Abstract

This specification defines a core subset of Mathematical Markup Language, or MathML, that is suitable for browser implementation. MathML is a markup language for describing mathematical notation and capturing both its structure and content. The goal of MathML is to enable mathematics to be served, received, and processed on the World Wide Web, just as HTML has enabled this functionality for text.

Status of This Document

This section describes the status of this document at the time of its publication. A list of current W3C publications and the latest revision of this technical report can be found in the W3C standards and drafts index at https://www.w3.org/TR/.

This document was published by the Math Working Group as a Candidate Recommendation Snapshot using the Recommendation track.

Publication as a Candidate Recommendation does not imply endorsement by W3C and its Members. A Candidate Recommendation Snapshot has received wide review, is intended to gather implementation experience, and has commitments from Working Group members to royalty-free licensing for implementations.

This Candidate Recommendation is not expected to advance to Proposed Recommendation any earlier than 30 September 2025.

This document was produced by a group operating under the W3C Patent Policy. W3C maintains a public list of any patent disclosures made in connection with the deliverables of the group; that page also includes instructions for disclosing a patent. An individual who has actual knowledge of a patent which the individual believes contains Essential Claim(s) must disclose the information in accordance with section 6 of the W3C Patent Policy.

This document is governed by the 03 November 2023 W3C Process Document.

This section is non-normative.

The [MATHML3] specification has several shortcomings that make it hard to implement consistently across web rendering engines or to extend with user-defined constructions, e.g.:

It is a huge and standalone specification.
It does not contain any detailed rendering rules.
It is not driven by browser-implementation.
It lacks automated testing.

This MathML Core specification intends to address these issues by being as accurate as possible on the visual rendering of mathematical formulas using additional rules from the TeXBook’s Appendix G [TEXBOOK] and from the Open Font Format [OPEN-FONT-FORMAT], [OPEN-TYPE-MATH-ILLUMINATED]. It also relies on modern browser implementations and web technologies [HTML] [SVG] [CSS2] [DOM], clarifying interactions with them when needed or introducing new low-level primitives to improve the web platform layering.

Parts of MathML3 that do not fit well in this framework or are less fundamental have been omitted. Instead, they are described in a separate and larger [MATHML4] specification. The details of which math feature will be included in future versions of MathML Core or implemented as polyfills is still open. This question and other potential improvements are tracked on GitHub.

By increasing the level of implementation details, focusing on a workable subset, following a browser-driven design and relying on automated web platform tests, this specification is expected to greatly improve MathML interoperability. Moreover, effort on MathML layering will enable users to implement the rest of the MathML 4 specification, or more generally to extend MathML Core, using modern web technologies such as shadow trees, custom elements or APIs from [HOUDINI].

The term MathML element refers to any element in the MathML namespace. The MathML elements defined in this specification are called the MathML Core elements and are listed below. Any MathML element that is not listed below is called an Unknown MathML element .

The grouping elements are maction, math, merror, mphantom, mprescripts, mrow, mstyle, semantics and unknown MathML elements.

The scripted elements are mmultiscripts, mover, msub, msubsup, msup, munder and munderover.

The radical elements are mroot and msqrt.

The attributes defined in this specification have no namespace and are called MathML attributes :

MathML specifies a single top-level or root math element, which encapsulates each instance of MathML markup within a document. All other MathML content must be contained in a <math> element.

The <math> element accepts the attributes described in 2.1.3 Global Attributes as well as the following attributes:

display
alttext

The display attribute, if present, must be an ASCII case-insensitive match to block or inline. The user agent stylesheet described in A. User Agent Stylesheet contains rules for this attribute that affect the default values for the display (block math or inline math) and math-style (normal or compact) properties. If the display attribute is absent or has an invalid value, the User Agent stylesheet treats it the same as inline.

This specification does not define any observable behavior that is specific to the alttext attribute.

Note

The alttext attribute may be used as alternative text by some legacy systems that do not implement math layout.

If the <math> element does not have its computed display property equal to block math or inline math then it is laid out according to the CSS specification where the corresponding value is described. Otherwise the layout algorithm of the mrow element is used to produce a math content box. That math content box is used as the content for the layout of the element, as described by CSS for display: block (if the computed value is block math) or display: inline (if the computed value is inline math). Additionally, if the computed display property is equal to block math then that math content box is rendered horizontally centered within the content box.

Note

T_EX's display mode $$...$$ and inline mode $...$ correspond to display="block" and display="inline" respectively.

In the following example, a math formula is rendered in display mode on a new line and taking full width, with the math content centered within the container:

<div style="width: 15em;">
  This mathematical formula with a big summation and the number pi
  <math display="block" style="border: 1px dotted black;">
    <mrow>
      <munderover>
        <mo>∑</mo>
        <mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow>
        <mrow><mo>+</mo><mn>∞</mn></mrow>
      </munderover>
      <mfrac>
        <mn>1</mn>
        <msup><mi>n</mi><mn>2</mn></msup>
      </mfrac>
    </mrow>
    <mo>=</mo>
    <mfrac>
      <msup><mi>π</mi><mn>2</mn></msup>
      <mn>6</mn>
    </mfrac>
  </math>
  is easy to prove.
</div>

$math example (display)$

As a comparison, the same formula would look as follows in inline mode. The formula is embedded in the paragraph of text without forced line breaking. The baselines specified by the layout algorithm of the mrow are used for vertical alignment. Note that the middle of sum and equal symbols or fractions are all aligned, but not with the alphabetical baseline of the surrounding text.

$math example (inline)$

Because good mathematical rendering requires use of mathematical fonts, the user agent stylesheet should set the font-family to the math value on the <math> element instead of inheriting it. Additionally, several CSS properties that can be set on a parent container such as font-style, font-weight, direction or text-indent etc are not expected to apply to the math formula and so the user agent stylesheet has rules to reset them by default.

math {
  direction: ltr;
  text-indent: 0;
  letter-spacing: normal;
  line-height: normal;
  word-spacing: normal;
  font-family: math;
  font-size: inherit;
  font-style: normal;
  font-weight: normal;
  display: inline math;
  math-shift: normal;
  math-style: compact;
  math-depth: 0;
}
math[display="block" i] {
  display: block math;
  math-style: normal;
}
math[display="inline" i] {
  display: inline math;
  math-style: compact;
}

In addition to CSS data types, some MathML attributes rely on the following MathML-specific types:

unsigned-integer: An <integer> value as defined in [CSS-VALUES-4], whose first character is neither U+002D HYPHEN-MINUS character (-) nor U+002B PLUS SIGN (+).
boolean: A string that is an ASCII case-insensitive match to true or false.

The following attributes are common to and may be specified on all MathML elements:

autofocus
class
data-*
dir
displaystyle
id
mathbackground
mathcolor
mathsize
nonce
scriptlevel
style
tabindex
on* event handler attributes
additional attributes, as described in 2.1.7 Attributes Reserved as Valid

The id , class , style , data-* , autofocus and nonce and tabindex attributes have the same syntax and semantics as defined for id, class, style, data-*, autofocus, nonce and tabindex attributes on HTML elements.

The dir attribute, if present, must be an ASCII case-insensitive match to ltr or rtl. In that case, the user agent is expected to treat the attribute as a presentational hint setting the element's direction property to the corresponding value. More precisely, an ASCII case-insensitive match to rtl is mapped to rtl while an ASCII case-insensitive match to ltr is mapped to ltr.

Note

The dir attribute is used to set the directionality of math formulas, which is often rtl in Arabic speaking world. However, languages written from right to left often embed math written from left to right and so the user agent stylesheet resets the direction property accordingly on the math elements.

In the following example, the dir attribute is used to render "𞸎 plus 𞸑 raised to the power of (٢ over, 𞸟 plus ١)" from right-to-left.

<math dir="rtl">
  <mrow>
    <mi>𞸎</mi>
    <mo>+</mo>
    <msup>
      <mi>𞸑</mi>
      <mfrac>
        <mn>٢</mn>
        <mrow>
          <mi>𞸟</mi>
          <mo>+</mo>
          <mn>١</mn>
        </mrow>
      </mfrac>
    </msup>
  </mrow>
</math>

$dir example$

All MathML elements support event handler content attributes, as described in event handler content attributes in HTML.

All event handler content attributes noted by HTML as being supported by all HTMLElements are supported by all MathML elements as well, as defined in the MathMLElement IDL.

The mathcolor and mathbackground attributes, if present, must have a value that is a <color>. In that case, the user agent is expected to treat these attributes as a presentational hint setting the element's color and background-color properties to the corresponding values. The mathcolor attribute describes the foreground fill color of MathML text, bars etc while the mathbackground attribute describes the background color of an element.

The mathsize attribute, if present, must have a value that is a valid <length-percentage>. In that case, the user agent is expected to treat the attribute as a presentational hint setting the element's font-size property to the corresponding value. The mathsize property indicates the desired height of glyphs in math formulas but also scales other parts (spacing, shifts, line thickness of bars etc) accordingly.

Note

The above attributes are implemented for compatibility with full MathML. Authors whose only target is MathML Core are encouraged to use CSS for styling.

The displaystyle attribute, if present, must have a value that is a boolean. In that case, the user agent is expected to treat the attribute as a presentational hint setting the element's math-style property to the corresponding value. More precisely, an ASCII case-insensitive match to true is mapped to normal while an ASCII case-insensitive match to false is mapped to compact. This attribute indicates whether formulas should try to minimize the logical height (value is false) or not (value is true) e.g. by changing the size of content or the layout of scripts.

The scriptlevel attribute, if present, must have value +, - or  where  is an unsigned-integer. In that case the user agent is expected to treat the scriptlevel attribute as a presentational hint setting the element's math-depth property to the corresponding value. More precisely, +, - and  are respectively mapped to add() add(<-U>) and .

displaystyle and scriptlevel values are automatically adjusted within MathML elements. To fully implement these attributes, additional CSS properties must be specified in the user agent stylesheet as described in A. User Agent Stylesheet. In particular, for all MathML elements a default font-size: math is specified to ensure that scriptlevel changes are taken into account.

In this example, an munder element is used to attach a script "A" to a base "∑". By default, the summation symbol is rendered with the font-size inherited from its parent and the A as a scaled down subscript. If displaystyle is true, the summation symbol is drawn bigger and the "A" becomes an underscript. If scriptlevel is reset to 0 on the "A", then it will use the same font-size as the top-level math root.

<math>
  <munder>
    <mo>∑</mo>
    <mi>A</mi>
  </munder>
  <munder displaystyle="true">
    <mo>∑</mo>
    <mi>A</mi>
  </munder>
  <munder>
    <mo>∑</mo>
    <mi scriptlevel="0">A</mi>
  </munder>
</math>

$displaystyle-scriptlevel example$

Note

T_EX's \displaystyle, \textstyle, \scriptstyle, and \scriptscriptstyle correspond to displaystyle and scriptlevel as true and 0, false and 0, false and 1, and false and 2, respectively.

