XML Entity Definitions for Characters

W3C Proposed Recommendation 11 February 2010

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David Carlisle, NAG
Patrick Ion, Mathematical Reviews, American Mathematical Society


This document defines several sets of names, so that to each name is assigned a Unicode character or sequence of characters. Each of these sets is expressed as a file of XML entity declarations.

Status of this Document

This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.

This document was produced by the W3C Math Working Group as a Proposed Recommendation and as part of the W3C Math Activity. The goals of the W3C Math Working Group are discussed in the W3C Math WG Charter. The authors of this document are W3C Math Working Group members.

A Proposed Recommendation is a specification that is under review by the W3C Advisory Committee for endorsement as a W3C Recommendation. It is is a mature document that has been widely reviewed and has been shown to be implementable. W3C encourages everybody to implement this specification.

Publication as a Proposed Recommendation does not imply endorsement by the W3C Membership. This is a draft document and may be updated, replaced or obsoleted by other documents at any time. It is inappropriate to cite this document as other than work in progress.

The previous version of this document was the W3C Last Call Working Draft of the XML Entity Definitions for Characters. The Last Call period ended on 8 December 2009. The Dispositions of comments on the last draft explains the changes that were made since that draft.

W3C asks members of the Advisory Committee to review this document and fill in the review form. The deadline for this is 11 March 2010. The document did not undergo Candidate Recommendation review, because no software is needed to implement the specification other than a standard XML parser.

Comments should be sent to the Public W3C Math mailing list (list archives; see also instructions). When sending an e-mail comment on the XML Entity Definitions for Characters, please put the text “XML-Entities” in the subject line, preferably like this: “[XML-Entities] …summary of comment ”.

This document was produced by a group operating under the 5 February 2004 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.

Table of Contents

1 Introduction
2 Sets of names
3 Unicode Character Ranges for Scientific Documents
4 Mathematical Alphanumeric Characters
5 Entities for Negated and Variant Characters
    5.1 Negated Mathematical Characters
    5.2 Variant Mathematical Characters


A Special Considerations
    A.1 Epsilon
    A.2 Phi
    A.3 Multiple Character Entities
B Changes
    B.1 Changes since 2009-11-17
    B.2 Changes between 2009-11-17 and 2008-07-21
    B.3 Changes between 2008-07-21 and 2007-12-14
C Differences between these entities and earlier W3C DTDs
    C.1 Differences from XHTML 1.0
    C.2 Differences from MathML 2.0 (second edition)
D Source Files
E References

1 Introduction

Notation and symbols have proved very important for human communication, especially in scientific documents. Mathematics has grown in part because its notation continually changes toward being succinct and suggestive. There have been many new signs developed for use in mathematical notation, and mathematicians have not held back from making use of many symbols originally introduced elsewhere. The result is that science in general, and particularly mathematics, makes use of a very large collection of symbols. It is difficult to write science fluently if these characters are not available for use. It is difficult to read science if corresponding glyphs are not available for presentation on specific display devices. In the majority of cases it is preferable to store characters directly as Unicode character data or as XML numeric character references.

However, in some environments it is more convenient to use the ASCII input mechanism provided by XML entity references. Many entity names are in common use, and this specification aims to provide standard mappings to Unicode for each of these names. It introduces no names that have not already been used in earlier specifications. Note that these names are short mnemonic names designed for input methods such as XML entity references, not the longer formal names that form part of the Unicode standard.

Specifically, the entity names in the sets starting with the letters "iso" were first standardized in SGML ([SGML]) and updated in [ISO9573-13-1991]. The W3C Math Working Group has been invited to take over the maintenance and development of these sets by the original standards committee (ISO/IECJTC1 SC34). The sets with names starting "mml" were first standardized in MathML [MathML2] and those starting with "xhtml" were first standardized in HTML [HTML4].

This document is the result of years of employing entity names on the Web. There were always a few named entities used for special characters in HTML, but a flood of new names came with the symbols of mathematics. This means that this document can be viewed as an extension and final revision of Chapter 6 of the MathML 2.0 [MathML2] recommendation. Now it presents a completed listing harmonizing the known uses of character entity names throughout the XML world and Unicode.

