| Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/doc/ref/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 18.9.2025 mit Größe 5 kB |
|
Quelle invsgp.xml Sprache: XML |
|||||||||
|
<Section Label="sect:IsInverseSemigroup"> <Heading>Inverse semigroups and monoids</Heading> <ManSection> <Func Name="InverseSemigroup" Arg="obj1, obj2, ... "/> <Returns>An inverse semigroup.</Returns> <Description> If <A>obj1</A>, <A>obj2</A>, ... are (any combination) of associative elements with unique semigroup inverses, semigroups of such elements, or collections of such elements, then <C>InverseSemigroup</C> returns the inverse semigroup generated by the union of <A>obj1</A>, <A>obj2</A>, .... This equals the semigroup generated by the union of <A>obj1</A>, <A>obj2</A>, ... and their inverses.<P/> For example if <C>S</C> and <C>T</C> are inverse semigroups, then <C>InverseSemigroup(S, f, Idempotents(T));</C> is the inverse semigroup generated by <C>Union(GeneratorsOfInverseSemigroup(S), [f], Idempotents(T)));</C>.<P/> As present, the only associative elements with unique semigroup inverses, which do not always generate a group, are partial permutations; see Chapter <Ref Chap="Partial permutations"/>. <Example> gap> S := InverseSemigroup( > PartialPerm( [ 1, 2, 3, 6, 8, 10 ], [ 2, 6, 7, 9, 1, 5 ] ) );; gap> f := PartialPerm( [ 1, 2, 3, 4, 5, 8, 10 ], > [ 7, 1, 4, 3, 2, 6, 5 ] );; gap> S := InverseSemigroup(S, f, Idempotents(SymmetricInverseSemigroup(5))); <inverse partial perm semigroup of rank 10 with 34 generators> gap> Size(S); 1233</Example> </Description> </ManSection> <ManSection> <Func Name="InverseMonoid" Arg="obj1, obj2, ... "/> <Returns>An inverse monoid.</Returns> <Description> If <A>obj1</A>, <A>obj2</A>, ... are (any combination) of associative elements with unique semigroup inverses, semigroups of such elements, or collections of such elements, then <C>InverseMonoid</C> returns the inverse monoid generated by the union of <A>obj1</A>, <A>obj2</A>, .... This equals the monoid generated by the union of <A>obj1</A>, <A>obj2</A>, ... and their inverses.<P/> As present, the only associative elements with unique semigroup inverses are partial permutations; see Chapter <Ref Chap="Partial permutations"/>.<P/> For example if <C>S</C> and <C>T</C> are inverse monoids, then <C>InverseMonoid(S, f, Idempotents(T));</C> is the inverse monoid generated by <C>Union(GeneratorsOfInverseMonoid(S), [f], Idempotents(T)));</C>. <Example> gap> S := InverseMonoid( > PartialPerm( [ 1, 2, 3, 6, 8, 10 ], [ 2, 6, 7, 9, 1, 5 ] ) );; gap> f := PartialPerm( [ 1, 2, 3, 4, 5, 8, 10 ], > [ 7, 1, 4, 3, 2, 6, 5 ] );; gap> S := InverseMonoid(S, f, Idempotents(SymmetricInverseSemigroup(5))); <inverse partial perm monoid of rank 10 with 35 generators> gap> Size(S); 1243</Example> </Description> </ManSection> <ManSection> <Attr Name="GeneratorsOfInverseSemigroup" Arg="S"/> <Returns>The generators of an inverse semigroup.</Returns> <Description> If <A>S</A> is an inverse semigroup, then <C>GeneratorsOfInverseSemigroup</C> returns the generators used to define <A>S</A>, i.e. an inverse semigroup generating set for <A>S</A>. <P/> The value of <C>GeneratorsOfSemigroup(<A>S</A>)</C>, for an inverse semigroup <A>S</A>, is the union of inverse semigroup generator and their inverses. So, <A>S</A> is the semigroup, as opposed to inverse semigroup, generated by the elements of <C>GeneratorsOfInverseSemigroup(<A>S</A>)</C> and their inverses. <P/> If <A>S</A> is an inverse monoid, then <C>GeneratorsOfInverseSemigroup</C> returns the generators used to define <A>S</A>, as described above, and the identity of <A>S</A>. <Example><![CDATA[ gap> S:=InverseMonoid( > PartialPerm( [ 1, 2 ], [ 1, 4 ] ), > PartialPerm( [ 1, 2, 4 ], [ 3, 4, 1 ] ) );; gap> GeneratorsOfSemigroup(S); [ <identity partial perm on [ 1, 2, 3, 4 ]>, [2,4](1), [2,4,1,3], [4,2](1), [3,1,4,2] ] gap> GeneratorsOfInverseSemigroup(S); [ [2,4](1), [2,4,1,3], <identity partial perm on [ 1, 2, 3, 4 ]> ] gap> GeneratorsOfMonoid(S); [ [2,4](1), [2,4,1,3], [4,2](1), [3,1,4,2] ]]]></Example> </Description> </ManSection> <ManSection> <Attr Name="GeneratorsOfInverseMonoid" Arg="S"/> <Returns>The generators of an inverse monoid.</Returns> <Description> If <A>S</A> is an inverse monoid, then <C>GeneratorsOfInverseMonoid</C> returns the generators used to define <A>S</A>, i.e. an inverse monoid generating set for <A>S</A>. <P/> There are four different possible generating sets which define an inverse monoid. More precisely, an inverse monoid can be generated as an inverse monoid, inverse semigroup, monoid, or semigroup. The different generating sets in each case can be obtained using <Ref Attr="GeneratorsOfInverseMonoid"/>, <Ref Attr="GeneratorsOfInverseSemigroup"/>, <Ref Attr="GeneratorsOfMonoid"/>, and <Ref Attr="GeneratorsOfSemigroup"/>, respectively. <Example> gap> S:=InverseMonoid( > PartialPerm( [ 1, 2 ], [ 1, 4 ] ), > PartialPerm( [ 1, 2, 4 ], [ 3, 4, 1 ] ) );; gap> GeneratorsOfInverseMonoid(S); [ [2,4](1), [2,4,1,3] ]</Example> </Description> </ManSection> <ManSection> <Oper Name="IsInverseSubsemigroup" Arg="S, T"/> <Returns><K>true</K> or <K>false</K>.</Returns> <Description> If the semigroup <A>T</A> is an inverse subsemigroup of the semigroup <A>S</A>, then this operation returns <K>true</K>. <Example> gap> T:=InverseSemigroup(RandomPartialPerm(4));; gap> IsInverseSubsemigroup(SymmetricInverseSemigroup(4), T); true gap> T:=Semigroup(Transformation( [ 1, 2, 4, 5, 6, 3, 7, 8 ] ), > Transformation( [ 3, 3, 4, 5, 6, 2, 7, 8 ] ), > Transformation([ 1, 2, 5, 3, 6, 8, 4, 4 ] ));; gap> IsInverseSubsemigroup(FullTransformationSemigroup(8), T); true</Example> </Description> </ManSection> </Section> ¤ Dauer der Verarbeitung: 0.22 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
|
2026-03-28