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McAlister triples
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| ############################################################################ | ||
| ## | ||
| #W semieunit.xml | ||
| #Y Copyright (C) 2016 Christopher Russell | ||
| ## | ||
| ## Licensing information can be found in the README file of this package. | ||
| ## | ||
| ############################################################################# | ||
| ## | ||
|
|
||
| <#GAPDoc Label="IsMcAlisterTripleSemigroup"> | ||
| <ManSection> | ||
| <Filt Name = "IsMcAlisterTripleSemigroup" Arg = "S"/> | ||
| <Returns><K>true</K> or <K>false</K>.</Returns> | ||
| <Description> | ||
| This function returns <K>true</K> if <A>S</A> is a McAlister triple semigroup. | ||
| A <E>McAlister triple semigroup</E> is a special representation of an | ||
| E-unitary inverse semigroup <Ref Oper="IsEUnitaryInverseSemigroup"/> | ||
| created by <Ref Oper="McAlisterTripleSemigroup"/>. | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> | ||
|
|
||
| <#GAPDoc Label="McAlisterTripleSemigroup"> | ||
| <ManSection> | ||
| <Oper Name = "McAlisterTripleSemigroup" Arg = "G, X, Y[, act]"/> | ||
| <Returns>A McAlister triple semigroup.</Returns> | ||
| <Description> | ||
|
|
||
| The following documentation covers the technical information needed to | ||
| create McAlister triple semigroups in GAP, the underlying theory can be | ||
| read in the introduction to Chapter | ||
| <Ref Chap = "McAlister triple semigroups"/>. | ||
| <P/> | ||
|
|
||
| In this implementation the partial order <C>X</C> of a McAlister triple is | ||
| represented by a <Ref Oper="Digraph" BookName="Digraphs"/> object <A>X</A>. | ||
| The digraph represents a partial order in the sense that vertices are the | ||
| elements of the partial order and the order relation is defined by | ||
| <C>A</C> <M>\leq</M> <C>B</C> if and only if there is an edge from <C>B</C> | ||
| to <C>A</C>. The semilattice <C>Y</C> of the McAlister triple should be an | ||
| induced subdigraph <A>Y</A> of <A>X</A> and the | ||
| <Ref Oper="DigraphVertexLabels" BookName="Digraphs"/> must correspond to | ||
| the vertices of <A>X</A> on which <A>Y</A> is induced. That means that | ||
| the following: | ||
| <P/> | ||
|
|
||
| <C><A>Y</A> = InducedSubdigraph(<A>X</A>, DigraphVertexLabels(<A>Y</A>)) | ||
| </C><P/> | ||
|
|
||
| must return <K>true</K>. Herein if we say that a vertex <C>A</C> of <A>X</A> | ||
| is 'in' <A>Y</A> then we mean there is a vertex of <A>Y</A> whose label is | ||
| <C>A</C>. Alerternatively the user may choose to give the argument | ||
| <A>Y</A> as the vertices of <A>X</A> on which <A>Y</A> is the induced | ||
| subdigraph. <P/> | ||
|
|
||
| A McAlister triple semigroup is created from a quadruple | ||
| <C>(<A>G</A>, <A>X</A>, <A>Y</A>, <A>act</A>)</C> where: | ||
|
|
||
| <List> | ||
| <Item> | ||
| <A>G</A> is a finite group. | ||
| </Item> | ||
| <Item> | ||
| <A>X</A> is a digraph satisfying | ||
| <Ref Prop= "IsPartialOrderDigraph" BookName="Digraphs"/>. | ||
| </Item> | ||
| <Item> | ||
| <A>Y</A> is a digraph satisfying | ||
| <Ref Prop="IsJoinSemilatticeDigraph" BookName="Digraphs"/> which | ||
| is an induced subdigraph of <A>X</A> satisfying the aforementioned | ||
| labeling criteria. Furthermore the <Ref Attr="OutNeighbours" | ||
| BookName="Digraphs"/> of each vertex of <A>X</A> which is in <A>Y</A> | ||
| must contain only vertices which are in <A>Y</A>. | ||
