# OpenMath Content Dictionary: magma1

Canonical URL:
http://www.openmath.org/cd/magma1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
magma1.ocd
CD as XML Encoded OpenMath:
magma1.omcd
Defines:
is_identity, multiplication, carrier, is_associative, is_commutative, is_submagma, left_divides, left_expression, magma, right_divides, right_expression, submagma
Date:
2004-06-01
Version:
1 (Revision 2)
Review Date:
2006-06-01
Status:
experimental

Basic functions for magma theory

Initiated by Arjeh M. Cohen 2003-10-03
Edited by AMC 2004-0302


## magma

Description:

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

Commented Mathematical property (CMP):
A magma is closed under its operation.
Formal Mathematical property (FMP):
$G=\mathrm{magma}\left(\mathrm{set},\mathrm{binop}\right)⇒x\in \mathrm{set}\wedge y\in \mathrm{set}⇒\mathrm{binop}\left(x,y\right)\in \mathrm{set}$
Example:
This example represents the magma which has as elements all integers, and the magma operation is addition of the square of the first argument to the second.
$\mathrm{magma}\left(\mathbb{Z},\lambda x,y.{x}^{2}+y\right)$
Signatures:
sts

 [Next: carrier] [Last: right_expression] [Top]

## carrier

Description:

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

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

 [Next: multiplication] [Previous: magma] [Top]

## multiplication

Description:

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

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

 [Next: is_commutative] [Previous: carrier] [Top]

## is_commutative

Description:

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

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

 [Next: is_associative] [Previous: multiplication] [Top]

## is_associative

Description:

The unary boolean function whose value is true iff the argument is an associative magma.

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

 [Next: is_submagma] [Previous: is_commutative] [Top]

## is_submagma

Description:

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

Commented Mathematical property (CMP):
If is_submagma(G,H) then H is a set of elements of G and H is closed under multiplication.
Signatures:
sts

 [Next: is_identity] [Previous: is_associative] [Top]

## is_identity

Description:

This symbols represents a binary boolean function, whose arguments should be a magma and an element of the element set of the magma. When applied to the arguments M and x, it returns true if the element x is an identity of the magma M, that is, x*y = y* x = y for all elements y of M.

Signatures:
sts

 [Next: submagma] [Previous: is_submagma] [Top]

## submagma

Description:

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

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

 [Next: left_divides] [Previous: is_identity] [Top]

## left_divides

Description:

This symbol is a ternary function. Its first argument should be a magma M and the second and third arguments should be elements of M. When applied to M, a, and b, it denotes the fact that a is a left_divisor of b in M. This means that there is v in M such that av=b.

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

 [Next: right_divides] [Previous: submagma] [Top]

## right_divides

Description:

This symbol is a ternary function. Its first argument should be a magma M and the second and third arguments should be elements of M. When applied to M, a, and b, it denotes the fact that a is a right_divisor of b in M. This means that there is v in M such that va = b.

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

 [Next: left_expression] [Previous: left_divides] [Top]

## left_expression

Description:

This symbol is a binary function. Its first argument should be a magma M, the second argument a list L of elements of M. When applied to M and L, it denotes the left product (L[1] * ( ... (L[n-1] * L[n]) ... )) of all elements in the list L.

Example:
$\mathrm{left_expression}\left(\mathrm{magma}\left(\mathbb{Z},×\right),\left(3,2\right)\right)=6$
Signatures:
sts

 [Next: right_expression] [Previous: right_divides] [Top]

## right_expression

Description:

This symbol is a binary function. Its first argument should be a magma M, the second argument a list L of elements of M When applied to M and L, it denotes the right product (( ... (L[1] * L[2]) * ... ) * L[n]) of all elements in the list L.

Example:
$\mathrm{right_expression}\left(\mathrm{magma}\left(\mathbb{Z},×\right),\left(3,2\right)\right)=6$
Signatures:
sts

 [First: magma] [Previous: left_expression] [Top]