The attributes intent and arg are reserved as valid attributes.

This specification does not define any observable behavior that is specific to the intent and arg attributes.

Note

These attributes are described in [MATHML4] and future versions of this specification may or may not define them. Authors should be aware that they are currently in development and subject to change.

MathML can be mixed with HTML and SVG as described in the relevant specifications [HTML] [SVG].

When evaluating the SVG requiredExtensions attribute, user agents must claim support for the language extension identified by the MathML namespace.

In this example, inline MathML and SVG elements are used inside an HTML document. SVG elements <switch> and <foreignObject> (with proper <requiredExtensions>) are used to embed a MathML formula with a text fallback, inside a diagram. HTML input element is used within the mtext to include an interactive input field inside a mathematical formula. See also 3.7 Semantics and Presentation for an example of SVG and HTML inside an annotation-xml element.

<svg style="font-size: 20px" width="400px" height="220px" viewBox="0 0 200 110">
  <g transform="translate(10,80)">
    <path d="M 0 0 L 150 0 A 75 75 0 0 0 0 0
             M 30 0 L 30 -60 M 30 -10 L 40 -10 L 40 0"
          fill="none" stroke="black"></path>
    <text transform="translate(10,20)">1</text>
    <switch transform="translate(35,-40)">
      <foreignObject width="200" height="50"
                     requiredExtensions="http://www.w3.org/1998/Math/MathML">
        <math>
          <msqrt>
            <mn>2</mn>
            <mi>r</mi>
            <mo>−</mo>
            <mn>1</mn>
          </msqrt>
        </math>
      </foreignObject>
      <text>\sqrt{2r - 1}</text>
    </switch>
  </g>
</svg>
<p>
  Fill the blank:
  <math>
    <msqrt>
      <mn>2</mn>
      <mtext><input onchange="..." size="2" type="text"></mtext>
      <mo>−</mo>
      <mn>1</mn>
    </msqrt>
    <mo>=</mo>
    <mn>3</mn>
  </math>
</p>

$html-svg example$

User agents must support various CSS features mentioned in this specification, including new ones described in 4. CSS Extensions for Math Layout. They must follow the computation rule for display: contents.

In this example, the MathML formula inherits the CSS color of its parent and uses the font-family specified via the style attribute.

<div style="width: 15em; color: blue">
  This mathematical formula with a big summation and the number pi
  <math display="block" style="font-family: STIX Two Math">
    <mrow>
      <munderover>
        <mo>∑</mo>
        <mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow>
        <mrow><mo>+</mo><mn>∞</mn></mrow>
      </munderover>
      <mfrac>
        <mn>1</mn>
        <msup><mi>n</mi><mn>2</mn></msup>
      </mfrac>
    </mrow>
    <mo>=</mo>
    <mfrac>
      <msup><mi>π</mi><mn>2</mn></msup>
      <mn>6</mn>
    </mfrac>
  </math>
  is easy to prove.
</div>

$style example$

All documents containing MathML Core elements must include CSS rules described in A. User Agent Stylesheet as part of user-agent level style sheet defaults. In particular, this adds !important rules to force writing mode to horizontal-lr on all MathML elements.

The float property does not create floating of elements whose parent's computed display value is block math or inline math, and does not take them out-of-flow.

The ::first-line and ::first-letter pseudo-elements do not apply to elements whose computed display value is block math or inline math, and such elements do not contribute a first formatted line or first letter to their ancestors.

The following CSS features are not supported and must be ignored:

Line breaking inside math formulas: white-space is treated as nowrap on all MathML elements.
Alignment properties: align-content, justify-content, align-self, justify-self have no effects on MathML elements.

Note

These features might be handled in future versions of this document. For now, authors are discouraged from setting a different value for these properties as that might lead to backward incompatibility issues.

User agents supporting Web application APIs must ensure that they keep the visual rendering of MathML synchronized with the [DOM] tree, in particular perform necessary updates when MathML attributes are modified dynamically.

All the nodes representing MathML elements in the DOM must implement, and expose to scripts, the following MathMLElement interface.

[Exposed=Window]
interface MathMLElement : Element { };
MathMLElement includes GlobalEventHandlers;
MathMLElement includes HTMLOrForeignElement;

The GlobalEventHandlers and HTMLOrForeignElement interfaces are defined in [HTML].

In the following example, a MathML formula is used to render the fraction "α over 2". When clicking the red α, it is changed into a blue β.

<script>
  function ModifyMath(mi) {
      mi.style.color = 'blue';
      mi.textContent = 'β';
  }
</script>
<math>
  <mrow>
    <mfrac>
      <mi style="color: red" onclick="ModifyMath(this)">α</mi>
      <mn>2</mn>
    </mfrac>
  </mrow>
</math>

$dom-idl example$

Issue

Because math fonts generally contain very tall glyphs such as big integrals, using typographic metrics is important to avoid excessive line spacing of text. As a consequence, user agents must take into account the USE_TYPO_METRICS flag from the OS/2 table [OPEN-FONT-FORMAT] when performing text layout.

MathML provides the ability for authors to allow for interactivity in supporting interactive user agents using the same concepts, approach and guidance to Focus as described in HTML, with modifications or clarifications regarding application for MathML as described in this section.

When an element is focused, all applicable CSS focus-related pseudo-classes as defined in Selectors Level 3 apply, as defined in that specification.

The contents of embedded math elements (including HTML elements inside token elements) contribute to the sequential focus order of the containing owner HTML document (combined sequential focus order).

The default display property is described in A. User Agent Stylesheet:

For the <math> root, it is equal to inline math or block math according to the value of the display attribute.
For Tabular MathML elements mtable, mtr, mtd it is respectively equal to inline-table, table-row and table-cell.
For all but the first children of the maction and semantics elements, it is equal to none.
For all the other MathML elements it is equal to block math.

In order to specify math layout in different writing modes, this specification uses concepts from [CSS-WRITING-MODES-4]:

abstract dimensions: block dimension, inline dimension, block axis, inline axis, block size, inline size.
flow-relative directions: inline-start, inline-end, block-start.
line-relative directions: line-left, line-over, line-under.

Note

Unless specified otherwise, the figures in this specification use a writing mode of horizontal-lr and ltr. See

Figure 4

Figure 5

and

Figure 6

for examples of other writing modes that are sometimes used for math layout.

Boxes used for MathML elements rely on several parameters in order to perform layout in a way that is compatible with CSS but also to take into account very accurate positions and spacing within math formulas:

Inline metrics. min-content inline size, max-content inline size and inline size from CSS. See Figure 1 .

Figure 1 Generic Box Model for MathML elements
Block metrics. The block size, first baseline set and last baseline set. The following baselines are defined for MathML boxes:
1. The alphabetic baseline which typically aligns with the bottom of uppercase Latin glyphs. The algebraic distance from the alphabetic baseline to the line-over edge of the box is called the line-ascent . The algebraic distance from the line-under edge to the alphabetic baseline of the box is called the line-descent .
2. The mathematical baseline , also called math axis , which typically aligns with the fraction bar, middle of fences and binary operators. It is shifted away from the alphabetic baseline by AxisHeight towards the line-over.
3. The ink-over baseline , indicating the line-over theorical limit of the math content drawn, excluding any extra space. If not specified, it is aligned with the line-over edge. The algebraic distance from the alphabetic baseline to the ink-over baseline is called the ink line-ascent .
4. The ink-under baseline , indicating the line-under theorical limit of the math content drawn, excluding any extra space. If not specified, it is aligned with the line-under edge. The algebraic distance from the ink-under baseline to the alphabetic baseline is called the ink line-descent .
Note
For math layout, it is very important to rely on the ink extent when positioning text. This is not the case for more complex notations (e.g. square root). Although ink-ascent and ink-descent are defined for all MathML elements they are really only used for the token elements. In other cases, they just match normal ascent and descent.
Unless specified otherwise, the last baseline set is equal to the first baseline set for MathML boxes.
An optional italic correction which provides a measure of how much the text of a box is slanted along the inline axis. See Figure 2 .

Figure 2 Examples of italic correction for italic f and large integral

If it is requested during calculation of min-content inline size and max-content inline size or during layout then 0 is used as a fallback value.
An optional top accent attachment which provides a reference offset on the inline axis of a box that should be used when positioning that box as an accent. See Figure 3 .

Figure 3 Example of top accent attachment for a circumflex accent

If it is requested during calculation of min-content inline size (respectively max-content inline size) then half the min-content inline size (respectively max-content inline size) is used as a fallback value. If it is requested during layout then half the inline size of the box is used as a fallback value.

Given a MathML box, the following offsets are defined:

The inline offset of a child box is the offset between the inline-start edge of the parent box and the inline-start edge of the child box.
The block offset of a child box is the offset between the block-start edge of the parent box and the block-start edge of the child box.
The line-left offset of a child box is the offset between the line-left edge of the parent box and the line-left edge of the child box.

Figure 4 Box model for writing mode `horizontal-tb` and `rtl` that may be used in e.g. Arabic math.

Figure 5 Box model for writing mode `vertical-lr` and `ltr` that may be used in e.g. Mongolian math.

Figure 6 Box model for writing mode `vertical-rl` and `ltr` that may be used in e.g. Japanese math.

Note

The position of child boxes and graphical items inside a MathML box are expressed using the inline offset and block offset. For convenience, the layout algorithms may describe offsets using flow-relative directions, line-relative directions or the alphabetic baseline. It is always possible to pass from one description to the other because position of child boxes is always performed after the metrics of the box and of its child boxes are calculated.

Here are examples of offsets obtained from line-relative metrics:

The inline offset of a child box is the line-left offset if the CSS property direction is ltr and is the inline size of the box − (line-left offset + inline size of the child box) otherwise.
The block offset of the alphabetic baseline of a box is the line-ascent if the CSS property writing-mode is horizontal-lr, vertical-rl or sideways-rl and is the line-descent otherwise.

Improve definition of ink ascent/descent?

Each MathML element has an associated math content box , which is calculated as described in this chapter's layout algorithms using the following structure:

Calculation of min-content inline size and max-content inline size of the math content.
Box layout:
1. Layout of in-flow child boxes.
2. Calculation of inline size, ink line-ascent, ink line-descent, line-ascent and line-ascent of the math content.
3. Calculation of offsets of child boxes within the math content box as well as sizes and offsets of extra graphical items (bars, radical symbol, etc).
4. Layout and positioning of absolutely-positioned and fixed-positioned boxes, as described in [CSS-POSITION-3].

The following extra steps must be performed:

After calculating the line-ascent and line-descent of a math content box, its block size is set to their sum.
By default, the box metrics, offsets, baselines, italic correction and top accent attachment of the content box are set to the one calculated for the math content box. However if a different size is specified for the content box (via width, height or related properties), that size is used instead and the inline-start and block-start edges of the math content box are aligned with the ones of the content box, unless otherwise specified in a specific layout algorithm.
The box metrics and offsets of the padding box are obtained from the content box by taking into account the corresponding padding properties as described in CSS.

