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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the declaration of operations for monoids.
##
#############################################################################
##
#P IsMonoid( <D> )
##
## <#GAPDoc Label="IsMonoid">
## <ManSection>
## <Filt Name="IsMonoid" Arg='D' Type='Synonym'/>
##
## <Description>
## A <E>monoid</E> is a magma-with-one (see <Ref Chap="Magmas"/>)
## with associative multiplication.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsMonoid", IsMagmaWithOne and IsAssociative );
#############################################################################
##
#F Monoid( <gen1>, <gen2> ... )
#F Monoid( <gens> )
#F Monoid( <gens>, <id> )
##
## <#GAPDoc Label="Monoid">
## <ManSection>
## <Heading>Monoid</Heading>
## <Func Name="Monoid" Arg='gen1, gen2 ...'
## Label="for various generators"/>
## <Func Name="Monoid" Arg='gens[, id]' Label="for a list"/>
##
## <Description>
## In the first form, <Ref Func="Monoid" Label="for various generators"/>
## returns the monoid generated by the arguments <A>gen1</A>, <A>gen2</A>,
## <M>\ldots</M>,
## that is, the closure of these elements under multiplication and taking
## the 0-th power.
## In the second form, <Ref Func="Monoid" Label="for a list"/> returns
## the monoid generated by the elements in the homogeneous list <A>gens</A>;
## a square matrix as only argument is treated as one generator,
## not as a list of generators.
## In the second form, the identity element <A>id</A> may be given as the
## second argument.
## <P/>
## It is <E>not</E> checked whether the underlying multiplication is
## associative, use <Ref Func="MagmaWithOne"/> and
## <Ref Prop="IsAssociative"/>
## if you want to check whether a magma-with-one is in fact a monoid.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "Monoid" );
#############################################################################
##
#F Submonoid( <M>, <gens> ) . . . . . . submonoid of <M> generated by <gens>
#F SubmonoidNC( <M>, <gens> )
##
## <#GAPDoc Label="Submonoid">
## <ManSection>
## <Func Name="Submonoid" Arg='M, gens'/>
## <Func Name="SubmonoidNC" Arg='M, gens'/>
##
## <Description>
## are just synonyms of <Ref Func="SubmagmaWithOne"/>
## and <Ref Func="SubmagmaWithOneNC"/>, respectively.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "Submonoid", SubmagmaWithOne );
DeclareSynonym( "SubmonoidNC", SubmagmaWithOneNC );
#############################################################################
##
#O MonoidByGenerators( <gens>[, <one>] ) . . . . monoid generated by <gens>
##
## <#GAPDoc Label="MonoidByGenerators">
## <ManSection>
## <Oper Name="MonoidByGenerators" Arg='gens[, one]'/>
##
## <Description>
## is the underlying operation of <Ref Func="Monoid" Label="for a list"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "MonoidByGenerators", [ IsCollection ] );
#############################################################################
##
#A AsMonoid( <C> ) . . . . . . . . . . . . collection <C> regarded as monoid
##
## <#GAPDoc Label="AsMonoid">
## <ManSection>
## <Oper Name="AsMonoid" Arg='C'/>
##
## <Description>
## If <A>C</A> is a collection whose elements form a monoid,
## then <Ref Oper="AsMonoid"/> returns this monoid.
## Otherwise <K>fail</K> is returned.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "AsMonoid", [IsCollection] );
#############################################################################
##
#O AsSubmonoid( <D>, <C> )
