OpenMath Content Dictionary: semigroup2

Canonical URL:
http://www.openmath.org/cd/semigroup2.ocd
CD Base:
http://www.openmath.org/cd
CD File:
semigroup2.ocd
CD as XML Encoded OpenMath:
semigroup2.omcd
Defines:
is_automorphism, is_endomorphism, is_homomorphism, is_isomorphism, isomorphic, left_multiplication, right_multiplication
Date:
2004-06-01
Version:
1 (Revision 1)
Review Date:
2006-06-01
Status:
experimental

Basic functions for semigroup theory

Initiated by Arjeh M. Cohen 2003-10-03


is_homomorphism

Description:

This symbol is a boolean function with three arguments. The first and arguments are semigroups M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a semigroup homomorphism from M to N.

Commented Mathematical property (CMP):
If is_homomorphism(M,N,f) then, for each pair of elements x, y of M, we have f(x * y) = f(x) * f(y).
Formal Mathematical property (FMP):
$\mathrm{is_homomorphism}\left(M,N,f\right)⇒\forall x,y.x\in \mathrm{carrier}\left(M\right)\wedge y\in \mathrm{carrier}\left(G\right)⇒f\left(xy\right)=f\left(y\right)f\left(x\right)$
Example:
$\mathrm{is_homomorphism}\left(M,N,f\right)$
Signatures:
sts

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is_isomorphism

Description:

This symbol is a boolean function with three arguments. The first and arguments are semigroups M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a semigroup isomorphism from M to N. This means that f is a homomorphism from M to N, that f is bijective, and that its inverse is a homomorphism from N to M.

Example:
$\mathrm{is_isomorphism}\left(M,N,f\right)$
Signatures:
sts

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is_endomorphism

Description:

This symbol is a boolean function with two arguments. The first argument is a semigroup M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes that f is a semigroup endomorphism from M to M.

Commented Mathematical property (CMP):
If is_endomorphism(M,f) then is_homomorphism(M,M,f)
Formal Mathematical property (FMP):
$\mathrm{is_endomorphism}\left(M,f\right)⇒\mathrm{is_homomorphism}\left(M,M,f\right)$
Example:
$\mathrm{is_endomorphism}\left(M,f\right)$
Signatures:
sts

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is_automorphism

Description:

This symbol is a boolean function with two arguments. The first is a semigroup M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes a semigroup automorphism f of M.

Commented Mathematical property (CMP):
If is_automorphism(M,f) then is_isomorphism(M,M,f)
Formal Mathematical property (FMP):
$\mathrm{is_automorphism}\left(M,f\right)⇒\mathrm{is_isomorphism}\left(M,M,f\right)$
Example:
$\mathrm{is_automorphism}\left(M,f\right)$
Signatures:
sts

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left_multiplication

Description:

This symbol is a function with two arguments, which should be a semigroup M and an element x of M. When applied to M and x, it denotes left multiplication on M by x.

Commented Mathematical property (CMP):
left_multiplication(M,x) (y) = x * y.
Formal Mathematical property (FMP):
$\forall M,x,y.\left(\mathrm{left_multiplication}\left(M,x\right)\right)\left(y\right)=\mathrm{multiplication}\left(M,x,y\right)$
Signatures:
sts

 [Next: right_multiplication] [Previous: is_automorphism] [Top]

right_multiplication

Description:

This symbol is a function with two arguments, which should be a semigroup M and an element x of M. When applied to M and x, it denotes right multiplication on M by x.

Commented Mathematical property (CMP):
right_multiplication(M,x) (y) = y * x.
Formal Mathematical property (FMP):
$\forall M,x,y.\left(\mathrm{right_multiplication}\left(M,x\right)\right)\left(y\right)=\mathrm{multiplication}\left(M,y,x\right)$
Signatures:
sts

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isomorphic

Description:

This symbol is a Boolean function with n arguments, n at least 2, which are semigroups. When applied to M_1, ..., M_n, it denotes the fact that there is an isomorphism from each M_i to each M_j.

Example:
$\mathrm{isomorphic}\left(M,N\right)$
Signatures:
sts

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