The baselines of the padding box are the same as the one of the content box.

If the content box has a top accent attachment then the padding box has the same property, increased by the inline-start padding. If the content box has an italic correction then the padding box has the same property, increased by the inline-end padding.
The box metrics and offsets of the border box are obtained from the padding box by taking into account the corresponding border-width property as described in CSS.

In general, the baselines of the border box are the same as the one of the padding box. However, if the line-over border is positive then the ink-over baseline is set to the line-over edge of the border box and if the line-under border is positive then the ink-under baseline is set to the line-under edge of the border box.

If the padding box has a top accent attachment then the border box has the same property, increased by the border-width of its inline-start egde. If the padding box has an italic correction then the border box has the same property, increased by the border-width of its inline-end egde.
The box metrics and offsets of the margin box are obtained from the border box by taking into account the corresponding margin properties as described in CSS.

The baselines of the margin box are the same as the one of the border box.

If the padding box has a top accent attachment then the margin box has the same property, increased by the inline-start margin. If the padding box has an italic correction then the margin box has the same property, increased by the inline-end margin.

During box layout, optional inline stretch size constraint and block stretch size constraint parameters may be used on embellished operators. The former indicates a target size that a core operator stretched along the inline axis should cover. The latter indicates an ink line-ascent and ink line-descent that a core operator stretched along the block axis should cover. Unless specified otherwise, these parameters are ignored during box layout and child boxes are laid out without any stretch size constraint.

Define what inline percentages resolve against

Define what block percentages resolve against

An anonymous box is a box without any associated element in the DOM tree and which is generated for layout purpose only. The properties of anonymous boxes are inherited from the enclosing non-anonymous box while non-inherited properties have their initial value. An anonymous <mrow> box is an anonymous box with display equal to block math and which is laid out as described in section 3.3.1.2 Layout of <mrow>.

If a MathML element generates an anonymous <mrow> box then it wraps its children in an anonymous <mrow> box. I.e., its subtree in the visual formatting model is made of an anonymous <mrow> box which itself contains the boxes associated to the children of this MathML element.

In the following example, the math and mrow elements are laid out as described in section 3.3.1.2 Layout of <mrow>. In particular, the <math> element adds proper spacing around its <mo>≠</mo> child and the <mrow> element stretches its <mo>|</mo> children vertically.

The mtd element has display: table-cell and the msqrt element displays a radical symbol around its children. However, they also place their children in a way that is similar to what is described in section 3.3.1.2 Layout of <mrow>: the <msqrt> element adds proper spacing around its <mo>+</mo> child while the <mtd> element stretches its <mo> children vertically. In order to make this possible, each of these two elements generates an anonymous <mrow> box.

<math>
  <mrow>
    <mo>|</mo>
    <mtable>
      <mtr>
        <mtd>
          <mi>x</mi>
        </mtd>
        <mtd>
          <mo>(</mo>
          <mfrac linethickness="0">
            <mn>5</mn>
            <mn>3</mn>
          </mfrac>
          <mo>)</mo>
        </mtd>
      </mtr>
      <mtr>
        <mtd>
          <msqrt>
            <mn>7</mn>
            <mo>+</mo>
            <mn>2</mn>
          </msqrt>
        </mtd>
        <mtd>
          <mi>y</mi>
        </mtd>
      </mtr>
    </mtable>
    <mo>|</mo>
  </mrow>
  <mo>≠</mo>
  <mn>0</mn>
</math>

$math example (display)$

MathML elements can overlap due to various spacing rules. They can as well contain extra graphical items (bars, radical symbol, etc). A MathML element with computed style display: block math or display: inline math generates a new stacking context. The painting order of in-flow children of such a MathML element is exactly the same as block elements. The extra graphical items are painted after text and background (right after step 7.2.4 for display: inline math and right after step 7.2 for display: block math).

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.

Note

In practice, most MathML token elements just contain simple text for variables, numbers, operators etc and don't need sophisticated layout. However, it can contain text with line breaks or arbitrary HTML5 phrasing elements.

The 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.

The <mtext> element accepts the attributes described in 2.1.3 Global Attributes.

In the following example, mtext is used to put conditional words in a definition:

<math>
  <mi>y</mi>
  <mo>=</mo>
  <mrow>
    <msup>
      <mi>x</mi>
      <mn>2</mn>
    </msup>
    <mtext>&nbsp;if&nbsp;</mtext>
    <mrow>
      <mi>x</mi>
      <mo>≥</mo>
      <mn>1</mn>
    </mrow>
    <mtext>&nbsp;and&nbsp;</mtext>
    <mn>2</mn>
    <mtext>&nbsp;otherwise.</mtext>
  </mrow>
</math>

$mtext example$

If the element does not have its computed display property equal to block math or inline math then it is laid out according to the CSS specification where the corresponding value is described. Otherwise, the layout below is performed.

If the <mtext> element contains only text content without forced line break or soft wrap opportunity then, the anonymous child node generated for that text is laid out as defined in the relevant CSS specification and:

The min-content inline size, max-content inline size and inline size of the math content box are equal to the one of the anonymous child.
The ink line-ascent and ink line-descent of the math content box are set so that its ink-over baseline, ink-under baseline and alphabetic baseline match the one of the ink bounding box of the text.
The line-ascent (respectively line-descent) of the math content box is set to the ink line-ascent (respectively ink line-descent).
If the text content is made of a single glyph and this glyph has an entry in the MathItalicsCorrectionInfo table then the specified value is used as the italic correction.
If the text content is made of a single glyph and this glyph has an entry in the MathTopAccentAttachment table then the specified value is used as the top accent attachment of the <mtext> element.

Otherwise, the mtext element is laid out as a block box and corresponding min-content inline size, max-content inline size, inline size, block size, first baseline set and last baseline set are used for the math content box.

The 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.

The <mi> element accepts the attributes described in 2.1.3 Global Attributes as well as the following attribute:

mathvariant

The layout algorithm is the same as the mtext element. The user agent stylesheet must contain the following property in order to implement automatic italic via the text-transform value introduced in 4.2 The math-auto transform:

mi {
  text-transform: math-auto;
}

The mathvariant attribute, if present, must be an ASCII case-insensitive match of normal. In that case, the user agent is expected to treat the attribute as a presentational hint setting the element's text-transform property to none. Otherwise it has no effects.

Note

In [MathML3], the mathvariant attribute was used to define logical classes of token elements, each class providing a collection of typographically-related symbolic tokens with specific meaning within a given mathematical expression.

In MathML Core, this attribute is only used to cancel automatic italic of the mi element. For other use cases, the proper Mathematical Alphanumeric Symbols [UNICODE] should be used instead. See also section C. Mathematical Alphanumeric Symbols.

In the following example, mi is used to render variables and function names. Note that per 4.2 The math-auto transform the default style text-transform: math-auto has no effect on the first <mi> ("cos" is made of three characters), makes the second <mi> render as math italic ("c" is made of a single character U+0063 Latin Small Letter C which is mapped to U+1D450 Mathematical Italic Small C per the italic table), has no effect on the third <mi> (overridden by mathvariant="normal", setting text-transform to none) or on the fourth <mi> (no mapping defined for U+221E Infinity in the italic table).

<math>
  <mi>cos</mi>
  <mo>,</mo>
  <mi>c</mi>
  <mo>,</mo>
  <mi mathvariant="normal">c</mi>
  <mo>,</mo>
  <mi>∞</mi>
</math>

$mi example$

The 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 <mn> element accepts the attributes described in 2.1.3 Global Attributes. Its layout algorithm is the same as the mtext element.

In the following example, mn is used to write a decimal number.

<math>
  <mn>3.141592653589793</mn>
</math>

$mn example$

The 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. This chapter uses the term "operator" to refer to operators in this broad sense.

The <mo> element accepts the attributes described in 2.1.3 Global Attributes as well as the following attributes:

This specification does not define any observable behavior that is specific to the fence and separator attributes.

Note

Authors may use the fence and separator to describe specific semantics of operators. The default values may be determined from the Operators_fence and Operators_separator tables, or equivalently the human-readable version of the operator dictionary.

In the following example, the mo element is used for the binary operator +. Default spacing is symmetric around that operator. A tighter spacing is used if you rely on the form attribute to force it to be treated as a prefix operator. Spacing can also be specified explicitly using the lspace and rspace attributes.

<math>
  <mn>1</mn>
  <mo>+</mo>
  <mn>2</mn>
  <mo form="prefix">+</mo>
  <mn>3</mn>
  <mo lspace="2em">+</mo>
  <mn>4</mn>
  <mo rspace="3em">+</mo>
  <mn>5</mn>
</math>

$mo example 1$

Another use case is for big operators such as summation. When displaystyle is true, such an operator is drawn larger but one can change that with the largeop attribute. When displaystyle is false, underscripts are actually rendered as subscripts but one can change that with the movablelimits attribute.

<math>
  <mrow displaystyle="true">
  <munder>
    <mo>∑</mo>
    <mn>5</mn>
  </munder>
  <munder>
    <mo largeop="false">∑</mo>
    <mn>6</mn>
  </munder>
  </mrow>
  <mrow>
    <munder>
      <mo>∑</mo>
      <mn>5</mn>
    </munder>
    <munder>
      <mo movablelimits="false">∑</mo>
      <mn>7</mn>
    </munder>
  </mrow>
</math>

$mo example 2$

Operators are also used for stretchy symbols such as fences, accents, arrows etc. In the following example, the vertical arrow stretches to the height of the mspace element. One can override default stretch behavior with the stretchy attribute e.g. to force an unstretched arrow. The symmetric attribute allows to indicate whether the operator should stretch symmetrically above and below the math axis (fraction bar). Finally the minsize and maxsize attributes add additional constraints over the stretch size.

<math>
  <mfrac>
    <mspace height="50px" depth="50px" width="10px" style="background: blue"/>
    <mspace height="25px" depth="25px" width="10px" style="background: green"/>
  </mfrac>
  <mo>↑</mo>
  <mo stretchy="false">↑</mo>
  <mo symmetric="true">↑</mo>
  <mo minsize="250px">↑</mo>
  <mo maxsize="50px">↑</mo>
</math>

$mo example 3$

Note that the default properties of operators are dictionary-based, as explained in 3.2.4.2 Dictionary-based attributes. For example a binary operator typically has default symmetric spacing around it while a fence is generally stretchy by default.

A MathML Core element is an embellished operator if it is:

An mo element;
a scripted element or an mfrac, whose first in-flow child exists and is an embellished operator;
a grouping element or mpadded, whose in-flow children consist (in any order) of one embellished operator and zero or more space-like elements.

The core operator of an embellished operator is the <mo> element defined recursively as follows:

The core operator of an mo element; is the element itself.
The core operator of an embellished scripted element or mfrac element is the core operator of its first in-flow child.
The core operator of an embellished grouping element or mpadded is the core operator of its unique embellished operator in-flow child.

