# 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.
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.
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:
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.
$z$
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
$\left(\mathbb{R},\mathbb{Z}\right)\to \mathbb{R}$
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:
$\mathrm{nary}\left(\mathrm{Object}\right)\to \mathrm{Object}$
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:
$\mathrm{nassoc}\left(\mathrm{AbelianSemiGroup}\right)\to \mathrm{AbelianSemiGroup}$
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:
$\mathrm{OMSymbol}\to \mathrm{error}$
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:
$\mathrm{structure}\left(\mathrm{set}\right)\to \mathrm{set}$
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:
$\mathrm{binder}$
Signatures:
sts

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:
$\mathrm{attribution}$
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
$\mathrm{nary}\left(\mathrm{Object}\right)\to \mathrm{Object}$
Signatures:
sts

## 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:
$\left(\mathrm{NumericalValue},\mathrm{NumericalValue}\right)\to \mathrm{NumericalValue}$
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:
$\mathrm{NumericalValue}\to \mathrm{SetNumericalValue}$
Signatures:
sts

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