Since there are so many character entity names, and the files specifying them are resources that may be subject to frequent lookup, a template catalog file has also been provided. Users are strongly encouraged to design their implementations so that relevant entity name tables are cached locally, since it is not expected that the listings provided with this specification will need changing for some long time.

2 Sets of names

This specification defines mappings to Unicode of many sets of names that have been defined by earlier specifications.

We first present two tables listing all the sets combined, first in Unicode order and then in alphabetic order:

Then there come tables documenting each of the entity sets. Each set has a link to the DTD entity declaration for the corresponding entity set, and also a link to an XSLT2 stylesheet that will implement a reverse mapping from characters to entity names (this is, of course, only possible for entity names that map to a single Unicode code point).

In addition to the stylesheets and entity files corresponding to each individual entity set, a combined stylesheet is provided, as well as two combined entity sets, each in two formats. First as a small file that references the individual entity sets listed above, and then as a larger file that directly contains a definition of each entity, with the definitions sorted in order of entity names, with duplicates removed. The first set, w3centities includes all the entity sets defined in this specification. the second, htmlmathml, is a slightly smaller set just comprising the entity sets included in HTML and MathML. These are the entities most likely to work (due to cached DTD or built in entity support) if used in a web context.

3 Unicode Character Ranges for Scientific Documents

Certain characters are of particular relevance to scientific document production. The following tables display Unicode ranges containing the characters that are most used in mathematics.

Note that each of the tables linked from this section contains 256 images and may take a while to load if the images have not been cached locally.

4 Mathematical Alphanumeric Characters

Many of the entities defined by this specification relate to the mathematical alphanumeric characters contained in the letter-like symbols block of Unicode Plane 0, or in the Mathematical Alphanumeric Symbols block in Unicode Plane 1. The following tables list all these symbols, highlighting those that are not in Plane 1, and giving entity names where appropriate.

5 Entities for Negated and Variant Characters

Each of the entity definitions in a majority of the specification expands to a single Unicode character. The definitions that expand to a sequence of two or more characters are outlined in this section.

5.1 Negated Mathematical Characters

In addition to the Unicode Characters so far listed, one may use the combining characters U+0338 (/), U+20D2 (|) and U+20E5 (\) to produce negated or canceled forms of characters. A combining character should be placed immediately after its "base" character, with no intervening markup or space, just as is the case for combining accents.

In principle, the negation characters may be applied to any Unicode character, although fonts designed for mathematics typically have some negated glyphs ready composed. A MathML renderer should be able to use these pre-composed glyphs in these cases. A compound character code either represents a UCS character that is already available, as in the case of U+003D U+0338 which amounts to U+2260, or it does not, as is the case for U+2202 U+0338. The common cases of negations, of the latter type, that have been identified are listed in the tables.

Note that it is the policy of the W3C and of Unicode that if a single character is already defined for what can be achieved with a combining character, that character must be used instead of the decomposed form. It is also intended that no new single characters representing what can be done by with existing compositions will be introduced. For further information on these matters see the Unicode Standard Annex 15, Unicode Normalization Forms [Unicode15], especially the discussion of Normalization Form C.

5.2 Variant Mathematical Characters

Unicode attempts to avoid having several character codes for simple font variants. For a code point to be assigned there should be more than a nuance in glyphs to be recorded. To record variants worth noting there is a special character in Unicode 3.2, U+FE00 (VARIATION SELECTOR-1), which acts as a postfix modifier. However the legally allowed combinations with this variation selector are restricted to a list recorded as part of Unicode. The VARIATION SELECTOR-1 character may only be applied to the characters listed here. The resulting combination is not regarded by Unicode as a separate character, but a variation on the base character. Unicode aware systems may render the combination as the base if the available fonts do not support the variant glyph shape.

A Special Considerations

A.1 Epsilon

Historically there has been much confusion and lack of agreement over variant forms for lower case epsilon.