The stretch axis of an embellished operator is inline if its core operator contains only text content made of a single character c, and that character has inline intrinsic stretch axis. Otherwise, the stretch axis of the embellished operator is block.

The same definitions apply for boxes in the visual formatting model where an anonymous <mrow> box is treated as a grouping element.

The form property of an embellished operator is either infix, prefix or postfix. The corresponding form attribute on the mo element, if present, must be an ASCII case-insensitive match to one of these values.

The algorithm for determining the form of an embellished operator is as follows:

If the form attribute is present and valid on the core operator, then its ASCII lowercased value is used.
If the embellished operator is the first in-flow child of a grouping element, mpadded or msqrt with more than one in-flow child (ignoring all space-like children) then it has form prefix.
Or, if the embellished operator is the last in-flow child of a grouping element, mpadded or msqrt with more than one in-flow child (ignoring all space-like children) then it has form postfix.
Or, if the embellished operator is an in-flow child of a scripted element, other than the first in-flow child, then it has form postfix.
Otherwise, the embellished operator has form infix.

The stretchy , symmetric , largeop , movablelimits properties of an embellished operator are either false or true. In the latter case, it is said that the embellished operator has the property. The corresponding stretchy , symmetric , largeop , movablelimits attributes on the mo element, if present, must be a boolean.

The lspace , rspace , minsize properties of an embellished operator are <length-percentage>. The maxsize property of an embellished operator is either a <length-percentage> or ∞. The lspace , rspace , minsize and maxsize attributes on the mo element, if present, must be a <length-percentage>.

The algorithm for determining the properties of an embellished operator is as follows:

If the corresponding stretchy, symmetric, largeop, movablelimits, lspace, rspace, maxsize or minsize attribute is present and valid on the core operator, then the ASCII lowercased value of this property is used.
Otherwise, run the algorithm for determining the form of an embellished operator.
If the core operator contains only text content Content, then set Category to the result of the algorithm to determine the category of an operator (Content, Form) where Form is the form calculated at the previous step.
If Category is Default and the form of embellished operator was not explicitly specified as an attribute on its core operator:
1. Set Category to the result of the algorithm to determine the category of an operator (Content, Form) where Form is infix.
2. If Category is Default, then run the algorithm again with Form set to postfix.
3. If Category is Default, then run the algorithm again with Form set to prefix.
Run the algorithm to set the properties of an operator from its category Category.

When used during layout, the values of stretchy, symmetric, largeop, movablelimits, lspace, rspace, minsize are obtained by the algorithm for determining the properties of an embellished operator with the following extra resolutions:

Percentage values for lspace, rspace are interpreted relative to the value read from the dictionary or to the fallback value above.
Interpretation of percentage values for minsize and maxsize are described in 3.2.4.3 Layout of operators.
Font-relative lengths for lspace, rspace, minsize and maxsize rely on the font style of the core operator, not the one of the embellished operator.

If the <mo> element does not have its computed display property equal to block math or inline math then it is laid out according to the CSS specification where the corresponding value is described. Otherwise, the layout below is performed.

The text of the operator must only be painted if the visibility of the <mo> element is visible. In that case, it must be painted with the color of the <mo> element.

Operators are laid out as follows:

If the content of the <mo> element is not made of a single character c then fall back to the layout algorithm of 3.2.1.1 Layout of <mtext>.
If the operator has the stretchy property:
- If the stretch axis of the operator is inline:
 1. If it is not possible to shape a stretchy glyph corresponding to c in the inline direction with the first available font then fall back to the layout algorithm of 3.2.1.1 Layout of <mtext>.
 2. The min-content inline size and max-content inline size of the math content are set to the one obtained by the layout algorithm of 3.2.1.1 Layout of <mtext>.
 3. If there is not any inline stretch size constraint T_inline then fall back to the layout algorithm of 3.2.1.1 Layout of <mtext>.
 4. The inline size and (ink) block metrics of the math content are given by algorithm to shape a stretchy glyph to inline dimension T_inline.
 5. The painting of the operator is performed by the algorithm to shape a stretchy glyph stretched to inline dimension T_inline and at position determined by the previous box metrics.
- Otherwise, the stretch axis of the operator is block. The following steps are performed:
 1. If it is not possible to shape a stretchy glyph corresponding to c in the block direction with the first available font then fall back to the layout algorithm of 3.2.1.1 Layout of <mtext>.
 2. The min-content inline size and max-content inline size of the math content are set to the preferred inline size of a glyph stretched along the block axis.
 3. If there is not any block stretch size constraint (U_ascent, U_descent) then fall back to the layout algorithm of 3.2.1.1 Layout of <mtext>.
 4. If the operator has the symmetric property then set the target sizes T_ascent and T_descent to S_ascent and S_descent respectively:
 - S_ascent = max( U_ascent − AxisHeight, U_descent + AxisHeight ) + AxisHeight
 - S_descent = max( U_ascent − AxisHeight, U_descent + AxisHeight ) − AxisHeight
 Otherwise set them to U_ascent and U_descent respectively.
 Note
 The property T_ascent − AxisHeight = T_descent + AxisHeight means that an operator stretching exactly T_ascent above the baseline and T_descent below the baseline would actually stretch symmetrically above and below the math axis. S_ascent and S_descent are the minimal values, that are respectively not less than U_ascent and U_descent, which satisfy this property.
 5. Let minsize and maxsize be the minsize and maxsize properties on the operator. Percentage values are interpreted relative to the height of the glyph for c. Let T = T_ascent + T_descent be the target size. If minsize < 0 then set minsize to 0. If maxsize < minsize then set maxsize to minsize. With 0 ≤ minsize ≤ maxsize:
 - If T ≤ 0 then set T_ascent to minsize / 2 + AxisHeight and then set T_descent to minsize − T_ascent.
 - Otherwise, if 0 < T < minsize then set T_ascent to max(0, (T_ascent − AxisHeight) × minsize / T + AxisHeight) and T_descent to minsize − T_ascent.
 - Otherwise, if maxsize < T then set T_ascent to max(0, (T_ascent − AxisHeight) × maxsize / T + AxisHeight) and T_descent to maxsize − T_ascent.
 Note
 The default maxsize is value ∞ is interpreted above as being larger than any other size, i.e. minsize ≤ maxsize is always true while maxsize < minsize and maxsize < T are always false.
 
 Note
 This step ensures that the condition minsize ≤ T ≤ maxsize holds. Additionnally, if the target values correspond to symmetric stretching with respect to the math axis then property T_ascent − AxisHeight = T_descent + AxisHeight is preserved.
 6. The inline size, ink line-ascent, ink line-descent, line-ascent and line-descent of the math content are obtained by the algorithm to shape a stretchy glyph to block dimension T_ascent + T_descent. The inline size of the math content is the width of the stretchy glyph. The stretchy glyph is shifted towards the line-under by a value Δ so that its center aligns with the center of the target: the ink ascent of the math content is the ascent of the stretchy glyph − Δ and the ink descent of the math content is the descent of the stretchy glyph + Δ. These centers have coordinates "½(ascent − descent)" so Δ = [(ascent of stretchy glyph − descent of stretchy glyph) − (T_ascent − T_descent)] / 2.
 7. The painting of the operator is performed by the algorithm to shape a stretchy glyph stretched to block dimension T_ascent + T_descent and at position determined by the previous box metrics shifted by Δ towards the line-over.
 Figure 7 Base size, size variants and glyph assembly for the left brace
If the operator has the largeop property and if math-style on the <mo> element is normal, then:
1. Use the MathVariants table to try and find a glyph of height at least DisplayOperatorMinHeight. If none is found, fall back to the largest non-base glyph. If none is found, fall back to the layout algorithm of 3.2.1.1 Layout of <mtext>.
2. The min-content inline size, max-content inline size, inline size and block metrics of the math content are given by the glyph found.
3. Paint the glyph.
Figure 8 Base and displaystyle sizes of the summation symbol
Otherwise fall back to the layout algorithm of 3.2.1.1 Layout of <mtext>.

If the algorithm to shape a stretchy glyph has been used for one of the step above, then the italic correction of the math content is set to the value returned by that algorithm.

The mspace empty element represents a blank space of any desired size, as set by its attributes.

The <mspace> element accepts the attributes described in 2.1.3 Global Attributes as well as the following attributes:

The width , height , depth , if present, must have a value that is a valid <length-percentage>.

If the width attribute is present, valid and not a percentage then that attribute is used as a presentational hint setting the element's width property to the corresponding value.
If the height attribute is absent, invalid or a percentage then the requested line-ascent is 0. Otherwise the requested line-ascent is the resolved value of the height attribute, clamping negative values to 0.
If both the height and depth attributes are present, valid and not a percentage then they are used as a presentational hint setting the element's height property to the concatenation of the strings "calc(", the height attribute value, " + ", the depth attribute value, and ")". If only one of these attributes is present, valid and not a percentage then it is treated as a presentational hint setting the element's height property to the corresponding value.

In the following example, mspace is used to force spacing within the formula (a 1px blue border is added to easily visualize the space):

<math>
  <mn>1</mn>
  <mspace width="1em"
          style="border-top: 1px solid blue"/>
  <mfrac>
    <mrow>
      <mn>2</mn>
      <mspace depth="1em"
              style="border-left: 1px solid blue"/>
    </mrow>
    <mrow>
      <mn>3</mn>
      <mspace height="2em"
              style="border-left: 1px solid blue"/>
    </mrow>
  </mfrac>
</math>

$mspace example$

If the <mspace> element does not have its computed display property equal to block math or inline math then it is laid out according to the CSS specification where the corresponding value is described. Otherwise, the <mspace> element is laid out as shown on Figure 9 . The min-content inline size, max-content inline size and inline size of the math content are equal to the resolved value of the width property. The block size of the math content is equal to the resolved value of the height property. The line-ascent of the math content is equal to the requested line-ascent determined above.

Figure 9 Box model for the `<mspace>` element

Note

The terminology height/depth comes from [MATHML3], itself inspired from [TEXBOOK].

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.

A MathML Core element is a space-like element if it is:

an mtext or mspace;
or a grouping element or mpadded all of whose in-flow children are space-like.

The same definitions apply for boxes in the visual formatting model where an anonymous <mrow> box is treated as a grouping element.

Note

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.

ms element is used to represent "string literals" in expressions meant to be interpreted by computer algebra systems or other systems containing "programming languages".

The <ms> element accepts the attributes described in 2.1.3 Global Attributes. Its layout algorithm is the same as the mtext element.

