<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- %% -->
<!-- %A  magma.msk                  GAP documentation                Thomas Breuer -->
<!-- %% -->
<!-- %A  @(#)<M>Id: magma.msk,v 1.10 2002/04/15 10:02:30 sal Exp </M> -->
<!-- %% -->
<!-- %Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland -->
<!-- %Y  Copyright (C) 2002 The GAP Group -->
<!-- %% -->
<Chapter Label="Magmas">
<Heading>Magmas</Heading>

This chapter deals with domains
(see&nbsp;<Ref Chap="Domains and their Elements"/>)
that are closed under multiplication <C>*</C>.
Following&nbsp;<Cite Key="Bourbaki70"/>, we call them <E>magmas</E> in &GAP;.
Together with the domains closed under addition <C>+</C>
(see&nbsp;<Ref Chap="Additive Magmas"/>),
they are the basic algebraic structures;
every semigroup,
monoid (see&nbsp;<Ref Chap="Semigroups"/>),
group (see&nbsp;<Ref Chap="Groups"/>),
ring (see&nbsp;<Ref Chap="Rings"/>),
or field (see&nbsp;<Ref Chap="Fields and Division Rings"/>) is a magma.
In the cases of a <E>magma-with-one</E> or <E>magma-with-inverses</E>,
additional multiplicative structure is present,
see&nbsp;<Ref Sect="Magma Categories"/>.
For functions to create free magmas, see&nbsp;<Ref Sect="Free Magmas"/>.


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Magma Categories">
<Heading>Magma Categories</Heading>

<#Include Label="IsMagma">
<#Include Label="IsMagmaWithOne">
<#Include Label="IsMagmaWithInversesIfNonzero">
<#Include Label="IsMagmaWithInverses">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Magma Generation">
<Heading>Magma Generation</Heading>

This section describes
functions that create magmas from generators
(see <Ref Func="Magma"/>, <Ref Func="MagmaWithOne"/>,
<Ref Func="MagmaWithInverses"/>),
the underlying operations for which methods can be installed
(see <Ref Func="MagmaByGenerators"/>, <Ref Func="MagmaWithOneByGenerators"/>,
<Ref Func="MagmaWithInversesByGenerators"/>),
functions for forming submagmas
(see <Ref Func="Submagma"/>, <Ref Func="SubmagmaWithOne"/>,
<Ref Func="SubmagmaWithInverses"/>),
and functions that form a magma equal to a given collection
(see <Ref Func="AsMagma"/>, <Ref Func="AsSubmagma"/>).
<P/>
<Ref Func="InjectionZeroMagma"/> creates a new magma which is the original
magma with a zero adjoined.

<#Include Label="Magma">
<#Include Label="MagmaWithOne">
<#Include Label="MagmaWithInverses">
<#Include Label="MagmaByGenerators">
<#Include Label="MagmaWithOneByGenerators">
<#Include Label="MagmaWithInversesByGenerators">
<#Include Label="Submagma">
<#Include Label="SubmagmaWithOne">
<#Include Label="SubmagmaWithInverses">
<#Include Label="AsMagma">
<#Include Label="AsSubmagma">
<#Include SYSTEM="mgmadj.xml">
</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Magmas Defined by Multiplication Tables">
<Heading>Magmas Defined by Multiplication Tables</Heading>

The most elementary (but of course usually not recommended) way to implement
a magma with only few elements is via a multiplication table.

<#Include Label="MagmaByMultiplicationTable">
<#Include Label="MagmaWithOneByMultiplicationTable">
<#Include Label="MagmaWithInversesByMultiplicationTable">
<#Include Label="MagmaElement">
<#Include Label="MultiplicationTable">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Attributes and Properties for Magmas">
<Heading>Attributes and Properties for Magmas</Heading>

<E>Note</E> that <Ref Func="IsAssociative"/> and <Ref Func="IsCommutative"/>
always refer to the multiplication of a domain.
If a magma <A>M</A> has also an <E>additive structure</E>,
e.g., if <A>M</A> is a ring (see&nbsp;<Ref Chap="Rings"/>),
then the addition <C>+</C> is always assumed to be associative and
commutative,
see&nbsp;<Ref Sect="Arithmetic Operations for Elements"/>.

<#Include Label="GeneratorsOfMagma">
<#Include Label="GeneratorsOfMagmaWithOne">
<#Include Label="GeneratorsOfMagmaWithInverses">
<#Include Label="Centralizer">
<#Include Label="Centre">
<#Include Label="Idempotents">
<#Include Label="IsAssociative">
<#Include Label="IsCentral">
<#Include Label="IsCommutative">
<#Include Label="MultiplicativeNeutralElement">
<#Include Label="MultiplicativeZero">
<#Include Label="SquareRoots">
<#Include Label="TrivialSubmagmaWithOne">

</Section>
</Chapter>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- %% -->
<!-- %E -->


