OpenMath Content Dictionary: group2

Canonical URL:
http://www.openmath.org/cd/group2.ocd
CD Base:
http://www.openmath.org/cd
CD File:
group2.ocd
CD as XML Encoded OpenMath:
group2.omcd
Defines:
conjugation, is_automorphism, is_endomorphism, is_homomorphism, is_isomorphism, isomorphic, left_multiplication, right_inverse_multiplication, right_multiplication
Date:
2004-06-01
Version:
1 (Revision 3)
Review Date:
2006-06-01
Status:
experimental

A CD of functions like homomorphisms for groups

Written by Arjeh M. Cohen 2004-02-20.
Edited AMC 2004-03-02

is_homomorphism

Description:

This symbol is a boolean function with three arguments. The first two arguments are groups 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 group 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):
is_homomorphism ( M , N , f ) x , y . x carrier ( M ) y carrier ( G ) f ( x y ) = f ( y ) f ( x )
Example:
is_homomorphism ( M , N , f )
Signatures:
sts


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is_isomorphism

Description:

This symbol is a boolean function with three arguments. The first and arguments are groups 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 group 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:
is_isomorphism ( M , N , f )
Signatures:
sts


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isomorphic

Description:

This symbol is a Boolean function with n arguments, n at least 2, which are groups. 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:
isomorphic ( M , N )
Signatures:
sts


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is_endomorphism

Description:

This symbol is a boolean function with two arguments. The first argument is a group 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 group endomorphism from M to M.

Commented Mathematical property (CMP):
If is_endomorphism(M,f) then is_homomorphism(M,M,f)
Formal Mathematical property (FMP):
is_endomorphism ( M , f ) is_homomorphism ( M , M , f )
Example:
is_endomorphism ( M , f )
Signatures:
sts


[Next: is_automorphism] [Previous: isomorphic] [Top]

is_automorphism

Description:

This symbol is a boolean function with two arguments. The first is a group 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 group automorphism f of M.

Commented Mathematical property (CMP):
If is_automorphism(M,f) then is_isomorphism(M,M,f)
Formal Mathematical property (FMP):
is_automorphism ( M , f ) is_isomorphism ( M , M , f )
Example:
is_automorphism ( M , f )
Signatures:
sts


[Next: left_multiplication] [Previous: is_endomorphism] [Top]

left_multiplication

Description:

This symbol is a function with two arguments, which should be a group 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):
M , x , y . ( left_multiplication ( M , x ) ) ( y ) = multiplication ( M , x , y )
Signatures:
sts


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

right_multiplication

Description:

This symbol is a function with two arguments, which should be a group 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):
M , x , y . ( right_multiplication ( M , x ) ) ( y ) = multiplication ( M , y , x )
Signatures:
sts


[Next: right_inverse_multiplication] [Previous: left_multiplication] [Top]

right_inverse_multiplication

Description:

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

Commented Mathematical property (CMP):
right_inverse_multiplication(M,x) (y) = y * x^(-1).
Formal Mathematical property (FMP):
M , x , y . ( right_inverse_multiplication ( M , x ) ) ( y ) = multiplication ( M , y , x -1 )
Signatures:
sts


[Next: conjugation] [Previous: right_multiplication] [Top]

conjugation

Description:

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

Commented Mathematical property (CMP):
conjugation(M,x) (y) = x * y * x^ {-1}.
Formal Mathematical property (FMP):
M , x , y . ( conjugation ( M , x ) ) ( y ) = expression ( M , x y x -1 )
Signatures:
sts


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