In the following example, ms is used to write a literal string of characters:

<math>
  <mi>s</mi>
  <mo>=</mo>
  <ms>"hello world"</ms>
</math>

$ms example$

Note

In MathML3, it was possible to use the lquote and rquote attributes to respectively specify the strings to use as opening and closing quotes. These are no longer supported and the quotes must instead be specified as part of the text of the <ms> element. One can add CSS rules to legacy documents in order to preserve visual rendering. For example, in left-to-right direction:

ms:before, ms:after {
  content: "\0022";
}
ms[lquote]:before {
  content: attr(lquote);
}
ms[rquote]:after {
  content: attr(rquote);
}

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.

The 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".

In the following example, mrow is used to group a sum "1 + 2/3" as a fraction numerator (first child of mfrac) and to construct a fenced expression (first child of msup) that is raised to the power of 5. Note that mrow alone does not add visual fences around its grouped content, one has to explicitly specify them using the mo element.

Within the mrow elements, one can see that vertical alignment of children (according to the alphabetic baseline or the mathematical baseline) is properly performed, fences are vertically stretched and spacing around the binary + operator automatically calculated.

<math>
  <msup>
    <mrow>
      <mo>(</mo>
      <mfrac>
        <mrow>
          <mn>1</mn>
          <mo>+</mo>
          <mfrac>
            <mn>2</mn>
            <mn>3</mn>
          </mfrac>
        </mrow>
        <mn>4</mn>
      </mfrac>
      <mo>)</mo>
    </mrow>
    <mn>5</mn>
  </msup>
</math>

$mrow example$

The <mrow> element accepts the attributes described in 2.1.3 Global Attributes. An <mrow> element with in-flow children child₁, child₂, …, child_N is laid out as shown on Figure 10 . The child boxes are put in a row one after the other with all their alphabetic baselines aligned.

Figure 10 Box model for the `<mrow>` element

Note

Because the box model ensures alignment of alphabetic baselines, fraction bars or symmetric stretchy operators will also be aligned along the math axis in the typical case when AxisHeight is the same for all in-flow children.

Figure 11 Symmetric and non-symmetric stretching of operators along the block axis

The algorithm for stretching operators along the block axis consists in the following steps:

If there is a block stretch size constraint or an inline stretch size constraint then the element being laid out is an embellished operator. Lay out the one in-flow child that is an embellished operator with the same stretch size constraint and all the other in-flow children without any stretch size constraint and stop.
Otherwise, split the list of in-flow children into a first list L_ToStretch containing embellished operators with a stretchy property and block stretch axis; and a second list L_NotToStretch.
Perform layout without any stretch size constraint on all the items of L_NotToStretch. If L_ToStretch is empty then stop. If L_NotToStretch is empty, perform layout with block stretch size constraint (0, 0) for all the items of L_ToStretch.
Calculate the unconstrained target sizes U_ascent and U_descent as respectively the maximum ink ascent and maximum ink descent of the margin boxes of in-flow children that have been laid out in the previous step.
Lay out or relayout all the elements of L_ToStretch with block stretch size constraint (U_ascent, U_descent).

If the box is not an anonymous <mrow> box and the associated element does not have its computed display property equal to block math or inline math then it is laid out according to the CSS specification where the corresponding value is described. Otherwise, the layout below is performed.

A child box is slanted if it is not an embellished operator and has nonzero italic correction.

Note

Large operators may have nonzero italic correction but that one is used when attaching scripts. More generally, all embellished operators are treated as non-slanted since the spacing around them is calculated as specified by lspace and rspace.

The min-content inline size (respectively max-content inline size) are calculated using the following algorithm:

Set add-space to true if the box corresponds to a math element or is not an embellished operator; and to false otherwise.
Set inline-offset to 0.
Set previous-italic-correction to 0.
For each in-flow child:
1. If the child is not slanted, then increment inline-offset by previous-italic-correction.
2. If the child is an embellished operator and add-space is true then increment inline-offset by its lspace property.
3. Increment inline-offset by the min-content inline size (respectively max-content inline size) of the child's margin box.
4. If the child is slanted then set previous-italic-correction to its italic correction. Otherwise set it to 0.
5. If the child is an embellished operator and add-space is true then increment inline-offset by its rspace property.
Increment inline-offset by previous-italic-correction.
Return inline-offset.

The in-flow children are laid out using the algorithm for stretching operators along the block axis.

The inline size of the math content is calculated like the min-content inline size and max-content inline size of the math content, using the inline size of the in-flow children's margin boxes instead.

The ink line-ascent (respectively line-ascent) of the math content is the maximum of the ink line-ascents (respectively line-ascents) of all the in-flow children's margin boxes. Similarly, the ink line-descent (respectively line-descent) of the math content is the maximum of the ink line-descents (respectively ink line-ascents) of all the in-flow children's margin boxes.

The in-flow children are positioned using the following algorithm:

Set add-space to true if the box corresponds to a math element or is not an embellished operator; and to false otherwise.
Set inline-offset to 0.
Set previous-italic-correction to 0.
For each in-flow child:
1. If the child is not slanted, then increment inline-offset by previous-italic-correction.
2. If the child is an embellished operator and add-space is true then increment inline-offset by its lspace property.
3. Set the inline offset of the child to inline-offset and its block offset such that the alphabetic baseline of the child is aligned with the alphabetic baseline.
4. Increment inline-offset by the inline size of the child's margin box.
5. If the child is slanted then set previous-italic-correction to its italic correction. Otherwise set it to 0.
6. If the child is an embellished operator and add-space is true then increment inline-offset by its rspace property.

The italic correction of the math content is set to the italic correction of the last in-flow child, which is the final value of previous-italic-correction.

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

If the <mfrac> element does not have its computed display property equal to block math or inline math then it is laid out according to the CSS specification where the corresponding value is described. Otherwise, the layout below is performed.

The <mfrac> element accepts the attributes described in 2.1.3 Global Attributes as well as the following attribute:

linethickness

The linethickness attribute indicates the fraction line thickness to use for the fraction bar. If present, it must have a value that is a valid <length-percentage>. If the attribute is absent or has an invalid value, FractionRuleThickness is used as the default value. A percentage is interpreted relative to that default value. A negative value is interpreted as 0.

The following example contains four fractions with different linethickness values. The bars are always aligned with the middle of plus and minus signs. The numerator and denominator are horizontally centered. The fractions that are not in displaystyle use smaller gaps and font-size.

<math>
  <mn>0</mn>
  <mo>+</mo>
  <mfrac displaystyle="true">
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mo>−</mo>
  <mfrac>
    <mn>1</mn>
    <mn>2</mn>
  </mfrac>
  <mo>+</mo>
  <mfrac linethickness="200%">
    <mn>1</mn>
    <mn>234</mn>
  </mfrac>
  <mo>−</mo>
  <mrow>
    <mo>(</mo>
    <mfrac linethickness="0">
      <mn>123</mn>
      <mn>4</mn>
    </mfrac>
    <mo>)</mo>
  </mrow>
</math>

$mfrac example$

The <mfrac> element sets displaystyle to false, or if it was already false increments scriptlevel by 1, within its children. It sets math-shift to compact within its second child. To avoid visual confusion between the fraction bar and another adjacent items (e.g. minus sign or another fraction's bar), a default 1-pixel space is added around the element. The user agent stylesheet must contain the following rules:

mfrac {
  padding-inline: 1px;
}
mfrac > * {
  math-depth: auto-add;
  math-style: compact;
}
mfrac > :nth-child(2) {
  math-shift: compact;
}

If the <mfrac> element has less or more than two in-flow children, its layout algorithm is the same as the mrow element. Otherwise, the first in-flow child is called numerator , the second in-flow child is called denominator and the layout algorithm is explained below.

Note

In practice, an <mfrac> element has two children that are in-flow. Hence the CSS rules basically perform scriptlevel, displaystyle and math-shift changes for the numerator and denominator.

If the fraction line thickness is nonzero, the <mfrac> element is laid out as shown on Figure 12 . The fraction bar must only be painted if the visibility of the <mfrac> element is visible. In that case, the fraction bar must be painted with the color of the <mfrac> element.

Figure 12 Box model for the `<mfrac>` element

The min-content inline size (respectively max-content inline size) of content is the maximum between the min-content inline size (respectively max-content inline size) of the numerator's margin box and the min-content inline size (respectively max-content inline size) of the denominator's margin box.

If there is an inline stretch size constraint or a block stretch size constraint then the numerator is also laid out with the same stretch size constraint, otherwise it is laid out without any stretch size constraint. The denominator is always laid out without any stretch size constraint.

The inline size of the math content is the maximum between the inline size of the numerator's margin box and the inline size of the denominator's margin box.

NumeratorShift is the maximum between:

FractionNumeratorShiftUp (respectively FractionNumeratorDisplayStyleShiftUp) if the math-style is compact (respectively normal).
The AxisHeight + half the fraction line thickness + FractionNumeratorGapMin (respectively FractionNumDisplayStyleGapMin) if the math-style is compact (respectively normal) + the ink line-descent of the numerator's margin box.

DenominatorShift is the maximum between:

FractionDenominatorShiftDown (respectively FractionDenominatorDisplayStyleShiftDown) if the math-style is compact (respectively normal).
Half the fraction line thickness + FractionDenominatorGapMin (respectively FractionDenomDisplayStyleGapMin) if the math-style is compact (respectively normal) + the ink line-ascent of the denominator's margin box − the AxisHeight.

The line-ascent of the math content is the maximum between:

Numerator Shift + the line-ascent of the numerator's margin box.
−Denominator Shift + the line-ascent of the denominator's margin box
AxisHeight plus half the fraction line thickness.

The line-descent of the math content is the maximum between:

−Numerator Shift + the line-descent of the numerator's margin box.
Denominator Shift + the line-descent of the denominator's margin box.
−AxisHeight plus half the fraction line thickness.
0

The inline offset of the numerator (respectively denominator) is half the inline size of the math content − half the inline size of the numerator's margin box (respectively denominator's margin box).

The alphabetic baseline of the numerator (respectively denominator) is shifted away from the alphabetic baseline by a distance of NumeratorShift (respectively DenominatorShift) towards the line-over (respectively line-under).

The math content box is placed within the content box so that their block-start edges are aligned and the middles of these edges are at the same position.

The inline size of the fraction bar is the inline size of the content box and its inline-start edge is the aligned with the one the content box. The center of the fraction bar is shifted away from the alphabetic baseline of the math content box by a distance of AxisHeight towards the line-over. Its block size is the fraction line thickness.

If the fraction line thickness is zero, the <mfrac> element is instead laid out as shown on Figure 13 .

Figure 13 Box model for the `<mfrac>` element without bar

The min-content inline size, max-content inline size and inline size of the math content are calculated the same as in 3.3.2.1 Fraction with nonzero line thickness.

If there is an inline stretch size constraint or a block stretch size constraint then the numerator is also laid out with the same stretch size constraint and otherwise it is laid out without any stretch size constraint. The denominator is always laid out without any stretch size constraint.

If the math-style is compact then TopShift and BottomShift are respectively set to StackTopShiftUp and StackBottomShiftDown. Otherwise math-style is normal and they are respectively set to StackTopDisplayStyleShiftUp and StackBottomDisplayStyleShiftDown.

