# OpenMath Content Dictionary: semigroup1

Canonical URL:
http://www.openmath.org/cd/semigroup1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
semigroup1.ocd
CD as XML Encoded OpenMath:
semigroup1.omcd
Defines:
carrier, expression, factor_of, is_commutative, is_subsemigroup, magma, multiplication, semigroup, subsemigroup
Date:
2004-06-01
Version:
3 (Revision 1)
Review Date:
2006-06-01
Status:
experimental

Basic functions for semigroup theory

Initiated by Arjeh M. Cohen 2003-05-17
Edited AMC 2004-0304


## semigroup

Description:

This symbol is a constructor for semigroups. It takes two arguments in the following order: a set to specify the elements in the semigroup, and a binary operation to specify the semigroup operation. The binary operation should act on elements of the set and return an element of the set.

Commented Mathematical property (CMP):
A semigroup is closed under its operation. A semigroup operation is associative.
Formal Mathematical property (FMP):
$S=\mathrm{semigroup}\left(\mathrm{set},\mathrm{binop}\right)⇒\left(x\in \mathrm{set}\wedge y\in \mathrm{set}⇒\mathrm{binop}\left(x,y\right)\in \mathrm{set}\right)\wedge \mathrm{binop}\left(x,\mathrm{binop}\left(y,z\right)\right)=\mathrm{binop}\left(\mathrm{binop}\left(x,y\right),z\right)$
Example:
This example represents the semigroup of all functions f: R -> R with function composition as the operation.
$\mathrm{semigroup}\left({\mathbb{R}}^{\mathbb{R}},o\right)$
Signatures:
sts

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## carrier

Description:

This symbol represents a unary function, whose argument should be a semigroup S (for instance constructed by semigroup). When applied to S, its value should be the set of elements of S.

Example:
The carrier of semigroup(S,*) is S.
$\mathrm{carrier}\left(\mathrm{semigroup}\left(S,\mathrm{times}\right)\right)=S$
Signatures:
sts

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## multiplication

Description:

This symbol represents a unary function, whose argument should be a semigroup S. It returns the multiplication map on S. We allow for the map to be n-ary.

Example:
The multiplication of semigroup(S,*) is *.
$\mathrm{multiplication}\left(\mathrm{semigroup}\left(S,\mathrm{times}\right)\right)=\mathrm{times}$
Signatures:
sts

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## is_commutative

Description:

The unary boolean function whose value is true iff the argument is a commutative semigroup.

Commented Mathematical property (CMP):
If is_commutative(S) then for all a,b in carrier(S) a*b = b*a
Formal Mathematical property (FMP):
$\mathrm{is_commutative}\left(S\right)⇒\forall a,b.a\in \mathrm{carrier}\left(S\right)\wedge b\in \mathrm{carrier}\left(S\right)⇒\left(\mathrm{multiplication}\left(S\right)\right)\left(a,b\right)=\left(\mathrm{multiplication}\left(S\right)\right)\left(b,a\right)$
Signatures:
sts

 [Next: is_subsemigroup] [Previous: multiplication] [Top]

## is_subsemigroup

Description:

The binary boolean function whose value is true iff the second argument is a subsemigroup of the second.

Commented Mathematical property (CMP):
If is_subsemigroup(S,T) then T is a set of elements of S and T is closed under multiplication.
Signatures:
sts

 [Next: magma] [Previous: is_commutative] [Top]

## magma

Description:

This symbol is a unary function. Its argument should be a semigroup S. When applied to S, it denotes the magma with the same element set and binary operation as S.

Example:
$\mathrm{magma}\left(\mathrm{semigroup}\left(X,\mathrm{times}\right)\right)=\mathrm{magma}\left(X,\mathrm{times}\right)$
Signatures:
sts

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## subsemigroup

Description:

This symbol is a constructor symbol with two arguments. The first argument is a semigroup S, the second a list or set, D, of elements of S. When applied to S and D, it denotes the subsemigroup of S generated by D.

Example:
$\mathrm{subsemigroup}\left(S,D\right)$
Example:
This example represents the subsemigroup of the multiplicative semigroup of the nonzero reals generated by the constants Pi and E:
$\mathrm{semigroup}\left(\mathrm{semigroup}\left(\left\{x\in \mathbb{R}|x\ne 0\right\},×\right),\left(\pi ,e\right)\right)$
Signatures:
sts

 [Next: factor_of] [Previous: magma] [Top]

## factor_of

Description:

This symbol is a ternary function. Its first argument should be a semigroup S and the second and third arguments should be elements of S. When applied to S, a, and b, it denotes the fact that a is a divisor of b in S. This means that there are u,v in carrier(S) such that uav=b.

Example:
$\mathrm{factor_of}\left(S,a,b\right)$
Signatures:
sts

 [Next: expression] [Previous: subsemigroup] [Top]

## expression

Description:

This symbol is a function with two arguments. Its first argument should be a semigroup G. The second should be an arithmetic expression A, whose operators are times and power, and whose leaves are members of the carrier of G. The second argument of power should be positive. When applied to G and A, it denotes the element (of G) that is obtained from the leaves of A by applying the multiplication and the power map of G instead of the times and power of the CD arith1 appearing in A.

Example:
$\mathrm{expression}\left(\mathrm{semigroup}\left(\mathbb{Z},+\right),2×3\right)=5$
Signatures:
sts

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