# OpenMath Content Dictionary: monoid3

Canonical URL:
http://www.openmath.org/cd/monoid3.ocd
CD Base:
http://www.openmath.org/cd
CD File:
monoid3.ocd
CD as XML Encoded OpenMath:
monoid3.omcd
Defines:
automorphism_group, concatenation, cyclic_monoid, direct_power, direct_product, emptyword, free_monoid, left_regular_representation, maps_monoid, strings
Date:
2004-06-01
Version:
3 (Revision 2)
Review Date:
2006-06-01
Status:
experimental

Monoid constructions

Initiated by Arjeh M. Cohen 2003-10-02
Edited AMC 2004-03-05
Edited AMC 2004-06-27


## cyclic_monoid

Description:

This symbol is a function of two natural numbers, the first of which should be positive. When evaluated at k and l, it denotes the cyclic monoid with a cycle of length l and a tail (including the identity element) of length k.

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

 [Next: maps_monoid] [Last: concatenation] [Top]

## maps_monoid

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 monoid 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

 [Next: left_regular_representation] [Previous: cyclic_monoid] [Top]

## left_regular_representation

Description:

This is a unary function whose argument must be a monoid M. When applied to M, it represents the map from M to the maps monoid 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 monoids from M to the maps monoid on M.
Formal Mathematical property (FMP):
$\forall M.\mathrm{is_homomorphism}\left(M,\mathrm{maps_monoid}\left(M\right),\mathrm{left_regular_representation}\left(M\right)\right)$
Signatures:
sts

 [Next: automorphism_group] [Previous: maps_monoid] [Top]

## automorphism_group

Description:

This is a function with a single argument which must be a monoid. 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 monoids. It refers to the direct product of its arguments.

Signatures:
sts

 [Next: direct_power] [Previous: automorphism_group] [Top]

## direct_power

Description:

This is a binary function whose first argument should be a monoid 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_monoid] [Previous: direct_product] [Top]

## free_monoid

Description:

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

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

 [Next: strings] [Previous: direct_power] [Top]

## strings

Description:

This symbol represents a unary function. The argument is a list or a set. When evaluated on such an argument, the function represents the set of all strings whose characters are entries of the list or set.

Signatures:
sts

 [Next: emptyword] [Previous: free_monoid] [Top]

## emptyword

Description:

This symbol represents a constant. It represents the empty string.

Signatures:
sts

 [Next: concatenation] [Previous: strings] [Top]

## concatenation

Description:

This symbol represents a binary concatenation operation on strings.

Signatures:
sts

 [First: cyclic_monoid] [Previous: emptyword] [Top]