Quelle semibipart.xml Sprache: XML

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#W semibipart.xml
#Y Copyright (C) 2013-14 James D. Mitchell
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## Licensing information can be found in the README file of this package.
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<#GAPDoc Label="IsBipartitionSemigroup">
 <ManSection>
 <Filt Name = "IsBipartitionSemigroup" Arg = "S"/>
 <Filt Name = "IsBipartitionMonoid" Arg = "S"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 A <E>bipartition semigroup</E> is simply a semigroup consisting of
 bipartitions. An object <A>obj</A> is a bipartition semigroup in &GAP; if
 it satisfies <Ref Prop = "IsSemigroup" BookName = "ref"/> and
 <Ref Filt = "IsBipartitionCollection"/>.

 A <E>bipartition monoid</E> is a monoid consisting of bipartitions.
 An object <A>obj</A> is a bipartition monoid in &GAP; if it satisfies
 <Ref Prop = "IsMonoid" BookName = "ref"/> and
 <Ref Filt = "IsBipartitionCollection"/>.

 Note that it is possible for a bipartition semigroup to have a
 multiplicative neutral element (i.e. an identity element) but not to
 satisfy <C>IsBipartitionMonoid</C>. For example,

 <Example><![CDATA[
gap> x := Bipartition([
> [1, 4, -2], [2, 5, -6], [3, -7], [6, 7, -9], [8, 9, -1],
> [10, -5], [-3], [-4], [-8], [-10]]);;
gap> S := Semigroup(x, One(x));
<commutative bipartition monoid of degree 10 with 1 generator>
gap> IsMonoid(S);
true
gap> IsBipartitionMonoid(S);
true
gap> S := Semigroup([
> Bipartition([
> [1, -3], [2, -8], [3, 8, -1], [4, -4], [5, -5], [6, -6],
> [7, -7], [9, 10, -10], [-2], [-9]]),
> Bipartition([
> [1, -1], [2, -2], [3, -3], [4, -4], [5, -5], [6, -6],
> [7, -7], [8, -8], [9, 10, -10], [-9]])]);;
gap> One(S);
fail
gap> MultiplicativeNeutralElement(S);
<bipartition: [ 1, -1 ], [ 2, -2 ], [ 3, -3 ], [ 4, -4 ], [ 5, -5 ],
[ 6, -6 ], [ 7, -7 ], [ 8, -8 ], [ 9, 10, -10 ], [ -9 ]>
gap> IsMonoid(S);
false]]></Example>
 In this example <C>S</C> cannot be converted into a monoid using <Ref
 Oper = "AsMonoid" BookName = "ref"/> since the <Ref Attr = "One"
 BookName = "ref"/> of any element in <C>S</C> differs from the
 multiplicative neutral element. 

 For more details see <Ref Filt = "IsMagmaWithOne" BookName = "ref"/>.
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="DegreeOfBipartitionSemigroup">
 <ManSection>
 <Attr Name = "DegreeOfBipartitionSemigroup" Arg = "S"/>
 <Returns>A non-negative integer.</Returns>
 <Description>
 The <E>degree</E> of a bipartition semigroup <A>S</A> is just the degree
 of any (and every) element of <A>S</A>.

 <Example><![CDATA[
gap> DegreeOfBipartitionSemigroup(JonesMonoid(8));
8]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsBlockBijectionSemigroup">
 <ManSection>
 <Prop Name = "IsBlockBijectionSemigroup" Arg = "S"/>
 <Filt Name = "IsBlockBijectionMonoid" Arg = "S"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 A <E>block bijection semigroup</E> is simply a semigroup consisting of
 block bijections. A <E>block bijection monoid</E> is a monoid consisting
 of block bijections.

 An object in &GAP; is a block bijection monoid if it satisfies
 <Ref Prop = "IsMonoid" BookName = "ref"/> and <Ref
 Prop = "IsBlockBijectionSemigroup"/>.

 See <Ref Prop = "IsBlockBijection"/>.
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsPartialPermBipartitionSemigroup">
 <ManSection>
 <Prop Name = "IsPartialPermBipartitionSemigroup" Arg = "S"/>
 <Filt Name = "IsPartialPermBipartitionMonoid" Arg = "S"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 A <E>partial perm bipartition semigroup</E> is simply a semigroup
 consisting of partial perm bipartitions. A <E>partial perm bipartition
 monoid</E> is a monoid consisting of partial perm bipartitions.

 An object in &GAP; is a partial perm bipartition monoid
 if it satisfies <Ref Prop = "IsMonoid" BookName = "ref"/> and
 <Ref Prop = "IsPartialPermBipartitionSemigroup"/>.

 See <Ref Prop = "IsPartialPermBipartition"/>.
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsPermBipartitionGroup">
 <ManSection>
 <Prop Name = "IsPermBipartitionGroup" Arg = "S"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 A <E>perm bipartition group</E> is simply a semigroup consisting of perm
 bipartitions.

 See <Ref Prop = "IsPermBipartition"/>.
 </Description>
 </ManSection>
<#/GAPDoc>

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