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products/Sources/formale Sprachen/GAP/pkg/semigroups/doc/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 29.7.2025 mit Größe 13 kB

Quelle semitrans.xml Sprache: XML

#############################################################################
##
#W semitrans.xml
#Y Copyright (C) 2011-14 James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##

<#GAPDoc Label="SmallestElementSemigroup">
<ManSection>
 <Attr Name = "SmallestElementSemigroup" Arg = "S"/>
 <Attr Name = "LargestElementSemigroup" Arg = "S"/>
 <Returns>A transformation.</Returns>
 <Description>
 These attributes return the smallest and largest element of the
 transformation semigroup <A>S</A>, respectively. Smallest means the first
 element in the sorted set of elements of <A>S</A> and largest means the last
 element in the set of elements. 

 It is not necessary to find the elements of the semigroup to determine the
 smallest or largest element, and this function has considerable better
 performance than the equivalent <C>Elements(<A>S</A>)[1]</C> and
 <C>Elements(<A>S</A>)[Size(<A>S</A>)]</C>.

 <Example><![CDATA[
gap> S := Monoid(
> Transformation([1, 4, 11, 11, 7, 2, 6, 2, 5, 5, 10]),
> Transformation([2, 4, 4, 2, 10, 5, 11, 11, 11, 6, 7]));
<transformation monoid of degree 11 with 2 generators>
gap> SmallestElementSemigroup(S);
IdentityTransformation
gap> LargestElementSemigroup(S);
Transformation( [ 11, 11, 10, 10, 7, 6, 5, 6, 2, 2, 4 ] )]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsTransitive">
<ManSection>
 <Prop Name = "IsTransitive" Arg = "S[, X]" Label = "for a transformation
 semigroup and a set"/>
 <Prop Name = "IsTransitive" Arg = "S[, n]" Label = "for a transformation
 semigroup and a pos int"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 A transformation semigroup <A>S</A> is <E>transitive</E> or <E>strongly
 connected</E> on the set <A>X</A> if for every <C>i, j</C> in <A>X</A>
 there is an element <C>s</C> in <A>S</A> such that <C>i ^ s = j</C>. 

 If the optional second argument is a positive integer <A>n</A>, then
 <C>IsTransitive</C> returns <K>true</K> if <A>S</A> is transitive on
 <C>[1 .. <A>n</A>]</C>, and <K>false</K> if it is not. 

 If the optional second argument is not provided, then the degree of
 <A>S</A> is used by default; see <Ref
 Attr = "DegreeOfTransformationSemigroup" BookName = "ref"/>.

 <Example><![CDATA[
gap> S := Semigroup([
> Bipartition([
> [1, 2], [3, 6, -2], [4, 5, -3, -4], [-1, -6], [-5]]),
> Bipartition([
> [1, -4], [2, 3, 4, 5], [6], [-1, -6], [-2, -3], [-5]])]);
<bipartition semigroup of degree 6 with 2 generators>
gap> AsSemigroup(IsTransformationSemigroup, S);
<transformation semigroup of size 11, degree 12 with 2 generators>
gap> IsTransitive(last);
false
gap> IsTransitive(AsSemigroup(Group((1, 2, 3))));
true]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="ComponentRepsOfTransformationSemigroup">
<ManSection>
 <Attr Name = "ComponentRepsOfTransformationSemigroup" Arg = "S"/>
 <Returns>The representatives of components of a transformation semigroup.</Returns>
 <Description>
 This function returns the representatives of the components of the
 action of the transformation semigroup <A>S</A> on the set of positive
 integers not greater than the degree of <A>S</A>. 

 The representatives are the least set of points such that every point can
 be reached from some representative under the action of <A>S</A>.

 <Example><![CDATA[
gap> S := Semigroup(
> Transformation([11, 11, 9, 6, 4, 1, 4, 1, 6, 7, 12, 5]),
> Transformation([12, 10, 7, 10, 4, 1, 12, 9, 11, 9, 1, 12]));;
gap> ComponentRepsOfTransformationSemigroup(S);
[ 2, 3, 8 ]]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="ComponentsOfTransformationSemigroup">
<ManSection>
 <Attr Name = "ComponentsOfTransformationSemigroup" Arg = "S"/>
 <Returns>The components of a transformation semigroup.</Returns>
 <Description>
 This function returns the components of the action of the transformation
 semigroup <A>S</A> on the set of positive integers not greater than the
 degree of <A>S</A>; the components of <A>S</A> partition this set.