| </Item> | ||
| <Item> | ||
| <A>act</A> is a function which takes as its first argument a vertex | ||
| of the digraph <A>X</A>, its second argument should be an element of | ||
| <A>G</A>, and it must return a vertex of <A>X</A>. | ||
| <A>act</A> must be a right action, meaning that | ||
| <A>act</A><C>(A,gh)=<A>act</A>(<A>act</A>(A,g),h)</C> holds for all | ||
| <C>A</C> in <A>X</A> and <C>g,h</C> <M>\in</M> <A>G</A>. Furthermore | ||
| the permutation represenation of this action must be a subgroup of the | ||
| automorphism group of <A>X</A>. That means we require the following | ||
| to return <K>true</K>: <P/> | ||
| <C>IsSubgroup(AutomorphismGroup(</C><A>X</A><C>), | ||
| Image(ActionHomomorphism(</C><A>G</A><C>, | ||
| DigraphVertices(</C><A>X</A><C>), </C><A>act</A><C>));</C> <P/> | ||
| Furthermore every vertex of <A>X</A> must be in the orbit of some | ||
| vertex of <A>X</A> which is in <A>Y</A>. Finally, <A>act</A> must fix | ||
| the vertex of <A>X</A> which is the minimal vertex of <A>Y</A>, i.e. | ||
| the unique vertex of <A>Y</A> whose only out-neighbour is itself. | ||
| </Item> | ||
| </List> | ||
|
|
||
| For user convienience, there are multiple versions of | ||
| <C>McAlisterTripleSemigroup</C>. When the argument <A>act</A> is ommitted | ||
| it is assumed to be <Ref Func= "OnPoints" BookName= "ref"/>. Additionally, | ||
| the semilattice argument <A>Y</A> may be replaced by a homogeneous list | ||
| <A>sub_ver</A> of vertices of <A>X</A>. When <A>sub_ver</A> is provided, | ||
| <C>McAlisterTripleSemigroup</C> is called with <A>Y</A> equalling | ||
| <C>InducedSubdigraph(<A>X</A>, <A>sub_ver</A>)</C> with the appropriate | ||
| labels. | ||
|
|
||
| <Example><![CDATA[ | ||
| gap> x := Digraph([[1], [1, 2], [1, 2, 3], [1, 4], [1, 4, 5]]); | ||
| <digraph with 5 vertices, 11 edges> | ||
| gap> y := InducedSubdigraph(x, [1, 4, 5]); | ||
| <digraph with 3 vertices, 6 edges> | ||
| gap> DigraphVertexLabels(y); | ||
| [ 1, 4, 5 ] | ||
| gap> A := AutomorphismGroup(x); | ||
| Group([ (2,4)(3,5) ]) | ||
| gap> S := McAlisterTripleSemigroup(A, x, y, OnPoints); | ||
| <McAlister triple semigroup over Group([ (2,4)(3,5) ])> | ||
| gap> T := McAlisterTripleSemigroup(A, x, y); | ||
| <McAlister triple semigroup over Group([ (2,4)(3,5) ])> | ||
| gap> S = T; | ||
| false | ||
| gap> IsIsomorphicSemigroup(S, T); | ||
| true | ||
| gap> GeneratorsOfSemigroup(T); | ||
| [ (1, ()), (4, ()), (5, ()), (1, (2,4)(3,5)) ] | ||
| gap> AsSemigroup(IsPartialPermSemigroup, T); | ||
| <inverse partial perm monoid of size 4, rank 4 with 4 generators>]]></Example> | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> | ||
|
|
||
| <#GAPDoc Label="McAlisterTripleSemigroupGroup"> | ||
| <ManSection> | ||
| <Attr Name = "McAlisterTripleSemigroupGroup" Arg = "S"/> | ||
| <Returns>A group.</Returns> | ||
| <Description> | ||
| Returns the group used to create the McAlister triple semigroup <A>S</A> | ||
| via <Ref Oper="McAlisterTripleSemigroup"/>. | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> | ||
|
|
||
| <#GAPDoc Label="McAlisterTripleSemigroupPartialOrder"> | ||
| <ManSection> | ||
| <Attr Name = "McAlisterTripleSemigroupPartialOrder" Arg = "S"/> | ||
| <Returns>A partial order digraph.</Returns> | ||
| <Description> | ||
| Returns the <Ref Prop="IsPartialOrderDigraph" BookName="Digraphs"/> used | ||
| to create the McAlister triple semigroup <A>S</A> via | ||