This specification uses the definitions below. Note that the name epsilon is used for the character used in textual Greek (U+03B5) and varepsilon used for the epsilon symbol character more commonly used in mathematics (U+03F5). Note that this usage is compatible with the naming of similar pairs of characters (for example theta, vartheta) but incompatible with the naming convention used in TeX, MathML2 and some earlier mappings of the ISO entity sets to Unicode.

EntitySetDescriptionUnicode Character
eacgrisogrk2=small epsilon, accent, GreekU+03ADU003ADGREEK SMALL LETTER EPSILON WITH TONOS
egrisogrk1=small epsilon, GreekU+03B5U003B5GREEK SMALL LETTER EPSILON
epsivisogrk3/straightepsilon, small epsilon, GreekU+03F5U003F5GREEK LUNATE EPSILON SYMBOL
straightepsilonmmlaliasalias ISOGRK3 epsiv
varepsilonmmlaliasalias ISOGRK3 epsiv
bepsiisoamsr/backepsilon R: such thatU+03F6U003F6GREEK REVERSED LUNATE EPSILON SYMBOL
backepsilonmmlaliasalias ISOAMSR bepsi
b.epsiisogrk4small epsilon, GreekU+1D6C6U1D6C6MATHEMATICAL BOLD SMALL EPSILON
b.epsivisogrk4variant epsilonU+1D6DCU1D6DCMATHEMATICAL BOLD EPSILON SYMBOL

A.2 Phi

The situation for phi is very similar to that of epsilon, although with the further complication that early versions of Unicode had the sample glyphs for U+03C6 and U+03D5 swapped from the current usage, and some older fonts still in use follow that older convention. The definitions used in this specification are as listed below.

EntitySetDescriptionUnicode Character
phiisogrk3/phi - small phi, GreekU+03C6U003C6GREEK SMALL LETTER PHI
phixhtml1-symbolgreek small letter phi
phgrisogrk1=small phi, Greek
straightphimmlaliasalias ISOGRK3 phivU+03D5U003D5GREEK PHI SYMBOL
phivisogrk3/varphi - straight phi
varphimmlaliasalias ISOGRK3 phiv
b.phiisogrk4small phi, GreekU+1D6D7U1D6D7MATHEMATICAL BOLD SMALL PHI

A.3 Multiple Character Entities

In addition to the combining and variant character combinations listed in the previous sections, the following table lists the remaining entity replacement texts that consist of more than one character.

EntitySetDescriptionUnicode Character
fjligisopubsmall fj ligatureU+0066 U+006AU00066-0006Afj ligature
ThickSpacemmlextraspace of width 5/18 emU+205F U+200AU0205F-0200Aspace of width 5/18 em
raceisoamsbreverse most positive, line belowU+223D U+0331U0223D-00331REVERSED TILDE with underline
acEisoamsbmost positive, two lines belowU+223E U+0333U0223E-00333INVERTED LAZY S with double underline
DownBrevemmlextrabreve, inverted (non-spacing)U+0020 U+0311U00311COMBINING INVERTED BREVE
tdotisotechthree dots aboveU+0020 U+20DBU020DBCOMBINING THREE DOTS ABOVE
TripleDotmmlaliasalias ISOTECH tdot
DotDotisotechfour dots aboveU+0020 U+20DCU020DCCOMBINING FOUR DOTS ABOVE

Unicode does not have an fj character, although the other common f ligatures such as fi (U+FB01) are contained in the Alphabetic Presentation Forms block. The fjlig entity is mapped to the pair of characters "fj"; modern typesetting engines should automatically use the fj ligature for this combination if the font supplies such a ligature.

Unicode has a range of space characters (including all multiples of 1/18 em up to 6/18, except for 5/18 em) thus the ThickSpace entity is mapped to a pair of space characters. An alternative would have been to use U+2005 (1/4 em), but 1/4 em is not equal to 5/18 em, so the above definition was chosen, despite the fact that the difference is unlikely to be visibly noticeable at most typeset font sizes.

The entities race and acE denote underlined characters for which Unicode does not have codepoints, thus combining underline characters have been used, in a way analogous to the use of combining strokes for negated operators.