The Gap is defined to be (BottomShift − the ink line-ascent of the denominator's margin box) + (TopShift − the ink line-descent of the numerator's margin box). If math-style is compact then GapMin is StackGapMin, otherwise math-style is normal and it is StackDisplayStyleGapMin. If Δ = GapMin − Gap is positive then TopShift and BottomShift are respectively increased by Δ/2 and Δ − Δ/2.

The line-ascent of the math content is the maximum between:

TopShift + the line-ascent of the numerator's margin box.
−BottomShift + the line-ascent of the denominator's margin box.

The line-descent of the math content is the maximum between:

−TopShift + the line-descent of the numerator's margin box.
BottomShift + the line-descent of the denominator's margin box.
0

The inline offsets of the numerator and denominator are calculated the same as in 3.3.2.1 Fraction with nonzero line thickness.

The alphabetic baseline of the numerator (respectively denominator) is shifted away from the alphabetic baseline by a distance of TopShift (respectively − BottomShift) towards the line-over (respectively line-under).

The math content box is placed within the content box so that their block-start edges are aligned and the middles of these edges are at the same position.

The radical elements construct an expression with a root symbol √ with a line over the content. The msqrt element is used for square roots, while the mroot element is used to draw radicals with indices, e.g. a cube root.

The <msqrt> and <mroot> elements accept the attributes described in 2.1.3 Global Attributes.

The following example contains a square root written with msqrt and a cube root written with mroot. Note that msqrt has several children and the square root applies to all of them. mroot has exactly two children: it is a root of index the second child (the number 3), applied to the first child (the square root). Also note these elements only change the font-size within the mroot index, but it is scaled down more than within the numerator and denumerator of the fraction.

<math>
  <mroot>
    <msqrt>
      <mfrac>
        <mn>1</mn>
        <mn>2</mn>
      </mfrac>
      <mo>+</mo>
      <mn>4</mn>
    </msqrt>
    <mn>3</mn>
  </mroot>
  <mo>+</mo>
  <mn>0</mn>
</math>

$msqrt-mroot example$

The <msqrt> and <mroot> elements sets math-shift to compact. The <mroot> element increments scriptlevel by 2, and sets displaystyle to "false" in all but its first child. The user agent stylesheet must contain the following rule in order to implement that behavior:

mroot > :not(:first-child) {
  math-depth: add(2);
  math-style: compact;
}
mroot, msqrt {
  math-shift: compact;
}

If the <msqrt> or <mroot> element do not have their computed display property equal to block math or inline math then they are laid out according to the CSS specification where the corresponding value is described. Otherwise, the layout below is performed.

If the <mroot> has less or more than two in-flow children, its layout algorithm is the same as the mrow element. Otherwise, the first in-flow child is called mroot base and the second in-flow child is called mroot index and its layout algorithm is explained below.

Note

In practice, an <mroot> element has two children that are in-flow. Hence the CSS rules basically perform scriptlevel and displaystyle changes for the index.

The <msqrt> element generates an anonymous <mrow> box called the msqrt base .

The radical symbol must only be painted if the visibility of the <msqrt> or <mroot> element is visible. In that case, the radical symbol must be painted with the color of that element.

The radical glyph is the glyph obtained for the character U+221A SQUARE ROOT.

The radical gap is given by RadicalVerticalGap if the math-style is compact and RadicalDisplayStyleVerticalGap if the math-style is normal.

The radical target size for the stretchy radical glyph is the sum of RadicalRuleThickness, radical gap and the ink height of the base.

The box metrics of the radical glyph and painting of the surd are given by the algorithm to shape a stretchy glyph to block dimension the target size for the radical glyph.

The <msqrt> element is laid out as shown on Figure 14 .

Figure 14 Box model for the `<msqrt>` element

The min-content inline size (respectively max-content inline size) of the math content is the sum of the preferred inline size of a glyph stretched along the block axis for the radical glyph and of the min-content inline size (respectively max-content inline size) of the msqrt base's margin box.

The inline size of the math content is the sum of the advance width of the box metrics of the radical glyph and of the inline size of the msqrt base's margin's box.

The line-ascent of the math content is the maximum between:

The line-ascent of the msqrt base's margin box.
The sum of the ink line-ascent of the msqrt base, the radical gap, the RadicalRuleThickness and RadicalExtraAscender.

The line-descent of the math content is the maximum between:

The line-descent of the msqrt base's margin box.
The sum of the line-ascent and line-descent for the box metrics of the radical glyph and of the RadicalExtraAscender, minus the line-ascent of the math content.

The inline size of the overbar is the inline size of the msqrt base's margin's box. The inline offsets of the msqrt base and overbar are also the same and equal to the width of the box metrics of the radical glyph.

The alphabetic baseline of the msqrt base is aligned with the alphabetic baseline. The block size of the overbar is RadicalRuleThickness. Its vertical center is shifted away from the alphabetic baseline by a distance towards the line-over equal to the line-ascent of the math content, minus the RadicalExtraAscender, minus half the RadicalRuleThickness.

Finally, the painting of the surd is performed:

At inline offset 0.
At block offset shifted away from the line-over edge of the overbar towards the line-under by a distance given by the ascent of the box metrics of the radical glyph.

The <mroot> element is laid out as shown on Figure 15 . The mroot index is first ignored and the mroot base and radical glyph are laid out as shown on figure Figure 14 using the same algorithm as in 3.3.3.2 Square root in order to produce a margin box B (represented in green).

Figure 15 Box model for the `<mroot>` element

The min-content inline size (respectively max-content inline size) of the math content is the sum of max(0, RadicalKernBeforeDegree), the mroot index's min-content inline size (respectively max-content inline size) of the mroot index's margin box, max(−min-content inline size, RadicalKernAfterDegree) (respectively max(−max-content inline size of the mroot index's margin box, RadicalKernAfterDegree)) and of the min-content inline size (respectively max-content inline size) of B.

Using the same clamping, AdjustedRadicalKernBeforeDegree and AdjustedRadicalKernAfterDegree are respectively defined as max(0, RadicalKernBeforeDegree) and is max(−inline size of the index's margin box, RadicalKernAfterDegree).

The inline size of the math content is the sum of AdjustedRadicalKernBeforeDegree, the inline size of the index's margin box, AdjustedRadicalKernAfterDegree and of the inline size of B.

The line-ascent of the math content is the maximum between:

−the line-descent of B + RadicalDegreeBottomRaisePercent × the block size of B + the line-ascent of the index's margin box
The line-ascent of B

The line-descent of the math content is the maximum between:

the line-descent of B − RadicalDegreeBottomRaisePercent × the block size of B + the line-ascent of the index's margin box
The line-descent of B

The inline offset of the index is AdjustedRadicalKernBeforeDegree. The inline-offset of the mroot base is the same + the inline size of the index's margin box.

The alphabetic baseline of B is aligned with the alphabetic baseline. The alphabetic baseline of the index is shifted away from the line-under edge by a distance of RadicalDegreeBottomRaisePercent × the block size of B + the line-descent of the index's margin box.

Note

In general, the kerning before the root index is positive while the kerning after it is negative, which means that the root element will have some inline-start space and that the root index will overlap the surd.

Historically, the mstyle element was introduced to make style changes that affect the rendering of its contents.

The <mstyle> element accepts the attributes described in 2.1.3 Global Attributes. Its layout algorithm is the same as the mrow element.

Note

<mstyle> is implemented for compatibility with full MathML. Authors whose only target is MathML Core are encouraged to use CSS for styling.

In the following example, mstyle is used to set the scriptlevel and displaystyle. Observe this is respectively affecting the font-size and placement of subscripts of their descendants. In MathML Core, one could just have used mrow elements instead.

<math>
  <munder>
    <mo movablelimits="true">*</mo>
    <mi>A</mi>
  </munder>
  <mstyle scriptlevel="1">
    <mstyle displaystyle="true">
      <munder>
        <mo movablelimits="true">*</mo>
        <mi>B</mi>
      </munder>
      <munder>
        <mo movablelimits="true">*</mo>
        <mi>C</mi>
      </munder>
    </mstyle>
    <munder>
      <mo movablelimits="true">*</mo>
      <mi>D</mi>
    </munder>
  </mstyle>
</math>

$mstyle example$

The merror element displays its contents as an ”error message”. The intent of this element is to provide a standard way for programs that generate MathML from other input to report syntax errors in their input.

In the following example, merror is used to indicate a parsing error for some LaTeX-like input:

<math>
  <mfrac>
    <merror>
      <mtext>Syntax error: \frac{1}</mtext>
    </merror>
    <mn>3</mn>
  </mfrac>
</math>

$merror example$

The <merror> element accepts the attributes described in 2.1.3 Global Attributes. Its layout algorithm is the same as the mrow element. The user agent stylesheet must contain the following rule in order to visually highlight the error message:

merror {
  border: 1px solid red;
  background-color: lightYellow;
}

The mpadded element renders the same as its in-flow child content, but with the size and relative positioning point of its content modified according to <mpadded>’s attributes.

The <mpadded> element accepts the attributes described in 2.1.3 Global Attributes as well as the following attributes:

The width , height , depth , lspace and voffset if present, must have a value that is a valid <length-percentage>.

In the following example, mpadded is used to tweak spacing around a fraction (a blue background is used to visualize it). Without attributes, it behaves like an mrow but the attributes allow to specify the size of the box (width, height, depth) and position of the fraction within that box (lspace and voffset).

<math>
  <mrow>
    <mn>1</mn>
    <mpadded style="background: lightblue;">
      <mfrac>
        <mn>23456</mn>
        <mn>78</mn>
      </mfrac>
    </mpadded>
    <mn>9</mn>
  </mrow>
  <mo>+</mo>
  <mrow>
    <mn>1</mn>
    <mpadded lspace="2em" voffset="-1em" height="1em" depth="3em" width="7em"
             style="background: lightblue;">
      <mfrac>
        <mn>23456</mn>
        <mn>78</mn>
      </mfrac>
    </mpadded>
    <mn>9</mn>
  </mrow>
</math>

$mpadded example$

The mpadded element generates an anonymous <mrow> box called the mpadded inner box with parameters called inner inline size, inner line-ascent and inner line-descent.

The requested <mpadded> parameters are determined as follows:

The requested width is the resolved value of the width property. If the width attribute is present, valid and not a percentage then that attribute is used as a presentational hint setting the element's width property to the corresponding value.
If the height attribute is absent, invalid or a percentage then the requested height is the inner line-ascent. Otherwise the requested height is the resolved value of the height attribute, clamping negative values to 0.
If the depth attribute is absent, invalid or a percentage then the requested depth is the inner line-ascent. Otherwise the requested depth is the resolved value of the depth attribute, clamping negative values to 0.
If the lspace attribute is absent, invalid or a percentage then the requested lspace is 0. Otherwise the requested lspace is the resolved value of the lspace attribute, clamping negative values to 0.
If the voffset attribute is absent, invalid or a percentage then the requested voffset is 0. Otherwise the requested voffset is the resolved value of the voffset attribute.
Note
Negative voffset values are not clamped to 0.

