematics is not static and that new characters are added when convenient. Characters that become well accepted will likely be eventually incorporated by the Unicode Consortium or other standards bodies, but that is often a lengthy process. In the meantime, a mechanism is necessary for accessing glyphs from non-standard fonts representing these characters.

The mglyph element is the means by which users can directly access glyphs for characters that are not defined by Unicode, or not known to the renderer. Similarly, the mglyph element can also be used to select glyph variants for existing Unicode characters, as might be desirable when a glyph variant has begun to differentiate itself as a new character by taking on a distinguished mathematical meaning.

The mglyph element names a specific character glyph, and is valid inside any MathML leaf content listed in Section 3.1.6 [Summary of Presentation Elements] (mi, etc.) or Section 4.2.2 [Containers] (ci, etc.) unless otherwise restricted by an attribute (e.g. base=2 to <cn>). In order for a visually-oriented renderer to render the character, the renderer must be told what font to use and what index within that font to use.

3.2.9.2 Attributes

mglyph elements accept the attributes listed in Section 3.2.2 [Mathematics style attributes common to token elements], and the additional attributes listed here.

The alt attribute provides an alternate name for the glyph. If the specified font can't be found, the renderer may use this name in a warning message or some unknown glyph notation. The name might also be used by an audio renderer or symbol processing system and should be chosen to be descriptive. The fontfamily and index uniquely identify the mglyph; two mglyphs with the same values for fontfamily and index should be considered identical by applications that must determine whether two characters/glyphs are identical. The alt attribute should not be part of the identity test.

The fontfamily and index attributes name a font and position within that font. All font properties apart from fontfamily are inherited. Variants of the font (e.g., bold) that may be inherited may be ignored if the variant of the font is not present.

Authors should be aware that rendering requires the fonts referenced by mglyph, which the MathML renderer may not have access to or may be not be supported by the system on which the renderer runs. For these reasons, authors are encouraged to use mglyph only when absolutely necessary, and not for stylistic purposes.

3.2.9.3 Example

The following example illustrates how a researcher might use the mglyph construct with an experimental font to work with braid group notation.

<mrow>
  <mi><mglyph fontfamily="my-braid-font" index="2" alt="23braid"/></mi>
  <mo>+</mo>
  <mi><mglyph fontfamily="my-braid-font" index="5" alt="132braid"/></mi>
  <mo>=</mo>
  <mi><mglyph fontfamily="my-braid-font" index="3" alt="13braid"/></mi>
</mrow>

This might render as:

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 Sub-Expressions (mrow)

3.3.1.1 Description

An mrow element is used to group together any number of sub-expressions, usually consisting of one or more mo elements acting as `operators' on one or more other expressions that 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 [Expression Inside Pair of Fences (mfenced)]), which can effectively form an mrow containing its arguments separated by commas.

3.3.1.2 Attributes

This element only permits id, xref, class and style attributes, as described in Section 2.4.5 [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 3.2.5 [Operator, Fence, Separator or Accent (mo)] 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 3.2.5 [Operator, Fence, Separator or Accent (mo)]), or for providing hints related to linebreaking when necessary (see Section 3.2.6 [Text (mtext)]), and the ability to use nested mrows to describe sub-expression structure (see below).

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 start tag. If there are attributes on the mrow element's start 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 Section 3.2.5 [Operator, Fence, Separator or Accent (mo)] and Section 3.2.7 [Space (mspace)]. See also the discussion of equivalence of MathML expressions in Chapter 7 [The MathML Interface].

3.3.1.3 Proper grouping of sub-expressions using mrow

Sub-expressions 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.

Precise rule for proper grouping

A precise rule for when and how to nest sub-expressions 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 sub-expressions 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 F [Operator Dictionary]. In all other cases, nested mrows should be used.

When forming a nested mrow (during generation of MathML) that 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 that occurs later in the suggested operator dictionary (Appendix F [Operator Dictionary]) can be assumed to have a higher precedence for this purpose.

Note that the above rule has no effect on whether any