| <Ref Oper="McAlisterTripleSemigroup"/>. | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> | ||
|
|
||
| <#GAPDoc Label="McAlisterTripleSemigroupSemilattice"> | ||
| <ManSection> | ||
| <Attr Name = "McAlisterTripleSemigroupSemilattice" Arg = "S"/> | ||
| <Returns>A join-semilattice digraph.</Returns> | ||
| <Description> | ||
| Returns the <Ref Prop="IsJoinSemilatticeDigraph" BookName="Digraphs"/> | ||
| used to create the McAlister triple semigroup <A>S</A> via | ||
| <Ref Oper="McAlisterTripleSemigroup"/>. | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> | ||
|
|
||
| <#GAPDoc Label="McAlisterTripleSemigroupAction"> | ||
| <ManSection> | ||
| <Attr Name = "McAlisterTripleSemigroupAction" Arg = "S"/> | ||
| <Returns>A function.</Returns> | ||
| <Description> | ||
| Returns the action used to create the McAlister triple semigroup | ||
| <A>S</A> via <Ref Oper="McAlisterTripleSemigroup"/>. | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> | ||
|
|
||
| <#GAPDoc Label="IsMcAlisterTripleSemigroupElement"> | ||
| <ManSection> | ||
| <Filt Name = "IsMcAlisterTripleSemigroupElement" Arg = "x"/> | ||
| <Filt Name = "IsMTSE" Arg = "x"/> | ||
| <Returns><K>true</K> or <K>false</K>.</Returns> | ||
| <Description> | ||
| Returns <K>true</K> if <A>x</A> is an element of a McAlister triple | ||
| semigroup; in particular, this returns <K>true</K> if <A>x</A> has been | ||
| created by <Ref Oper="McAlisterTripleSemigroupElement"/>. The functions | ||
| <C>IsMTSE</C> and <C>IsMcAlisterTripleSemigroupElement</C> are synonyms. | ||
| The mathematical description of these objects can be found in the | ||
| introduction to Chapter <Ref Chap = "McAlister triple semigroups"/>. | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> | ||
|
|
||
| <#GAPDoc Label="McAlisterTripleSemigroupElement"> | ||
| <ManSection> | ||
| <Oper Name = "McAlisterTripleSemigroupElement" Arg = "S, A, g"/> | ||
| <Oper Name = "MTSE" Arg = "S, A, g"/> | ||
| <Returns>A McAlister triple semigroup element.</Returns> | ||
| <Description> | ||
| Returns the <E>McAlister triple semigroup element</E> of the McAlister | ||
| triple semigroup <A>S</A> which corresponds to a label <A>A</A> of a | ||
| vertex from the <Ref Attr="McAlisterTripleSemigroupSemilattice"/> of | ||
| <A>S</A> and a group element <A>g</A> of the <Ref Attr= | ||
| "McAlisterTripleSemigroupGroup"/> of <A>S</A>. The pair | ||
| <A>(A,g)</A> only represents an element of <A>S</A> if the following | ||
| holds: | ||
|
|
||
| <A>A</A> acted on by the inverse of <A>g</A> (via <Ref | ||
| Attr="McAlisterTripleSemigroupAction"/>) is a vertex of the | ||
| join-semilattice of <A>S</A>. <P/> | ||
|
|
||
| The functions <C>MTSE</C> and <C>McAlisterTripleSemigroupElement</C> | ||
| are synonyms. | ||
| <Example><![CDATA[ | ||
| gap> x := Digraph([[1], [1, 2], [1, 2, 3], [1, 4], [1, 4, 5]]); | ||
| <digraph with 5 vertices, 11 edges> | ||
| gap> y := InducedSubdigraph(x, [1, 2, 3]); | ||
| <digraph with 3 vertices, 6 edges> | ||
| gap> A := AutomorphismGroup(x); | ||
| Group([ (2,4)(3,5) ]) | ||
| gap> S := McAlisterTripleSemigroup(A, x, y, OnPoints); | ||
| <McAlister triple semigroup over Group([ (2,4)(3,5) ])> | ||
| gap> T := McAlisterTripleSemigroup(A, x, y); | ||
| <McAlister triple semigroup over Group([ (2,4)(3,5) ])> | ||
| gap> S = T; | ||
| false | ||
| gap> IsIsomorphicSemigroup(S, T); | ||
| true | ||
| gap> a := MTSE(S, 1, (2, 4)(3, 5)); | ||
| (1, (2,4)(3,5)) | ||
