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##
#W froidure-pin.xml
#Y Copyright (C) 2017-2022 Michael Young
##
## Licensing information can be found in the README file of this package.
##
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##
<#GAPDoc Label="CanUseFroidurePin">
<ManSection>
<Prop Name = "CanUseFroidurePin" Arg = "obj"/>
<Prop Name = "CanUseGapFroidurePin" Arg = "obj"/>
<Prop Name = "CanUseLibsemigroupsFroidurePin" Arg = "obj"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
Every semigroup satisfying <C>CanUseFroidurePin</C> is a valid input
for the Froidure-Pin algorithm; see Section <Ref Subsect="FroidurePin"/>
for more details.<P/>
<#GAPDoc Label="AsListCanonical">
<ManSection>
<Attr Name="AsListCanonical" Arg="S"/>
<Attr Name="EnumeratorCanonical" Arg="S"/>
<Oper Name="IteratorCanonical" Arg="S"/>
<Returns>A list, enumerator, or iterator.</Returns>
<Description>
When the argument <A>S</A> is a semigroup satisfying
<Ref Prop="CanUseFroidurePin"/>, <C>AsListCanonical</C>
returns a list of the elements of <A>S</A> in the order they are
enumerated by the Froidure-Pin Algorithm. This is the same as the order
used to index the elements of <A>S</A> in <Ref
Attr="RightCayleyDigraph"/> and <Ref
Attr="LeftCayleyDigraph"/>. <P/>
<C>EnumeratorCanonical</C> and <C>IteratorCanonical</C> return an
enumerator and an iterator where the elements are
ordered in the same way as <C>AsListCanonical</C>. Using
<C>EnumeratorCanonical</C> or <C>IteratorCanonical</C> will often use
less memory than <C>AsListCanonical</C>, but may have slightly worse
performance if the elements of the semigroup are looped over repeatedly.
<C>EnumeratorCanonical</C> returns the same list as
<C>AsListCanonical</C> if <C>AsListCanonical</C> has ever been called for
<A>S</A>.<P/>
If <A>S</A> is an acting semigroup, then the value returned by
<C>AsList</C> may not equal the value returned by <C>AsListCanonical</C>.
<C>AsListCanonical</C> exists so that there is a method for obtaining the
elements of <A>S</A> in the particular order used by
<Ref Attr="RightCayleyDigraph"/> and
<Ref Attr="LeftCayleyDigraph"/>.<P/>
<#GAPDoc Label="Enumerate">
<ManSection>
<Oper Name="Enumerate" Arg="S[, limit]"/>
<Returns>A semigroup (the argument).</Returns>
<Description>
If <A>S</A> is a semigroup with representation
<Ref Prop="CanUseFroidurePin"/> and <A>limit</A> is a positive
integer, then this operation can be used to enumerate at least
<A>limit</A> elements of <A>S</A>, or <C>Size(<A>S</A>)</C> elements if
this is less than <A>limit</A>, using the Froidure-Pin Algorithm. <P/>
If the optional second argument <A>limit</A> is not given, then the
semigroup is enumerated until all of its elements have been found. <P/>
<Example><![CDATA[
gap> S := FullTransformationMonoid(7);
<full transformation monoid of degree 7>
gap> Enumerate(S, 1000);
<full transformation monoid of degree 7>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="IsEnumerated">
<ManSection>
<Oper Name="IsEnumerated" Arg="S"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
If <A>S</A> is a semigroup with representation
<Ref Prop="CanUseFroidurePin"/>, then this operation returns
<K>true</K> if the Froidure-Pin Algorithm has been run to completion
(i.e. all of the elements of <A>S</A> have been found) and <K>false</K>
if <A>S</A> has not been fully enumerated.
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="PositionCanonical">
<ManSection>
<Oper Name="PositionCanonical" Arg="S, x"/>
<Returns>A positive integer or <K>fail</K>.</Returns>
<Description>
When the argument <A>S</A> is a semigroup satisfying
<Ref Prop="CanUseFroidurePin"/> and <A>x</A> is an element of
<A>S</A>, <C>PositionCanonical</C>
returns the position of <A>x</A> in <C>AsListCanonical(<A>S</A>)</C> or
equivalently the position of <A>x</A> in
<C>EnumeratorCanonical(<A>S</A>)</C>.<P/>
See also <Ref Attr="AsListCanonical"/> and
<Ref Attr="EnumeratorCanonical"/>.
<Example><![CDATA[
gap> S := FullTropicalMaxPlusMonoid(2, 3);
<monoid of 2x2 tropical max-plus matrices with 13 generators>
gap> x := Matrix(IsTropicalMaxPlusMatrix, [[1, 3], [2, 1]], 3);
Matrix(IsTropicalMaxPlusMatrix, [[1, 3], [2, 1]], 3)
gap> PositionCanonical(S, x);
234
gap> EnumeratorCanonical(S)[234] = x;
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="RightCayleyDigraph">
<ManSection>
<Attr Name="RightCayleyDigraph" Arg="S"/>
<Attr Name="LeftCayleyDigraph" Arg="S"/>
<Returns>A digraph.</Returns>
<Description>
When the argument <A>S</A> is a semigroup satisfying
<Ref Prop="CanUseFroidurePin"/>,
<C>RightCayleyDigraph</C> returns the right
Cayley graph of <A>S</A>, as a <Ref Oper="Digraph" BookName="digraphs"/>
<C>digraph</C> where vertex <C>OutNeighbours(digraph)[i][j]</C> is
<C>PositionCanonical(<A>S</A>, AsListCanonical(<A>S</A>)[i] *
GeneratorsOfSemigroup(<A>S</A>)[j])</C>.
The digraph returned by <C>LeftCayleyDigraph</C> is defined analogously.<P/>
The digraph returned by this attribute belongs to the category
<Ref Filt="IsCayleyDigraph" BookName="digraphs"/>, the semigroup <A>S</A>
and the generators used to create the digraph can be recovered from the
digraph using <Ref Attr="SemigroupOfCayleyDigraph" BookName="digraphs"/>
and <Ref Attr="GeneratorsOfCayleyDigraph" BookName="digraphs"/>.
<Example><![CDATA[
gap> S := FullTransformationMonoid(2);
<full transformation monoid of degree 2>
gap> RightCayleyDigraph(S);
<immutable multidigraph with 4 vertices, 12 edges>
gap> LeftCayleyDigraph(S);
<immutable multidigraph with 4 vertices, 12 edges>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
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