For reasons explained further in [Charmod-norm], it is not advisable to to start the replacement text of an entity with a combining character, as then potentially different results may be produced depending on the order in which entity expansion and Unicode normalisation are performed. As far as possible this specification uses non-combining characters, however, in the cases tdot, TripleDot and DotDot Unicode only has combining forms of the accents, and so the entity replacement text starts with a space, to avoid the possibility that the expansion of the entity combines with preceding text.

B Changes

B.1 Changes since 2009-11-17

Various editorial improvments, including using Unicode U+1234 notation more consistently rather than displaying the internal IDs of the form U01234.

The combined entities file distributed with the 2009-11-17 draft introduced an error that if two entity names differed only by case, only one was included. This has been corrected.

The combined entity set htmlmathml corresponding to the entities usable in HTML and MathML is now explicitly provided. The predefined set, corresponding to the entities predefined in XML is now documented (it was previously used internally).

The entities xvee and xwedge had the correct Unicode assignments (U+22C1 and U+22C0) but the entity descriptions have been swapped, xvee is logical or and xwedge is logical and. This error in [ISO9573-13-1991] was reported in 1999, Proposed Technical Corrigendum but not previously fixed. The entity files are unaffected by this change.

The entity NotGreaterFullEqual which had been erroneously assigned to a negated less than operator (U+2266 U+0338) has been corrected to be the negated greater than operator (U+2267 U+0338).

A sample catalog is now provided to redirect references to the entity files to copies on the local machine rather than the W3C server.

B.2 Changes between 2009-11-17 and 2008-07-21

The html5-uppercase set is now documented.

The entities ohm and angst have changed to U+03A9 and U+00C5 to match NFC. See w3c bugzilla entry.

The entity race, which had been erroneously assigned U+29DA, is now assigned the combination U+223D U+0331. (U+223D isn't quite the shape shown in the original ISO document which is a rotated S rather than a rotated tilde, but this appears to be the closest character in Unicode 5.2.)

The entities bsolhsub and suphsol which were previously mapped to two-character combinations U+005C U+2282 and U+2283 U+002F are now mapped to the Unicode 5 characters that were added specifically to support these entities, U+27C8 and U+27C9.

The source files have all been updated to match Unicode 5.2.

The entity ThickSpace now maps to the pair U+205F U+200A rather than the triple U+2009 U+200A U+200A (4/18 + 1/18)em rather than (3/18 + 1/18 + 1/18)em.

The entity UnderBar maps to the spacing character _ rather than the combining character U+0332.

The entity OverBar maps to the spacing character U+203E (like the XHTML entity oline) rather than the macron character U+00AF.

The entities epsiv and varepsilon are now mapped to the epsilon symbol U+03F5 rather than being aliases for the entity epsilon, U+03B5.

The entities phiv and varphi are now mapped to the phi symbol U+03D5 rather than being aliases for the entity phi, U+03C6.

B.3 Changes between 2008-07-21 and 2007-12-14

The following entity definitions have changed at this draft:

phi, lang, rang, OverParenthesis, UnderParenthesis, OverBrace, UnderBrace, lbbrk, rbbrk.

C Differences between these entities and earlier W3C DTDs

C.1 Differences from XHTML 1.0

Differences between the XHTML entity definitions described here and the entity set described in the XHTML 1.0 DTD.

lang and rang
U+27E8 and U+27E9; XHTML 1.0 used U+2329 and U+232A (which have canonical decomposition to U+3008 and U+3009).


The current drafts of [HTML5] use entity definitions derived from this specification.

C.2 Differences from MathML 2.0 (second edition)

The differences between MathML 2 and the current entity definitions are listed below.