 <Example><![CDATA[
gap> S := Semigroup(
> Transformation([11, 11, 9, 6, 4, 1, 4, 1, 6, 7, 12, 5]),
> Transformation([12, 10, 7, 10, 4, 1, 12, 9, 11, 9, 1, 12]));;
gap> ComponentsOfTransformationSemigroup(S);
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] ]]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="CyclesOfTransformationSemigroup">
<ManSection>
 <Attr Name = "CyclesOfTransformationSemigroup" Arg = "S"/>
 <Returns>The cycles of a transformation semigroup.</Returns>
 <Description>
 This function returns the cycles, or strongly connected components, of the
 action of the transformation semigroup <A>S</A> on the set of positive
 integers not greater than the degree of <A>S</A>.
 <Example><![CDATA[
gap> S := Semigroup(
> Transformation([11, 11, 9, 6, 4, 1, 4, 1, 6, 7, 12, 5]),
> Transformation([12, 10, 7, 10, 4, 1, 12, 9, 11, 9, 1, 12]));;
gap> CyclesOfTransformationSemigroup(S);
[ [ 1, 11, 12, 5, 4, 6, 10, 7, 9 ], [ 2 ], [ 3 ], [ 8 ] ]]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsSynchronizingSemigroup">
 <ManSection>
 <Prop Name = "IsSynchronizingSemigroup" Arg = "S"
 Label = "for a transformation semigroup" />
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 For a positive integer <A>n</A>, <C>IsSynchronizingSemigroup</C> returns
 <K>true</K> if the semigroup of transformations <A>S</A> contains a
 transformation with constant value on <C>[1 .. <A>n</A>]</C>
 where <C>n</C> is the degree of the semigroup.
 See also
 <Ref Oper = "ConstantTransformation" BookName = "ref"/>.
 

 Note that the semigroup consisting of the identity transformation is the
 unique transformation semigroup with degree <C>0</C>. In this special
 case, the function <C>IsSynchronizingSemigroup</C> will return
 <K>false</K>.

 <Example><![CDATA[
gap> S := Semigroup(
> Transformation([1, 1, 8, 7, 6, 6, 4, 1, 8, 9]),
> Transformation([5, 8, 7, 6, 10, 8, 7, 6, 9, 7]));;
gap> IsSynchronizingSemigroup(S);
true
gap> S := Semigroup(
> Transformation([3, 8, 1, 1, 9, 9, 8, 7, 9, 6]),
> Transformation([7, 6, 8, 7, 5, 6, 8, 7, 8, 9]));;
gap> IsSynchronizingSemigroup(S);
false
gap> Representative(MinimalIdeal(S));
Transformation( [ 8, 7, 7, 7, 8, 8, 7, 8, 8, 8 ] )]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphOfAction">
 <ManSection>
 <Oper Name = "DigraphOfAction" Arg = "S, list, act"
 Label = "for a transformation semigroup, list, and action" />
 <Returns>A digraph, or <K>fail</K>.</Returns>
 <Description>
 If <A>S</A> is a transformation semigroup and <A>list</A> is list such
 that <A>S</A> acts on the items in <A>list</A> via the function
 <A>act</A>, then <C>DigraphOfAction</C> returns a digraph representing
 the action of <A>S</A> on the items in <A>list</A> and any further items
 output by <C><A>act</A>(<A>list</A>[i], <A>S</A>.j)</C>.

 If <C><A>act</A>(<A>list</A>[i], <A>S</A>.j)</C> is the <C>k</C>-th item
 in <A>list</A>, then in the output digraph there is an edge from the
 vertex <C>i</C> to the vertex <C>k</C> labelled by <C>j</C>.
 

 The values in <A>list</A> and the additional values generated are stored
 in the vertex labels of the output digraph; see <Ref
 Oper="DigraphVertexLabels" BookName="Digraphs" />, and the edge labels
 are stored in the <Ref Oper="DigraphEdgeLabels" BookName="Digraphs" />
 

 The digraph returned by <C>DigraphOfAction</C> has no multiple edges; see
 <Ref Prop="IsMultiDigraph" BookName="Digraphs"/>.

 <Example><![CDATA[
gap> S := Semigroup(Transformation([2, 4, 3, 4, 7, 1, 6]),
> Transformation([3, 3, 2, 3, 5, 1, 5]));
<transformation semigroup of degree 7 with 2 generators>
gap> list := Concatenation(List([1 .. 7], x -> [x]),
> Combinations([1 .. 7], 2));
[ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ], [ 1, 2 ],
 [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], [ 2, 3 ],
 [ 2, 4 ], [ 2, 5 ], [ 2, 6 ], [ 2, 7 ], [ 3, 4 ], [ 3, 5 ],
 [ 3, 6 ], [ 3, 7 ], [ 4, 5 ], [ 4, 6 ], [ 4, 7 ], [ 5, 6 ],
 [ 5, 7 ], [ 6, 7 ] ]
gap> D := DigraphOfAction(S, list, OnSets);
<immutable digraph with 28 vertices, 54 edges>
gap> OnSets([2, 5], S.1);
[ 4, 7 ]
gap> Position(DigraphVertexLabels(D), [2, 5]);
16
gap> DigraphVertexLabel(D, 25);
[ 4, 7 ]
gap> DigraphEdgeLabel(D, 16, 25);
1]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphOfActionOnPoints">
 <ManSection>
 <Attr Name = "DigraphOfActionOnPoints" Arg = "S"
 Label = "for a transformation semigroup" />
 <Attr Name = "DigraphOfActionOnPoints" Arg = "S, n"
 Label = "for a transformation semigroup and an integer" />
 <Returns>A digraph.</Returns>
 <Description>
 If <A>S</A> is a transformation semigroup and <A>n</A> is a non-negative
 integer, then <C>DigraphOfActionOnPoints(<A>S</A>, <A>n</A>)</C> returns
 a digraph representing the <Ref Func="OnPoints" BookName="ref"/> action
 of <A>S</A> on the set <C>[1 .. <A>n</A>]</C>.
 

