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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- %% -->
<!-- %A magma.xml GAP documentation Thomas Breuer -->
<!-- %% -->
<!-- %% -->
<!-- %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 <Ref Chap="Domains and their Elements"/>)
that are closed under multiplication <C>*</C>.
Following <Cite Key="Bourbaki70"/>, we call them <E>magmas</E> in &GAP;.
Together with the domains closed under addition <C>+</C>
(see <Ref Chap="Additive Magmas"/>),
they are the basic algebraic structures;
every semigroup,
monoid (see <Ref Chap="Semigroups"/>),
group (see <Ref Chap="Groups"/>),
ring (see <Ref Chap="Rings"/>),
or field (see <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 <Ref Sect="Magma Categories"/>.
For functions to create free magmas, see <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 Oper="MagmaByGenerators"/>, <Ref Oper="MagmaWithOneByGenerators"/>,
<Ref Oper="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 Attr="AsMagma"/>, <Ref Oper="AsSubmagma"/>).
<P/>
<Ref Attr="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 Prop="IsAssociative"/> and <Ref Prop="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 <Ref Chap="Rings"/>),
then the addition <C>+</C> is always assumed to be associative and
commutative,
see <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 -->
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