##
## <#GAPDoc Label="AsSubmonoid">
## <ManSection>
## <Oper Name="AsSubmonoid" Arg='D, C'/>
##
## <Description>
## Let <A>D</A> be a domain and <A>C</A> a collection.
## If <A>C</A> is a subset of <A>D</A> that forms a monoid then
## <Ref Oper="AsSubmonoid"/>
## returns this monoid, with parent <A>D</A>.
## Otherwise <K>fail</K> is returned.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "AsSubmonoid", [ IsDomain, IsCollection ] );
#############################################################################
##
#A GeneratorsOfMonoid( <M> ) . . . . . . . monoid generators of monoid <M>
##
## <#GAPDoc Label="GeneratorsOfMonoid">
## <ManSection>
## <Attr Name="GeneratorsOfMonoid" Arg='M'/>
##
## <Description>
## Monoid generators of a monoid <A>M</A> are the same as
## magma-with-one generators
## (see <Ref Attr="GeneratorsOfMagmaWithOne"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "GeneratorsOfMonoid", GeneratorsOfMagmaWithOne );
#############################################################################
##
#A TrivialSubmonoid( <M> ) . . . . . . . . . trivial submonoid of monoid <M>
##
## <#GAPDoc Label="TrivialSubmonoid">
## <ManSection>
## <Attr Name="TrivialSubmonoid" Arg='M'/>
##
## <Description>
## is just a synonym for <Ref Attr="TrivialSubmagmaWithOne"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "TrivialSubmonoid", TrivialSubmagmaWithOne );
#############################################################################
##
#F FreeMonoid( [<wfilt>,]<rank>[, <name>] )
#F FreeMonoid( [<wfilt>,][<name1>[, <name2>[, ...]]] )
#F FreeMonoid( [<wfilt>,]<names> )
#F FreeMonoid( [<wfilt>,]infinity[, <name>][, <init>] )
