3. Presentation Elements

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 substantially more attributes in order to provide the fine layout control necessary for commutative diagrams, block matrices and so on.

3.5.1 <mtable> -- table or matrix

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.

Attributes of <mtable>:

 Name values default align (top | bottom | center | baseline | axis) [ rownumber ] axis rowalign (top | bottom | center | baseline | axis) + baseline columnalign (left | center | right) + center groupalign group-alignment-list-list {left} alignmentscope (true | false) + true rowspacing number +   (exs) 1.0 columnspacing number +   (ems) 0.8 rowlines (none | solid | dashed) + none columnlines (none | solid | dashed) + none frame none | solid | dashed none framespacing number number   (ems exs) 0.4 0.5 equalrows true | false true equalcolumns true | false true displaystyle true | false false

Note that the default values for rowlines, columnlines and frame is the literal string "none", meaning no lines is the default and not that there is no default.

As described in section 2.3.3, 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.

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.

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) 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) is not changed for the table cells.

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 <mtr> -- row in a table or matrix

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 <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.

Attributes of <mtr>:

 Name values default rowalign top | bottom | center | baseline | axis inherited columnalign (left | center | right) + inherited groupalign group-alignment-list-list inherited

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.

3.5.3 <mtd> -- one entry in a table or matrix

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.

Attributes of <mtd>:

 Name values default rowspan number 1 columnspan number 1 rowalign top | bottom | center | baseline | axis inherited columnalign left | center | right inherited groupalign group-alignment-list inherited

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.

3.5.4 <maligngroup/> and <malignmark/> -- alignment markers

Description:

These are spacelike elements (see Section 3.2.6) 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".

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 <mrow>s 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 spacelike 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 insure 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 "spacelike elements" in Section 3.2.6.

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.

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```

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

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 "spacelike elements" in Section 3.2.6).

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 the 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 and Chapter 5 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 it 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.)

Attributes of <malignmark/>:

 Name values default edge left | right left

<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 attriubte 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.

Attributes of <maligngroup/>:

 Name values default groupalign left | center | right | decimalpoint inherited

<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 <mrow>s), <mstyle>, <mpadded>, <mphantom>, or <mfenced>, descending into only the first argument of each "scripting" element (<msub>, <msup>, <msubsup>, <munder>, <mover>, <munderover>, <mmultiscripts>) or of each <msqrt>, <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.

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, `|` 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:

 Element type groupalign attribute syntax default value group-alignment-list-list {left} group-alignment-list-list inherited from attribute group-alignment-list inherited from within attribute group-alignment inherited from within attribute

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.

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> ⁢ </mo>               <maligngroup/>               <mi> x </mi>            </mrow>            <maligngroup/>            <mo> + </mo>            <mrow>               <maligngroup/>               <mn> 55 </mn>               <mo> ⁢ </mo>               <maligngroup/>               <mi> y </mi>            </mrow>         </mrow>         <maligngroup/>         <mo> = </mo>         <maligngroup/>         <mn> 0 </mn>      </mtd>      <mtd>         <mrow>            <mrow>               <maligngroup/>               <mn> 3.1 </mn>               <mo> ⁢ </mo>               <maligngroup/>               <mi> x </mi>            </mrow>            <maligngroup/>            <mo> - </mo>            <mrow>               <maligngroup/>               <mn> 0.7 </mn>               <mo> ⁢ </mo>               <maligngroup/>               <mi> y </mi>            </mrow>         </mrow>         <maligngroup/>         <mo> = </mo>         <maligngroup/>         <mrow>            <mo> - </mo>            <mn> 1.1 </mn>         </mrow>      </mtd>```

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 descussed 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 converted content elements to presentation elements which render the same way. Thus, as a general rule, even if a 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.

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 uses both the alignment elements, the groupalign attribute described in this section, and the columnalign attribute described under <mtable> (Section 3.5.1).

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; 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.

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