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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 3 kB

Quelle semipperm.xml Sprache: XML

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

<#GAPDoc Label="ComponentRepsOfPartialPermSemigroup">
<ManSection>
  <Attr Name = "ComponentRepsOfPartialPermSemigroup" Arg = "S"/>
  <Returns>The representatives of components of a partial perm semigroup.</Returns>
  <Description>
    This function returns the representatives of the components of the
    action of the partial perm semigroup <A>S</A> on the set of positive
    integers where it is defined. <P/>

    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>
gap> S := Semigroup([
> PartialPerm([1, 2, 3, 5, 6, 7, 8, 11, 12, 16, 19],
>             [9, 18, 20, 11, 5, 16, 8, 19, 14, 13, 1]),
> PartialPerm([1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 18, 19, 20],
>             [13, 1, 8, 5, 4, 14, 11, 12, 9, 20, 2, 18, 7, 3, 19])]);;
gap> ComponentRepsOfPartialPermSemigroup(S);
[ 6, 10, 15, 17 ]</Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="ComponentsOfPartialPermSemigroup">
<ManSection>
  <Attr Name = "ComponentsOfPartialPermSemigroup" Arg = "S"/>
  <Returns>The components of a partial perm semigroup.</Returns>
  <Description>
    This function returns the components of the action of the partial perm
    semigroup <A>S</A> on the set of positive integers where it is defined; the
    components of <A>S</A> partition this set.

    <Example>
gap> S := Semigroup([
> PartialPerm([1, 2, 3, 5, 6, 7, 8, 11, 12, 16, 19],
>             [9, 18, 20, 11, 5, 16, 8, 19, 14, 13, 1]),
> PartialPerm([1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 18, 19, 20],
>             [13, 1, 8, 5, 4, 14, 11, 12, 9, 20, 2, 18, 7, 3, 19])]);;
gap> ComponentsOfPartialPermSemigroup(S);
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 20 ],
  [ 15 ], [ 17 ] ]</Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="CyclesOfPartialPermSemigroup">
<ManSection>
  <Attr Name = "CyclesOfPartialPermSemigroup" Arg = "S"/>
  <Returns>The cycles of a partial perm semigroup.</Returns>
  <Description>
    This function returns the cycles, or strongly connected components, of the
    action of the partial perm semigroup <A>S</A> on the set of positive
    integers where it is defined.

    <Example>
gap> S := Semigroup([
> PartialPerm([1, 2, 3, 5, 6, 7, 8, 11, 12, 16, 19],
>             [9, 18, 20, 11, 5, 16, 8, 19, 14, 13, 1]),
> PartialPerm([1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 18, 19, 20],
>             [13, 1, 8, 5, 4, 14, 11, 12, 9, 20, 2, 18, 7, 3, 19])]);;
gap> CyclesOfPartialPermSemigroup(S);
[ [ 18, 7, 16 ], [ 1, 9, 12, 14, 2, 20, 19, 3, 8, 11 ], [ 4, 5 ] ]
</Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="CyclesOfPartialPerm">
<ManSection>
  <Attr Name = "CyclesOfPartialPerm" Arg = "x"/>
  <Returns>The cycles of a partial perm.</Returns>
  <Description>
    This function returns the cycles, or strongly connected components, of the
    action of the partial perm <A>x</A> on the set of positive
    integers where it is defined.

    <Example>
gap> x := PartialPerm([3, 1, 4, 2, 5, 0, 0, 6, 0, 7]);
[8,6][10,7](1,3,4,2)(5)
gap> CyclesOfPartialPerm(x);
[ [ 5 ], [ 1, 3, 4, 2 ] ]</Example>
  </Description>
</ManSection>
<#/GAPDoc>

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