##
## <#GAPDoc Label="FreeMonoid">
## <ManSection>
## <Heading>FreeMonoid</Heading>
## <Func Name="FreeMonoid" Arg='[wfilt, ]rank[, name]'
## Label="for given rank"/>
## <Func Name="FreeMonoid" Arg='[wfilt, ][name1[, name2[, ...]]]'
## Label="for various names"/>
## <Func Name="FreeMonoid" Arg='[wfilt, ]names'
## Label="for a list of names"/>
## <Func Name="FreeMonoid" Arg='[wfilt, ]infinity[, name][, init]'
## Label="for infinitely many generators"/>
##
## <Description>
## <C>FreeMonoid</C> returns a free monoid. The number of
## generators, and the labels given to the generators, can be specified in
## several different ways.
## Warning: the labels of generators are only an aid for printing,
## and do not necessarily distinguish generators;
## see the examples at the end of
## <Ref Func="FreeSemigroup" Label="for given rank"/>
## for more information.
## <List>
## <Mark>
## 1: For a given rank, and an optional generator name prefix
## </Mark>
## <Item>
## Called with a nonnegative integer <A>rank</A>,
## <Ref Func="FreeMonoid" Label="for given rank"/> returns
## a free monoid on <A>rank</A> generators.
## The optional argument <A>name</A> must be a string;
## its default value is <C>"m"</C>. <P/>
##
## If <A>name</A> is not given but the <C>generatorNames</C> option is,
## then this option is respected as described in
## Section <Ref Sect="Generator Names"/>. <P/>
##
## Otherwise, the generators of the returned free monoid are labelled
## <A>name</A><C>1</C>, ..., <A>name</A><C>k</C>,
## where <C>k</C> is the value of <A>rank</A>. <P/>
## </Item>
## <Mark>2: For given generator names</Mark>
## <Item>
## Called with various nonempty strings,
## <Ref Func="FreeMonoid" Label="for various names"/> returns
## a free monoid on as many generators as arguments, which are labelled
## <A>name1</A>, <A>name2</A>, etc.
## </Item>
## <Mark>3: For a given list of generator names</Mark>
## <Item>
## Called with a finite list <A>names</A> of
## nonempty strings,
## <Ref Func="FreeMonoid" Label="for a list of names"/> returns
## a free monoid on <C>Length(<A>names</A>)</C> generators, whose
## <C>i</C>-th generator is labelled <A>names</A><C>[i]</C>.
## </Item>
## <Mark>
## 4: For the rank <K>infinity</K>,
## an optional default generator name prefix,
## and an optional finite list of generator names
## </Mark>
## <Item>
## Called in the fourth form,
## <Ref Func="FreeMonoid" Label="for infinitely many generators"/>
## returns a free monoid on infinitely many generators.
## The optional argument <A>name</A> must be a string; its default value is
## <C>"m"</C>,
## and the optional argument <A>init</A> must be a finite list of
## nonempty strings; its default value is an empty list.
## The generators are initially labelled according to the list <A>init</A>,
## followed by
## <A>name</A><C>i</C> for each <C>i</C> in the range from
## <C>Length(<A>init</A>)+1</C> to <K>infinity</K>.
## </Item>
## </List>
##
## If the optional first argument <A>wfilt</A> is given, then it must be either
## <C>IsSyllableWordsFamily</C>, <C>IsLetterWordsFamily</C>,
## <C>IsWLetterWordsFamily</C>, or <C>IsBLetterWordsFamily</C>.
## This filter specifies the representation used for the elements of
## the free monoid
## (see <Ref Sect="Representations for Associative Words"/>).
## If no such filter is given, a letter representation is used.
## <P/>
## For more on associative words see
## Chapter <Ref Chap="Associative Words"/>.
##
## <Example><![CDATA[
## gap> FreeMonoid(5);
## <free monoid on the generators [ m1, m2, m3, m4, m5 ]>
## gap> FreeMonoid(4, "gen");
## <free monoid on the generators [ gen1, gen2, gen3, gen4 ]>
## gap> FreeMonoid(3 : generatorNames := "turbo");
## <free monoid on the generators [ turbo1, turbo2, turbo3 ]>
## gap> FreeMonoid(2 : generatorNames := ["u", "v", "w"]);
## <free monoid on the generators [ u, v ]>
## gap> FreeMonoid(); FreeMonoid(0); FreeMonoid([]);
## <free monoid of rank zero>
## <free monoid of rank zero>
## <free monoid of rank zero>
## gap> FreeMonoid("a", "b", "c");
## <free monoid on the generators [ a, b, c ]>
## gap> FreeMonoid(["x", "y"]);
## <free monoid on the generators [ x, y ]>
## gap> FreeMonoid(infinity);
## <free monoid with infinity generators>
## gap> F := FreeMonoid(infinity, "g", ["a", "b"]);
## <free monoid with infinity generators>
## gap> GeneratorsOfMonoid(F){[1..4]};
## [ a, b, g3, g4 ]
## gap> GeneratorsOfMonoid(FreeMonoid(infinity, "gen")){[1..3]};
## [ gen1, gen2, gen3 ]
## gap> GeneratorsOfMonoid(FreeMonoid(infinity, [ "f", "g" ])){[1..3]};
## [ f, g, m3 ]
## gap> FreeMonoid(IsSyllableWordsFamily, 50);
## <free monoid with 50 generators>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "FreeMonoid" );
#############################################################################
##
#P IsFinitelyGeneratedMonoid( <M> )
##
## <#GAPDoc Label="IsFinitelyGeneratedMonoid">
## <ManSection>
## <Prop Name="IsFinitelyGeneratedMonoid" Arg='M'/>
##
## <Description>
## tests whether the monoid <A>M</A> can be generated by a finite number of
## generators. (This property is mainly used to obtain finiteness
## conditions.)
## <P/>
## Note that this is a pure existence statement. Even if a monoid is known
## to be generated by a finite number of elements, it can be very hard or
## even impossible to obtain such a generating set if it is not known.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsFinitelyGeneratedMonoid", IsMonoid );
InstallTrueMethod( IsMonoid, IsFinitelyGeneratedMonoid );
# make IsFinitelyGeneratedMonoid equivalent to IsMonoid and IsFinitelyGeneratedMagma
InstallTrueMethod( IsFinitelyGeneratedMonoid, IsMonoid and IsFinitelyGeneratedMagma );
InstallTrueMethod( HasIsFinitelyGeneratedMonoid, IsMonoid and HasIsFinitelyGeneratedMagma );
InstallTrueMethod( IsFinitelyGeneratedMagma, IsFinitelyGeneratedMonoid );
InstallTrueMethod( HasIsFinitelyGeneratedMagma, HasIsFinitelyGeneratedMonoid );
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