ISOPUB (and MathML 1) defined an fj ligature; Unicode does not have a specific character and the entity was dropped from MathML2. It is re-instated here for maximum compatibility with [SGML].
U+03C6 GREEK SMALL LETTER PHI (the definition used in HTML4); MathML2 used U+03D5 GREEK PHI SYMBOL.
epsiv, varepsilon, phiv, varphi
these have been changed to map to the symbol character (to match other uses of the var prefix such as vartheta).
U+0237; MathML 2 used U+006A (j) as there was no dotless j before Unicode 4.1.
trpezium, elinters
U+23E2 and U+23E7; MathML 2 used U+FFFD (REPLACEMENT CHARACTER) as these characters were added at Unicode 5.0 specifically to support these entities.
ohm, angst
As noted above, the definitions of these entities have been changed so that the definitions use characters that are in NFC normal form.
bsolhsub and suphsol
U+27C8 and U+27C9; MathML2 used U+005C U+02282 and U+2283 U+002F.
U+2267 U+0338 ; MathML2 used the erroneous definition U+2266 U+0338.

The following bracket symbols have been added to the Mathematical symbols block in Unicode versions between 3.1 and 5.1. MathML2 used similar characters intended for CJK punctuation.

lang, langle, LeftAngleBracket and rang, rangle, RightAngleBracket
U+27E8 and U+27E9; MathML2 used U+2329 and U+232A (which have canonical decomposition to U+3008 and U+3009).
Lang and Rang
U+27EA and U+27EB; MathML2 used U+300A and U+300B.
lbbrk and rbbrk
U+2772 and U+2773; MathML2 used U+3014 and U+3015.
loang and roang
U+27EC and U+27ED; MathML2 used U+3018 and U+3019.
lobrk and robrk
U+27E6 and U+27E7; MathML2 used U+301A and U+301B.
OverBrace and UnderBrace
U+23DE and U+23DF; MathML2 used U+FE37 and U+FE38.
OverParenthesis and UnderParenthesis
U+23DC and U+23DD; MathML2 used U+FE35 and U+FE36.
LeftDoubleBracket and RightDoubleBracket
U+27E6 and U+27E7; MathML2 used U+301A and U+301B.


[MathML3] uses the entity sets defined by this specification, so there will be no differences between MathML and the entities defined here once MathML3 is finalized.

D Source Files

All data files used to construct the entity declarations, XSLT character maps, and HTML tables referenced from this document are available from http://www.w3.org/2003/entities/2007xml/.

E References

ISO/IEC 8879:1986, Information processing — Text and office systems — Standard Generalized Markup Language (SGML)
ISO/IEC TR :1991, Information technology — SGML support facilities Techniques for using SGML — Part 13: Public entity sets for mathematics and science
The Unicode Consortium. The Unicode Standard, Version 5.2.0, defined by: The Unicode Standard, Version 5.2 (Mountain View, CA: The Unicode Consortium, 2009. ISBN 978-1-936213-00-9). (http://www.unicode.org/versions/Unicode5.2.0/)
Unicode Standard Annex 15, Version 5.2.0; Unicode Normalization Forms, The Unicode Consortium, 2009-09-03. (http://www.unicode.org/reports/tr15/tr15-31.html)
Barbara Beeton, Asmus Freytag, Murray Sargent III, Unicode Support for Mathematics, Unicode Technical Report #25 2008-08-14. (http://www.unicode.org/unicode/reports/tr25/)
David Carlisle, Patrick Ion, Robert Miner, Nico Poppelier, Mathematical Markup Language (MathML) Version 2.0 (Second Edition) W3C Recommendation 21 October 2003 (http://www.w3.org/TR/2003/REC-MathML2-20031021/)
David Carlisle, Patrick Ion, Robert Miner, Mathematical Markup Language (MathML) Version 3.0 W3C Candidate Recommendation 15 December 2009 (http://www.w3.org/TR/2009/CR-MathML3-20091215/)
Dave Raggett, Arnaud Le Hors, Ian Jacobs, HTML 4.01 Specification W3C Recommendation 24 December 1999 (http://www.w3.org/TR/1999/REC-html401-19991224)
Ian Hickson, David Hyat, HTML 5, A vocabulary and associated APIs for HTML and XHTML W3C Working Draft 25 August 2009 (http://www.w3.org/TR/html5/)
François Yergeau, Martin J. Dürst, Richard Ishida, Addison Phillips, Misha Wolf, Tex Texin, Character Model for the World Wide Web 1.0: Normalization W3C Working Draft 27 October 2005 (http://www.w3.org/TR/charmod-norm/)