OpenMath Content Dictionary: sts

Canonical URL:

http://www.openmath.org/cd/sts.ocd

CD Base:

http://www.openmath.org/cd

CD File:

sts.ocd

CD as XML Encoded OpenMath:

sts.omcd

Defines:

NumericalValue, Object, SetNumericalValue, attribution, binder, error, mapsto, nary, nassoc, structure, type

Date:

2004-03-30

Version:

4 (Revision 1)

Review Date:

2017-12-31

Status:

official

---

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs
            

Definitions of the symbols used by the OpenMath Small Type System.

---

type

Role:

semantic-attribution

Description:

A symbol to be used within an OpenMath attribute to specify the type of the object.

Example:

The variable z is attributed with a type for complex numbers.

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMATTR>
    <OMATP>
       <OMS cd="sts" name="type"/>
       <OMS name="C" cd="setname1"/>
    </OMATP>
    <OMV name="z"/>
  </OMATTR>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <semantics>
  <ci>z</ci>
  <annotation-xml cd="sts" name="type"><csymbol cd="setname1">C</csymbol></annotation-xml>
 </semantics>
</math>

Prefix

Attrib([ type C ], z)

Popcorn

$z{sts.type -> setname1.C}

Rendered Presentation MathML

$$z$$

Signatures:

sts

---

[Next: mapsto] [Last: SetNumericalValue] [Top]

---

mapsto

Role:

application

Description:

This symbol represents the construction of a function type.

The first n-1 children denote the types of the arguments, the last denotes the return type.

Example:

The signature of the power function: (Real , Integer) -> Real

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="sts" name="mapsto"/>
    <OMS cd="setname1" name="R"/>
    <OMS cd="setname1" name="Z"/>
    <OMS cd="setname1" name="R"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="sts">mapsto</csymbol>
  <csymbol cd="setname1">R</csymbol>
  <csymbol cd="setname1">Z</csymbol>
  <csymbol cd="setname1">R</csymbol>
 </apply>
</math>

Prefix

mapsto (R, Z, R)

Popcorn

sts.mapsto(setname1.R, setname1.Z, setname1.R)

Rendered Presentation MathML

$$\left(\mathbb{R},\mathbb{Z}\right)\to \mathbb{R}$$

Signatures:

sts

---

[Next: nary] [Previous: type] [Top]

---

nary

Role:

application

Description:

Constructs a child of mapsto which denotes an arbitrary number of copies of the argument of nary.

Example:

The signature for list, an n-ary function:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMA>
  <OMS name="mapsto" cd="sts"/>
   <OMA>
     <OMS name="nary" cd="sts"/>
     <OMS name="Object" cd="sts"/>
   </OMA>
  <OMS name="Object" cd="sts"/>
 </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="sts">mapsto</csymbol>
  <apply><csymbol cd="sts">nary</csymbol><csymbol cd="sts">Object</csymbol></apply>
  <csymbol cd="sts">Object</csymbol>
 </apply>
</math>

Prefix

mapsto (nary (Object) , Object)

Popcorn

sts.mapsto(sts.nary(sts.Object), sts.Object)

Rendered Presentation MathML

$$\mathrm{nary}\left(\mathrm{Object}\right)\to \mathrm{Object}$$

Signatures:

sts

---

[Next: nassoc] [Previous: mapsto] [Top]

---

nassoc

Role:

application

Description:

Constructs a child of mapsto which denotes an arbitrary number of copies of the argument of nassoc. The operator is associative on these arguments which means that repeated uses may be flattened/unflattened.

Example:

The signature for plus, an associative n-ary function:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS name="mapsto" cd="sts"/>
    <OMA>
      <OMS name="nassoc" cd="sts"/> 
      <OMV name="AbelianSemiGroup"/>
    </OMA>
    <OMV name="AbelianSemiGroup"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="sts">mapsto</csymbol>
  <apply><csymbol cd="sts">nassoc</csymbol><ci>AbelianSemiGroup</ci></apply>
  <ci>AbelianSemiGroup</ci>
 </apply>
</math>

Prefix

mapsto (nassoc ( AbelianSemiGroup) , AbelianSemiGroup)

Popcorn

sts.mapsto(sts.nassoc($AbelianSemiGroup), $AbelianSemiGroup)

Rendered Presentation MathML

$$\mathrm{nassoc}\left(\mathrm{AbelianSemiGroup}\right)\to \mathrm{AbelianSemiGroup}$$

Signatures:

sts

---

[Next: error] [Previous: nary] [Top]

---

error

Role:

constant

Description:

The error symbol is the 'return type' of error symbols in the error signature file.

Example:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="sts" name="mapsto"/>
    <OMV name="OMSymbol"/>
    <OMS name="error" cd="sts"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="sts">mapsto</csymbol><ci>OMSymbol</ci><csymbol cd="sts">error</csymbol></apply>
</math>

Prefix

mapsto ( OMSymbol, error)

Popcorn

sts.mapsto($OMSymbol, sts.error)

Rendered Presentation MathML

$$\mathrm{OMSymbol}\to \mathrm{error}$$

Signatures:

sts

---

[Next: structure] [Previous: nassoc] [Top]

---

structure

Role:

application

Description:

The structure element is used to represent a structure of a particular (algebraic) type.

