Quellcode-Bibliothek semipbr.xml Sprache: XML

#############################################################################
##
#W semipbr.xml
#Y Copyright (C) 2015 James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##

<#GAPDoc Label="IsPBRSemigroup">
 <ManSection>
 <Filt Name = "IsPBRSemigroup" Arg = "S"/>
 <Filt Name = "IsPBRMonoid" Arg = "S"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 A <E>PBR semigroup</E> is simply a semigroup consisting of
 PBRs. An object <A>obj</A> is a PBR semigroup in &GAP; if
 it satisfies <Ref Prop = "IsSemigroup" BookName = "ref"/> and
 <Ref Filt = "IsPBRCollection"/>.

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

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

 <Example><![CDATA[
gap> x := PBR([[-2, -1, 3], [-2, 2], [-3, -2, 1, 2, 3]],
> [[-3, -2, -1, 2, 3], [-3, -2, -1, 2, 3], [-1]]);;
gap> S := Semigroup(x, One(x));
<commutative pbr monoid of degree 3 with 1 generator>
gap> IsMonoid(S);
true
gap> IsPBRMonoid(S);
true
gap> S := Semigroup([
> PBR([[-2, 1], [-3, 2], [-1, 3], [-4, 4, 5], [-4, 4, 5]],
> [[-1, 3], [-2, 1], [-3, 2], [-4, 4, 5], [-5]]),
> PBR([[-2, 1], [-1, 2], [-3, 3], [-4, 4, 5], [-4, 4, 5]],
> [[-1, 2], [-2, 1], [-3, 3], [-4, 4, 5], [-5]]),
> PBR([[-1, 1, 3], [-2, 2], [-1, 1, 3], [-4, 4, 5], [-4, 4, 5]],
> [[-1, 1, 3], [-2, 2], [-3], [-4, 4, 5], [-5]])]);
<pbr semigroup of degree 5 with 3 generators>
gap> One(S);
fail
gap> MultiplicativeNeutralElement(S);
PBR([ [ -1, 1 ], [ -2, 2 ], [ -3, 3 ], [ -4, 4, 5 ], [ -4, 4, 5 ] ],
 [ [ -1, 1 ], [ -2, 2 ], [ -3, 3 ], [ -4, 4, 5 ], [ -5 ] ])
gap> IsPBRMonoid(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="DegreeOfPBRSemigroup">
 <ManSection>
 <Attr Name = "DegreeOfPBRSemigroup" Arg = "S"/>
 <Returns>A non-negative integer.</Returns>
 <Description>
 The <E>degree</E> of a PBR semigroup <A>S</A> is just the
 degree of any (and every) element of <A>S</A>.

 <Example><![CDATA[
gap> S := Semigroup(
> PBR([[-1, 1], [-2, 2], [-3, 3]],
> [[-1, 1], [-2, 2], [-3, 3]]),
> PBR([[1, 2], [1, 2], [-3, 3]],
> [[-2, -1], [-2, -1], [-3, 3]]),
> PBR([[-1, 1], [2, 3], [2, 3]],
> [[-1, 1], [-3, -2], [-3, -2]]));
<pbr semigroup of degree 3 with 3 generators>
gap> DegreeOfPBRSemigroup(S);
3]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsomorphismPBRSemigroup">
 <ManSection>
 <Attr Name = "IsomorphismPBRSemigroup" Arg = "S"/>
 <Returns>An isomorphism.</Returns>
 <Description>
 If <A>S</A> is a semigroup, then <C>IsomorphismPBRSemigroup</C> returns
 an isomorphism from <A>S</A> to a PBR semigroup. When <A>S</A> is a
 transformation or bipartition semigroup of degree <C>n</C>,
 <C>IsomorphismPBRSemigroup</C> returns the
 natural embedding of <A>S</A> into the full pbr monoid on <C>n</C>
 points. 

 When <A>S</A> is any other type of semigroup, this function returns the
 composition of an isomorphism from <A>S</A> to a transformation
 semigroup, and an isomorphism from that transformation semigroup into a
 PBR semigroup.
 

 See <Ref Oper = "AsPBRSemigroup"/>.
 <Example><![CDATA[
gap> S := InverseSemigroup([
> PartialPerm([2, 6, 7, 0, 0, 9, 0, 1, 0, 5]),
> PartialPerm([3, 8, 1, 9, 0, 4, 10, 5, 0, 6])]);;
gap> AsSemigroup(IsPBRSemigroup, S);
<pbr semigroup of degree 11 with 4 generators>]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="FullPBRMonoid">
 <ManSection>
 <Oper Name = "FullPBRMonoid" Arg = "n"/>
 <Returns>A PBR monoid.</Returns>
 <Description>
 If <A>n</A> is a positive integer not greater than <C>2</C>, then this
 operation returns the monoid consisting of all of the partitioned binary
 relations (PBRs) of degree <A>n</A>; called the <E>full PBR monoid</E>.
 There are <C>2 ^ ((2 * n) ^ 2)</C> PBRs of degree <A>n</A>. The full PBR
 monoid of degree <A>n</A> is currently too large to compute when
 <M><A>n</A> \geq 3</M>. 

 The full PBR monoid is not regular in general.

 <Example><![CDATA[
gap> S := FullPBRMonoid(1);
<pbr monoid of degree 1 with 4 generators>
gap> S := FullPBRMonoid(2);
<pbr monoid of degree 2 with 10 generators>]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

quality99%

¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.0.29Bemerkung: (vorverarbeitet) ¤

*Bot Zugriff

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.