If the <mpadded> element does not have its computed display property equal to block math or inline math then it is laid out according to the CSS specification where the corresponding value is described. Otherwise, it is laid out as shown on Figure 16 .

Figure 16 Box model for the `<mpadded>` element

The min-content inline size (respectively max-content inline size) of the math content is the requested width calculated in 3.3.6.1 Inner box and requested parameters but using the min-content inline size (respectively max-content inline size) of the mpadded inner box instead of the "inner inline size".

The inline size of the math content is the requested width calculated in 3.3.6.1 Inner box and requested parameters.

The line-ascent of the math content is the requested height. The line-descent of the math content is the requested depth.

The mpadded inner box is placed so that its alphabetic baseline is shifted away from the alphabetic baseline by the requested voffset towards the line-over.

Historically, the mphantom element was introduced to render its content invisibly, but with the same metrics size and other dimensions, including alphabetic baseline position that its contents would have if they were rendered normally.

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

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

$mphantom example$

The <mphantom> element accepts the attributes described in 2.1.3 Global Attributes. Its layout algorithm is the same as the mrow element. The user agent stylesheet must contain the following rule in order to hide the content:

mphantom {
  visibility: hidden;
}

Note

<mphantom> is implemented for compatibility with full MathML. Authors whose only target is MathML Core are encouraged to use CSS for styling.

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

In addition to sub-/superscript elements, MathML has overscript and underscript elements that place scripts above and below the base. These elements can be used to place limits on large operators, or for placing accents and lines above or below the base.

The msub , msup and msubsup elements are used to attach subscript and superscript to a MathML expression. They accept the attributes described in 2.1.3 Global Attributes.

The following example shows basic use of subscripts and superscripts. The font-size is automatically scaled down within the scripts.

<math>
  <msub>
    <mn>1</mn>
    <mn>2</mn>
  </msub>
  <mo>+</mo>
  <msup>
    <mn>3</mn>
    <mn>4</mn>
  </msup>
  <mo>+</mo>
  <msubsup>
    <mn>5</mn>
    <mn>6</mn>
    <mn>7</mn>
  </msubsup>
</math>

$msub-msup-msubsup example$

If the <msub>, <msup> or <msubsup> elements do not have their computed display property equal to block math or inline math then they are laid out according to the CSS specification where the corresponding value is described. Otherwise, the layout below is performed.

If the <msub> element has less or more than two in-flow children, its layout algorithm is the same as the mrow element. Otherwise, the first in-flow child is called the msub base , the second in-flow child is called the msub subscript and the layout algorithm is explained in 3.4.1.2 Base with subscript.

If the <msup> element has less or more than two in-flow children, its layout algorithm is the same as the mrow element. Otherwise, the first in-flow child is called the msup base , the second in-flow child is called the msup superscript and the layout algorithm is explained in 3.4.1.3 Base with superscript.

If the <msubsup> element has less or more than three in-flow children, its layout algorithm is the same as the mrow element. Otherwise, the first in-flow child is called the msubsup base , the second in-flow child is called the msubsup subscript , its third in-flow child is called the msubsup superscript and the layout algorithm is explained in 3.4.1.4 Base with subscript and superscript.

The <msub> element is laid out as shown on Figure 17 . LargeOpItalicCorrection is the italic correction of the msub base if it is an embellished operator with the largeop property and 0 otherwise.

Figure 17 Box model for the `<msub>` element

The min-content inline size (respectively max-content inline size) of the math content is the min-content inline size (respectively max-content inline size) of the msub base's margin box − LargeOpItalicCorrection + min-content inline size (respectively max-content inline size) of the msub subscript's margin box + SpaceAfterScript.

If there is an inline stretch size constraint or a block stretch size constraint then the msub base is also laid out with the same stretch size constraint and otherwise it is laid out without any stretch size constraint. The scripts are always laid out without any stretch size constraint.

The inline size of the math content is the inline size of the msub base's margin box − LargeOpItalicCorrection + the inline size of the msub subscript's margin box + SpaceAfterScript.

SubShift is the maximum between:

SubscriptShiftDown
the ink line-ascent of the subscript's margin box − SubscriptTopMax
SubscriptBaselineDropMin + the ink line-descent of the msub base's margin box

The line-ascent of the math content is the maximum between:

The line-ascent of the msub base's margin box.
The line-ascent of the subscript's margin box − SubShift.

The line-descent of the math content is the maximum between:

The line-descent of the msub base's margin box.
The line-descent of the subscript's margin box + SubShift.

The inline offset of the msub base is 0 and the inline offset of the msub subscript is the inline size of the msub base's margin box − LargeOpItalicCorrection.

The msub base is placed so that its alphabetic baseline matches the alphabetic baseline. The msub subscript is placed so that its alphabetic baseline is shifted away from the alphabetic baseline by SubShift towards the line-under.

The <msup> element is laid out as shown on Figure 18 . ItalicCorrection is the italic correction of the msup base if it is not an embellished operator with the largeop property and 0 otherwise.

Figure 18 Box model for the `<msup>` element

The min-content inline size (respectively max-content inline size) of the math content is the min-content inline size (respectively max-content inline size) of the msup base's margin box + ItalicCorrection + the min-content inline size (respectively max-content inline size) of the msup superscript's margin box + SpaceAfterScript.

If there is an inline stretch size constraint or a block stretch size constraint then the msup base is also laid out with the same stretch size constraint and otherwise it is laid out without any stretch size constraint. The scripts are always laid out without any stretch size constraint.

The inline size of the math content is the inline size of the msup base's margin box + ItalicCorrection + the inline size of the msup superscript's margin box + SpaceAfterScript.

SuperShift is the maximum between:

The value SuperscriptShiftUpCramped if the element has a computed math-shift property equal to compact, or SuperscriptShiftUp otherwise.
SuperscriptBottomMin + the ink line-descent of the superscript's margin box
The ink line-ascent of the msup base's margin box − SuperscriptBaselineDropMax

The line-ascent of the math content is the maximum between:

The line-ascent of the msup base's margin box.
The line-ascent of the superscript's margin box + SuperShift.

The line-descent of the math content is the maximum between:

The line-descent of the msup base's margin box.
The line-descent of the superscript's margin box − SuperShift.

The inline offset of the msup base is 0 and the inline offset of msup superscript is the inline size of the msup base's margin box + ItalicCorrection.

The msup base is placed so that its alphabetic baseline matches the alphabetic baseline. The msup superscript is placed so that its alphabetic baseline is shifted away from the alphabetic baseline by SuperShift towards the line-over.

The <msubsup> element is laid out as shown on Figure 18 . LargeOpItalicCorrection and SubShift are set as in 3.4.1.2 Base with subscript. ItalicCorrection and SuperShift are set as in 3.4.1.3 Base with superscript.

Figure 19 Box model for the `<msubsup>` element

The min-content inline size (respectively max-content inline size and inline size) of the math content is the maximum between the min-content inline size (respectively max-content inline size and inline size) of the math content calculated in 3.4.1.2 Base with subscript and 3.4.1.3 Base with superscript.

If there is an inline stretch size constraint or a block stretch size constraint then the msubsup base is also laid out with the same stretch size constraint and otherwise it is laid out without any stretch size constraint. The scripts are always laid out without any stretch size constraint.

SubSuperGap is the gap between the two scripts along the block axis and is defined by (SubShift − the ink line-ascent of the msubsup subscript's margin box) + (SuperShift − the ink line-descent of the msubsup superscript's margin box). If SubSuperGap is not at least SubSuperscriptGapMin then the following steps are performed to ensure that the condition holds:

Let Δ be SuperscriptBottomMaxWithSubscript − (SuperShift − the ink line-descent of the msubsup superscript's margin box). If Δ > 0 then set Δ to the minimum between Δ set SubSuperscriptGapMin − SubSuperGap and increase SuperShift (and so SubSuperGap too) by Δ.
Let Δ be SubSuperscriptGapMin − SubSuperGap. If Δ > 0 then increase SubscriptShift (and so SubSuperGap too) by Δ.

The ink line-ascent (respectively line-ascent, ink line-descent, line-descent) of the math content is set to the maximum of the ink line-ascent (respectively line-ascent, ink line-descent, line-descent) of the math content calculated in 3.4.1.2 Base with subscript and 3.4.1.3 Base with superscript but using the adjusted values SubShift and SuperShift above.

The inline offset and block offset of the msubsup base and scripts are performed the same as described in 3.4.1.2 Base with subscript and 3.4.1.3 Base with superscript.

Note

Even when the msubsup subscript (respectively msubsup superscript) is an empty box, <msubsup> does not generally render the same as 3.4.1.3 Base with superscript (respectively 3.4.1.2 Base with subscript) because of the additional constraint on SubSuperGap. Moreover, positioning the empty msubsup subscript (respectively msubsup superscript) may also change the total size.

In order to keep the algorithm simple, no attempt is made to handle empty scripts in a special way.

The munder , mover and munderover elements are used to attach accents or limits placed under or over a MathML expression.

The <munderover> element accepts the attribute described in 2.1.3 Global Attributes as well as the following attributes:

accent
accentunder

Similarly, the <mover> element (respectively <munder> element) accepts the attribute described in 2.1.3 Global Attributes as well as the accent attribute (respectively the accentunder attribute).

accent , accentunder attributes, if present, must have values that are booleans. If these attributes are absent or invalid, they are treated as equal to false. User agents must implement them as described in 3.4.4 Displaystyle, scriptlevel and math-shift in scripts.

The following example shows basic use of under- and overscripts. The font-size is automatically scaled down within the scripts, unless they are meant to be accents.

<math>
  <munder>
    <mn>1</mn>
    <mn>2</mn>
  </munder>
  <mo>+</mo>
  <mover>
    <mn>3</mn>
    <mn>4</mn>
  </mover>
  <mo>+</mo>
  <munderover>
    <mn>5</mn>
    <mn>6</mn>
    <mn>7</mn>
  </munderover>
  <mo>+</mo>
  <munderover accent="true">
    <mn>8</mn>
    <mn>9</mn>
    <mn>10</mn>
  </munderover>
  <mo>+</mo>
  <munderover accentunder="true">
    <mn>11</mn>
    <mn>12</mn>
    <mn>13</mn>
  </munderover>
</math>

$munder-over-munderover example$

If the <munder>, <mover> or <munderover> elements do not have their computed display property equal to block math or inline math then they are laid out according to the CSS specification where the corresponding value is described. Otherwise, the layout below is performed.

If the <munder> element has less or more than two in-flow children, its layout algorithm is the same as the mrow element. Otherwise, the first in-flow child is called the munder base and the second in-flow child is called the munder underscript .