Example:

The signature of a function which given a set will return an element of that set:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="sts" name="mapsto"/>
    <OMA>
      <OMS cd="sts" name="structure"/>
      <OMV name="set"/>
    </OMA>
    <OMV name="set"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="sts">mapsto</csymbol>
  <apply><csymbol cd="sts">structure</csymbol><ci>set</ci></apply>
  <ci>set</ci>
 </apply>
</math>

Prefix

mapsto (structure ( set) , set)

Popcorn

sts.mapsto(sts.structure($set), $set)

Rendered Presentation MathML

$$\mathrm{structure}\left(\mathrm{set}\right)\to \mathrm{set}$$

Signatures:

sts

---

[Next: binder] [Previous: error] [Top]

---

binder

Role:

constant

Description:

An `OMBIND' object has three parts: a "binder" such as "lambda" or "for all", a (list of) bound variables, and an expression. The use of `binder' in a signature indicates that we are describing something which can only be used as the first child of an OMBIND construct.

Example:

The signature of forall is:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMS name="binder" cd="sts"/>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><csymbol cd="sts">binder</csymbol></math>

Prefix

binder

Popcorn

sts.binder

Rendered Presentation MathML

$$\mathrm{binder}$$

Signatures:

sts

---

[Next: attribution] [Previous: structure] [Top]

---

attribution

Role:

constant

Description:

An `attribution' object consists of pairs of keys and values. The use of the symbol `attribution' in a signature indicates that the symbol is to be used as a key.

Example:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMS name="attribution" cd="sts"/>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><csymbol cd="sts">attribution</csymbol></math>

Prefix

attribution

Popcorn

sts.attribution

Rendered Presentation MathML

$$\mathrm{attribution}$$

Signatures:

sts

---

[Next: Object] [Previous: binder] [Top]

---

Object

Role:

constant

Description:

Denotes any OpenMath object.

Example:

The signature for list, to show that list has the signature: Object* -> Object

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMA>
  <OMS name="mapsto" cd="sts"/>
   <OMA>
     <OMS name="nary" cd="sts"/>
     <OMS name="Object" cd="sts"/>
   </OMA>
  <OMS name="Object" cd="sts"/>
 </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="sts">mapsto</csymbol>
  <apply><csymbol cd="sts">nary</csymbol><csymbol cd="sts">Object</csymbol></apply>
  <csymbol cd="sts">Object</csymbol>
 </apply>
</math>

Prefix

mapsto (nary (Object) , Object)

Popcorn

sts.mapsto(sts.nary(sts.Object), sts.Object)

Rendered Presentation MathML

$$\mathrm{nary}\left(\mathrm{Object}\right)\to \mathrm{Object}$$

Signatures:

sts

---

[Next: NumericalValue] [Previous: attribution] [Top]

---

NumericalValue

Role:

constant

Description:

Denotes an OpenMath object that is to be thought of as something that represents a numerical value, or a numerical value.

Example:

The generic signature for the function power:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMA>
  <OMS name="mapsto" cd="sts"/>
  <OMS name="NumericalValue" cd="sts"/> 
  <OMS name="NumericalValue" cd="sts"/> 
  <OMS name="NumericalValue" cd="sts"/> 
 </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="sts">mapsto</csymbol>
  <csymbol cd="sts">NumericalValue</csymbol>
  <csymbol cd="sts">NumericalValue</csymbol>
  <csymbol cd="sts">NumericalValue</csymbol>
 </apply>
</math>

Prefix

mapsto (NumericalValue, NumericalValue, NumericalValue)

Popcorn

sts.mapsto(sts.NumericalValue, sts.NumericalValue, sts.NumericalValue)

Rendered Presentation MathML

$$\left(\mathrm{NumericalValue},\mathrm{NumericalValue}\right)\to \mathrm{NumericalValue}$$

Signatures:

sts

---

[Next: SetNumericalValue] [Previous: Object] [Top]

---

SetNumericalValue

Role:

constant

Description:

Denotes an OpenMath object that is to be thought of as something that represents a set of numerical values, or a set of numerical values.

Example:

The generic signature for the function arctan from transc3:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMA>
  <OMS name="mapsto" cd="sts"/>
  <OMS name="NumericalValue" cd="sts"/> 
  <OMS name="SetNumericalValue" cd="sts"/> 
 </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="sts">mapsto</csymbol>
  <csymbol cd="sts">NumericalValue</csymbol>
  <csymbol cd="sts">SetNumericalValue</csymbol>
 </apply>
</math>

Prefix

mapsto (NumericalValue, SetNumericalValue)

Popcorn

sts.mapsto(sts.NumericalValue, sts.SetNumericalValue)

Rendered Presentation MathML

$$\mathrm{NumericalValue}\to \mathrm{SetNumericalValue}$$

Signatures:

sts

---

[First: type] [Previous: NumericalValue] [Top]