OpenMath Content Dictionary: semigroup3

Canonical URL:
http://www.openmath.org/cd/semigroup3.ocd
CD Base:
http://www.openmath.org/cd
CD File:
semigroup3.ocd
CD as XML Encoded OpenMath:
semigroup3.omcd
Defines:
automorphism_group, cyclic_semigroup, direct_power, direct_product, free_semigroup, left_regular_representation, maps_semigroup
Date:
2004-06-01
Version:
3 (Revision 1)
Review Date:
2006-06-01
Status:
experimental

Semigroup constructions

Initiated by Arjeh M. Cohen 2003-10-02

cyclic_semigroup

Description:

This symbol denotes the cyclic semigroup with a cycle of length l and a tail of length k.

Example:
$\mathrm{size}\left(\mathrm{carrier}\left(\mathrm{cyclic_semigroup}\left(k,l\right)\right)\right)=k+l$
Commented Mathematical property (CMP):
The size of cyclic_semigroup(k,l) equals k+l.
Formal Mathematical property (FMP):
$\mathrm{size}\left(\mathrm{cyclic_semigroup}\left(k,l\right)\right)=k+l$
Signatures:
sts

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maps_semigroup

Description:

This is a unary function whose argument must be a set X or a positive integer. When applied to X, it refers to the semigroup of all functions from X to X if X is a set and to {1,...,X} if X is an integer, whose binary operation is composition of maps and whose identity element is the identity map on the set X, respectively {1,...,X}.

Signatures:
sts

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left_regular_representation

Description:

This is a unary function whose argument must be a semigroup M. When applied to M, it represents the map from M to the maps semigroup on M that assigns to m left multiplication by m on M.

Commented Mathematical property (CMP):
The left regular representation on M applied to the element x of M represents left multiplication by x on M
Formal Mathematical property (FMP):
$\forall M,x.\left(\mathrm{left_regular_representation}\left(M\right)\right)\left(x\right)=\mathrm{left_multiplication}\left(M,x\right)$
Commented Mathematical property (CMP):
The left regular representation is a homomorphism of semigroups from M to the maps semigroup on M.
Formal Mathematical property (FMP):
$\forall M.\mathrm{is_homomorphism}\left(M,\mathrm{maps_semigroup}\left(M\right),\mathrm{left_regular_representation}\left(M\right)\right)$
Signatures:
sts

 [Next: automorphism_group] [Previous: maps_semigroup] [Top]

automorphism_group

Description:

This is a function with a single argument which must be a semigroup. It refers to the automorphism group of its argument.

Signatures:
sts

 [Next: direct_product] [Previous: left_regular_representation] [Top]

direct_product

Description:

This is an n-ary function whose arguments must be semigroups. It refers to the direct product of its arguments.

Signatures:
sts

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direct_power

Description:

This is a binary function whose first argument should be a semigroup M and whose second argument should be a natural number n. It refers to the direct product of n copies of M.

Signatures:
sts

 [Next: free_semigroup] [Previous: direct_product] [Top]

free_semigroup

Description:

This symbol represents a binary function. The argument is a list or a set. When evaluated on such an argument, the function represents the free semigroup generated by the entries of the list or set.

Example:
The free semigroup on the letters a, b:
$\mathrm{free_semigroup}\left(\left(a,b\right)\right)$
Signatures:
sts

 [First: cyclic_semigroup] [Previous: direct_power] [Top]