If the <mover> element has less or more than two in-flow children, its layout algorithm is the same as the mrow element. Otherwise, the first in-flow child is called the mover base and the second in-flow child is called the mover overscript .

If the <munderover> element has less or more than three in-flow children, its layout algorithm is the same as the mrow element. Otherwise, the first in-flow child is called the munderover base , the second in-flow child is called the munderover underscript and its third in-flow child is called the munderover overscript .

If the <munder>, <mover> or <munderover> elements have a computed math-style property equal to compact and their base is an embellished operator with the movablelimits property, then their layout algorithms are respectively the same as the ones described for <msub>, <msup> and <msubsup> in 3.4.1.2 Base with subscript, 3.4.1.3 Base with superscript and 3.4.1.4 Base with subscript and superscript.

Otherwise, the <munder>, <mover> and <munderover> layout algorithms are respectively described in 3.4.2.3 Base with underscript, 3.4.2.4 Base with overscript and 3.4.2.5 Base with underscript and overscript.

The algorithm for stretching operators along the inline axis is as follows.

If there is an inline stretch size constraint or block stretch size constraint then the element being laid out is an embellished operator. Lay out the base with the same stretch size constraint.
Split the list of in-flow children that have not been laid out yet into a first list L_ToStretch containing embellished operators with a stretchy property and inline stretch axis; and a second list L_NotToStretch.
Perform layout without any stretch size constraint on all the items of L_NotToStretch. If L_ToStretch is empty then stop. If L_NotToStretch is empty, perform layout with inline stretch size constraint 0 for all the items of L_ToStretch.
Calculate the target size T to the maximum inline size of the margin boxes of child boxes that have been laid out in the previous step.
Lay out or relayout all the elements of L_ToStretch with inline stretch size constraint T.

The <munder> element is laid out as shown on Figure 20 . LargeOpItalicCorrection is the italic correction of the munder base if it is an embellished operator with the largeop property and 0 otherwise.

Figure 20 Box model for the `<munder>` element

The min-content inline size (respectively max-content inline size) of the math content are calculated like the inline size of the math content below but replacing the inline sizes of the munder base's margin box and munder underscript's margin box with the min-content inline size (respectively max-content inline size) of the munder base's margin box and munder underscript's margin box.

The in-flow children are laid out using the algorithm for stretching operators along the inline axis.

The inline size of the math content is calculated by determining the absolute difference between:

The maximum of:
- Half the inline size of the munder base's margin box.
- Half the inline size of the munder underscript's margin box − half LargeOpItalicCorrection.
The minimum of:
- −half the inline size of the munder base's margin box.
- −half the inline size of the munder underscript's margin box − half LargeOpItalicCorrection.

If m is the minimum calculated in the second item above then the inline offset of the munder base is −m − half the inline size of the base's margin box. The inline offset of the munder underscript is −m − half the inline size of the munder underscript's margin box − half LargeOpItalicCorrection.

Parameters UnderShift and UnderExtraDescender are determined by considering three cases in the following order:

The munder base is an embellished operator with the largeop property. UnderShift is the maximum of
- LowerLimitBaselineDropMin
- LowerLimitGapMin + the ascent of the munder underscript.
UnderExtraDescender is 0.
The munder base is an embellished operator with the stretchy property and stretch axis inline. UnderShift is the maximum of:
- StretchStackBottomShiftDown
- StretchStackGapAboveMin + the ascent of the munder underscript.
UnderExtraDescender is 0.
Otherwise, UnderShift is equal to UnderbarVerticalGap if the accentunder attribute is not an ASCII case-insensitive match to true and to zero otherwise. UnderExtraAscender is UnderbarExtraDescender.

The line-ascent of the math content is the maximum between:

The line-ascent of the munder base's margin box.
The line-ascent of the munder underscript's margin box − UnderShift.

The line-descent of the math content is the maximum between:

The line-descent of the munder base's margin box.
The line-descent of the munder underscript's margin box + UnderShift + UnderExtraAscender.

The alphabetic baseline of the munder base is aligned with the alphabetic baseline. The alphabetic baseline of the munder underscript is shifted away from the alphabetic baseline and towards the line-under by a distance equal to the ink line-descent of the munder base's margin box + UnderShift.

The math content box is placed within the content box so that their block-start edges are aligned and the middles of these edges are at the same position.

The <mover> element is laid out as shown on Figure 21 . LargeOpItalicCorrection is the italic correction of the mover base if it is an embellished operator with the largeop property and 0 otherwise.

Figure 21 Box model for the `<mover>` element

The min-content inline size (respectively max-content inline size) of the math content are calculated like the inline size of the math content below but replacing the inline sizes of the mover base's margin box and mover overscript's margin box with the min-content inline size (respectively max-content inline size) of the mover base's margin box and mover overscript's margin box.

The in-flow children are laid out using the algorithm for stretching operators along the inline axis.

The TopAccentAttachment is the top accent attachment of the mover overscript or half the inline size of the mover overscript's margin box if it is undefined.

The inline size of the math content is calculated by applying the algorithm for stretching operators along the inline axis for layout and determining the absolute difference between:

The maximum of:
- Half the inline size of the mover base's margin box.
- The inline size of the mover overscript's margin box − TopAccentAttachment + half LargeOpItalicCorrection.
The minimum of:
- −half the inline size of the mover base's margin box.
- −TopAccentAttachment + half LargeOpItalicCorrection.

If m is the minimum calculated in the second item above then the inline offset of the mover base is −m − half the inline size of the base's margin. The inline offset of the mover overscript is −m − half the inline size of the mover overscript's margin box + half LargeOpItalicCorrection.

Parameters OverShift and OverExtraDescender are determined by considering three cases in the following order:

The mover base is an embellished operator with the largeop property. OverShift is the maximum of
- UpperLimitBaselineRiseMin
- UpperLimitGapMin + the descent of the mover overscript.
OverExtraAscender is 0.
The mover base is an embellished operator with the stretchy property and stretch axis inline. OverShift is the maximum of:
- StretchStackTopShiftUp
- StretchStackGapBelowMin + the descent of the mover overscript.
OverExtraDescender is 0.
Otherwise, OverShift is equal to
1. OverbarVerticalGap if the accent attribute is not an ASCII case-insensitive match to true.
2. Or AccentBaseHeight minus the line-ascent of the mover base's margin box if this difference is nonnegative.
3. Or 0 otherwise.
OverExtraAscender is OverbarExtraAscender.

Note

For accent overscripts and bases with line-ascents that are at most AccentBaseHeight, the rule from [OPEN-FONT-FORMAT] [TEXBOOK] is actually to align the alphabetic baselines of the overscripts and of the bases. This assumes that accent glyphs are designed in such a way that their ink bottoms are more or less AccentBaseHeight above their alphabetic baselines. Hence, the previous rule will guarantee that all the overscript bottoms are aligned while still avoiding collision with the bases. However, MathML can have arbitrary accent overscripts, so a more general and simpler rule is provided above: Ensure that the bottom of overscript is at least AccentBaseHeight above the alphabetic baseline of the base.

The line-ascent of the math content is the maximum between:

The line-ascent of the mover base's margin box.
The line-ascent of the mover overscript's margin box + OverShift + OverExtraAscender.

The line-descent of the math content is the maximum between:

The line-descent of the mover base's margin box.
The line-descent of the mover overscript's margin box − OverShift.

The alphabetic baseline of the mover base is aligned with the alphabetic baseline. The alphabetic baseline of the mover overscript is shifted away from the alphabetic baseline and towards the line-over by a distance equal to the ink line-ascent of the base + OverShift.

The math content box is placed within the content box so that their block-start edges are aligned and the middles of these edges are at the same position.

The general layout of <munderover> is shown on Figure 22 . The LargeOpItalicCorrection, UnderShift, UnderExtraDescender, OverShift, OverExtraDescender parameters are calculated the same as in 3.4.2.3 Base with underscript and 3.4.2.4 Base with overscript.

Figure 22 Box model for the `<munderover>` element

The min-content inline size, max-content inline size and inline size of the math content are calculated as an absolute difference between a maximum inline offset and minimum inline offset. These extrema are calculated by taking the extremum value of the corresponding extrema calculated in 3.4.2.3 Base with underscript and 3.4.2.4 Base with overscript. The inline offsets of the munderover base, munderover underscript and munderover overscript are calculated as in these sections but using the new minimum m (minimum of the corresponding minima).

Like in these sections, the in-flow children are laid out using the algorithm for stretching operators along the inline axis.

The line-ascent and line-descent of the math content are also calculated by taking the extremum value of the extrema calculated in 3.4.2.3 Base with underscript and 3.4.2.4 Base with overscript.

Finally, the alphabetic baselines of the munderover base, munderover underscript and munderover overscript are calculated as in sections 3.4.2.3 Base with underscript and 3.4.2.4 Base with overscript.

The math content box is placed within the content box so that their block-start edges are aligned and the middles of these edges are at the same position.

Note

When the underscript (respectively overscript) is an empty box, the base and overscript (respectively underscript) are laid out similarly to 3.4.2.4 Base with overscript (respectively 3.4.2.3 Base with underscript) but the position of the empty underscript (respectively overscript) may add extra space. In order to keep the algorithm simple, no attempt is made to handle empty scripts in a special way.

Presubscripts and tensor notations are represented by the mmultiscripts element. The mprescripts element is used as a separator between the postscripts and prescripts. These two elements accept the attributes described in 2.1.3 Global Attributes.

The following example shows basic use of prescripts and postscripts, involving a mprescripts. Empty mrow elements are used at positions where no scripts are rendered. The font-size is automatically scaled down within the scripts.

<math>
  <mmultiscripts>
    <mn>1</mn>
    <mn>2</mn>
    <mn>3</mn>
    <mrow></mrow>
    <mn>5</mn>
    <mprescripts/>
    <mn>6</mn>
    <mrow></mrow>
    <mn>8</mn>
    <mn>9</mn>
  </mmultiscripts>
</math>

$mmultiscripts example$

If the <mmultiscripts> or <mprescripts> elements do not have their computed display property equal to block math or inline math then they are laid out according to the CSS specification where the corresponding value is described. Otherwise, the layout below is performed.

The <mprescripts> element is laid out as an mrow element.

A valid <mmultiscripts> element contains the following in-flow children:

A first in-flow child, called the mmultiscripts base , that is not an mprescripts element.
Followed by an even number of in-flow children called mmultiscripts postscripts , none of them being a mprescripts element. These scripts form a (possibly empty) list subscript, superscript, subscript, superscript, subscript, superscript, etc. Each consecutive couple of children subscript, superscript is called a subscript/superscript pair .
Optionally followed by an mprescripts element and an even number of in-flow children called mmultiscripts prescripts , none of them being a mprescripts element. These scripts form a (possibly empty) list of subscript/superscript pair.

If an <mmultiscripts> element is not valid then it is laid out the same as the mrow element. Otherwise the layout algorithm is performed as in "https://www.w3.org/TR/mathml-

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