 If the optional argument <A>n</A> is not specified, then by default the
 degree of <A>S</A> will be chosen for <A>n</A>; see <Ref
 Attr="DegreeOfTransformationSemigroup" BookName="ref" />.
 

 The digraph returned by <C>DigraphOfActionOnPoints</C> has <A>n</A>
 vertices, where the vertex <C>i</C> corresponds to the point <C>i</C>.
 For each point <C>i</C> in <C>[1 .. <A>n</A>]</C>, and for each generator
 <C>f</C> in <C>GeneratorsOfSemigroup(<A>S</A>)</C>, there is an edge from
 the vertex <C>i</C> to the vertex <C>i ^ f</C>. See <Ref
 Attr="GeneratorsOfSemigroup" BookName="ref" /> for further information.
 

 <Example><![CDATA[
gap> S := Semigroup(Transformation([2, 4, 2, 4, 7, 1, 6]),
> Transformation([3, 3, 2, 3, 5, 1, 5]));
<transformation semigroup of degree 7 with 2 generators>
gap> D1 := DigraphOfActionOnPoints(S);
<immutable digraph with 7 vertices, 12 edges>
gap> OnPoints(2, S.1);
4
gap> D2 := DigraphOfActionOnPoints(S, 4);
<immutable digraph with 4 vertices, 7 edges>
gap> D2 = InducedSubdigraph(D1, [1 .. 4]);
true
gap> DigraphOfActionOnPoints(S, 5);
<immutable digraph with 5 vertices, 8 edges>]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsConnectedTransformationSemigroup">
 <ManSection>
 <Prop Name = "IsConnectedTransformationSemigroup" Arg = "S"
 Label = "for a transformation semigroup" />
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 A transformation semigroup <A>S</A> is connected if the digraph returned
 by the function <C>DigraphOfActionOnPoints</C> is connected. See
 <Ref Prop = "IsConnectedDigraph" BookName = "digraphs" /> and <Ref Attr
 = "DigraphOfActionOnPoints" Label = "for a transformation semigroup"/>.
 The function <C>IsConnectedTransformationSemigroup</C> returns <K>true</K>
 if the semigroup <A>S</A> is connected and <K>false</K> otherwise.

 <Example><![CDATA[
gap> S := Semigroup([
> Transformation([2, 4, 3, 4]),
> Transformation([3, 3, 2, 3, 3])]);
<transformation semigroup of degree 5 with 2 generators>
gap> IsConnectedTransformationSemigroup(S);
true]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="FixedPointsOfTransformationSemigroup">
 <ManSection>
 <Attr Name="FixedPointsOfTransformationSemigroup" Arg="S"
 Label="for a transformation semigroup"/>
 <Returns>A set of positive integers.</Returns>
 <Description>
 If <A>S</A> is a transformation semigroup, then
 <C>FixedPointsOfTransformationSemigroup(<A>S</A>)</C> returns the set of
 points <C>i</C> in <C>[1 .. DegreeOfTransformationSemigroup(<A>S</A>)]</C>
 such that <C>i ^ f = i</C> for all <C>f</C> in <A>S</A>. 
 <Example><![CDATA[
gap> f := Transformation([1, 4, 2, 4, 3, 7, 7]);
Transformation( [ 1, 4, 2, 4, 3, 7, 7 ] )
gap> S := Semigroup(f);
<commutative transformation semigroup of degree 7 with 1 generator>
gap> FixedPointsOfTransformationSemigroup(S);
[ 1, 4, 7 ]]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="WreathProduct">
 <ManSection>
 <Oper Name = "WreathProduct" Arg = "M, S"/>
 <Returns>A transformation semigroup.</Returns>
 <Description>
 If <A>M</A> is a transformation monoid (or a permutation group) of degree
 <C>m</C>, and <A>S</A> is a transformation semigroup (or permutation
 group) of degree <C>s</C>, the operation
 <C>WreathProduct(<A>M</A>, <A>S</A>)</C> returns the wreath product of
 <A>M</A> and <A>S</A>, in terms of an embedding in the full
 transformation monoid of degree <C>m * s</C>.

 <Example><![CDATA[
gap> T := FullTransformationMonoid(3);;
gap> C := Group((1, 3));;
gap> W := WreathProduct(T, C);;
gap> Size(W);
39366
gap> WW := WreathProduct(C, T);;
gap> Size(WW);
216
]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

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