| gap> b := MTSE(S, 2, ()); | ||
| (2, ()) | ||
| gap> a * a; | ||
| (1, ()) | ||
| gap> IsMTSE(a * a); | ||
| true | ||
| gap> a = MTSE(T, 1, (2, 4)(3, 5)); | ||
| false | ||
| gap> a * b; | ||
| (1, (2,4)(3,5))]]></Example> | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> | ||
|
|
||
| <#GAPDoc Label="EUnitaryInverseCover"> | ||
| <ManSection> | ||
| <Attr Name = "EUnitaryInverseCover" Arg = "S"/> | ||
| <Returns>A homomorphism between semigroups.</Returns> | ||
| <Description> | ||
| If the argument <A>S</A> is an inverse semigroup then this function | ||
| returns a finite E-unitary inverse cover of <A>S</A>. A finite E-unitary | ||
| cover of <A>S</A> is a surjective idempotent separating homomorphism from | ||
| a finite semigroup satisfying <Ref Prop="IsEUnitaryInverseSemigroup"/> | ||
| to <A>S</A>. A semigroup homomorphism is said to be idempotent separating | ||
| if no two idempotents are mapped to the same element of the image. | ||
| <Example><![CDATA[ | ||
| gap> S := InverseSemigroup([PartialPermNC([1, 2], [2, 1]), | ||
| > PartialPermNC([1], [1])]); | ||
| <inverse partial perm semigroup of rank 2 with 2 generators> | ||
| gap> cov := EUnitaryInverseCover(S); | ||
| MappingByFunction( <inverse partial perm semigroup of rank 4 with 2 | ||
| generators>, <inverse partial perm semigroup of rank 2 with 2 | ||
| generators>, function( x ) ... end ) | ||
| gap> IsEUnitaryInverseSemigroup(Source(cov)); | ||
| true | ||
| gap> S = Range(cov); | ||
| true]]></Example> | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> | ||
|
|
||
| <#GAPDoc Label="IsFInverseSemigroup"> | ||
| <ManSection> | ||
| <Prop Name = "IsFInverseSemigroup" Arg = "S"/> | ||
| <Returns><K>true</K> or <K>false</K>.</Returns> | ||
| <Description> | ||
| This functions determines whether a given semigroup is an F-inverse | ||
| semigroup. An F-inverse semigroup is a semigroup which satisfies | ||
| <Ref Prop="IsEUnitaryInverseSemigroup"/> as well as being isomorphic | ||
| to some <Ref Oper="McAlisterTripleSemigroup"/> where the | ||
| <Ref Attr="McAlisterTripleSemigroupPartialOrder"/> satisfies | ||
| <Ref Prop="IsJoinSemilatticeDigraph" BookName="Digraphs"/>. McAlister | ||
| triple semigroups are a represenation of E-unitary inverse semigroups and | ||
| more can be read about them in Chapter | ||
| <Ref Chap="McAlister triple semigroups"/>. | ||
| <Example><![CDATA[ | ||
| gap> S := InverseMonoid([PartialPermNC([1, 2], [1, 2]), | ||
| > PartialPermNC([1, 2, 3], [1, 2, 3]), | ||
| > PartialPermNC([1, 2, 4], [1, 2, 4]), | ||
| > PartialPermNC([1, 2], [2, 1]), PartialPermNC([1, 2, 3], [2, 1, 3]), | ||
| > PartialPermNC([1, 2, 4], [2, 1, 4])]);; | ||
| gap> IsEUnitaryInverseSemigroup(S); | ||
| true | ||
| gap> IsFInverseSemigroup(S); | ||
| false | ||
| gap> IsFInverseSemigroup(IdempotentGeneratedSubsemigroup(S)); | ||
| true]]></Example> | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> | ||
|
|
||
| <#GAPDoc Label="IsFInverseMonoid"> | ||
| <ManSection> | ||
| <Prop Name = "IsFInverseMonoid" Arg = "S"/> | ||
| <Returns><K>true</K> or <K>false</K>.</Returns> | ||
| <Description> | ||
| This function determines whether a given semigroup is an F-inverse | ||
| monoid. A semigroup is an F-inverse monoid when it satisfies | ||
| <Ref Filt="IsMonoid" BookName = "ref"/> and | ||
| <Ref Prop="IsFInverseSemigroup"/>. | ||
| </Description> | ||
| </ManSection> | ||
| <#/GAPDoc> |
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