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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- %% -->
<!-- %A semigrp.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="Semigroups">
<Heading>Semigroups and Monoids</Heading>
This chapter describes functions for creating semigroups and monoids
and determining information about them.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:IsSemigroup">
<Heading>Semigroups</Heading>
<#Include Label="IsSemigroup">
<#Include Label="Semigroup">
<#Include Label="Subsemigroup">
<#Include Label="IsSubsemigroup">
<#Include Label="SemigroupByGenerators">
<#Include Label="AsSemigroup">
<#Include Label="AsSubsemigroup">
<#Include Label="GeneratorsOfSemigroup">
<#Include Label="IsGeneratorsOfSemigroup">
<#Include Label="FreeSemigroup">
<#Include Label="SemigroupByMultiplicationTable">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:IsMonoid">
<Heading>Monoids</Heading>
<#Include Label="IsMonoid">
<#Include Label="Monoid">
<#Include Label="Submonoid">
<#Include Label="MonoidByGenerators">
<#Include Label="AsMonoid">
<#Include Label="AsSubmonoid">
<#Include Label="GeneratorsOfMonoid">
<#Include Label="TrivialSubmonoid">
<#Include Label="FreeMonoid">
<#Include Label="MonoidByMultiplicationTable">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<#Include SYSTEM "invsgp.xml">
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Properties of Semigroups">
<Heading>Properties of Semigroups</Heading>
The following functions determine information
about semigroups.
<#Include Label="IsRegularSemigroup">
<#Include Label="IsRegularSemigroupElement">
<#Include Label="InversesOfSemigroupElement">
<#Include Label="IsSimpleSemigroup">
<#Include Label="IsZeroSimpleSemigroup">
<#Include Label="IsZeroGroup">
<#Include Label="IsReesCongruenceSemigroup">
<#Include Label="IsInverseSemigroup">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Ideals of semigroups">
<Heading>Ideals of semigroups</Heading>
Ideals of semigroups are the same as ideals of the semigroup when
considered as a magma.
For documentation on ideals for magmas, see <Ref Func="Magma"/>.
<#Include Label="SemigroupIdealByGenerators">
<#Include Label="ReesCongruenceOfSemigroupIdeal">
<#Include Label="IsLeftSemigroupIdeal">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Congruences on semigroups">
<Heading>Congruences on semigroups</Heading>
An equivalence or a congruence on a semigroup is the
equivalence or congruence on the semigroup considered as a magma.
So, to deal with equivalences and congruences on semigroups,
magma functions are used.
For documentation on equivalences and congruences on magmas,
see <Ref Func="Magma"/>.
<#Include Label="IsSemigroupCongruence">
<#Include Label="IsReesCongruence">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Quotients">
<Heading>Quotients</Heading>
Given a semigroup and a congruence on the semigroup, one
can construct a new semigroup: the quotient semigroup.
The following functions deal with quotient semigroups in &GAP;.
<#Include Label="[1]{semiquo}">
<#Include Label="IsQuotientSemigroup">
<#Include Label="HomomorphismQuotientSemigroup">
<#Include Label="QuotientSemigroupPreimage">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Green's Relations">
<Heading>Green's Relations</Heading>
<#Include Label="[1]{semirel}">
<#Include Label="GreensRRelation">
<#Include Label="IsGreensRelation">
<#Include Label="IsGreensClass">
<#Include Label="IsGreensLessThanOrEqual">
<#Include Label="RClassOfHClass">
<#Include Label="EggBoxOfDClass">
<#Include Label="DisplayEggBoxOfDClass">
<#Include Label="GreensRClassOfElement">
<#Include Label="GreensRClasses">
<#Include Label="GroupHClassOfGreensDClass">
<#Include Label="IsGroupHClass">
<#Include Label="IsRegularDClass">
<#Include Label="DisplaySemigroup">
</Section>
<#Include SYSTEM "reesmat.xml">
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
</Chapter>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- %% -->
<!-- %E -->
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