Quelle semigraph.xml Sprache: XML

#############################################################################
##
#W semigraph.xml
#Y Copyright (C) 2014 James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##

<#GAPDoc Label="GraphInverseSemigroup">
<ManSection>
 <Oper Name="GraphInverseSemigroup" Arg="E"/>
 <Returns>A graph inverse semigroup.</Returns>
 <Description>
 If <A>E</A> is a digraph (i.e. it satisfies <Ref Prop="IsDigraph"
 BookName="Digraphs"/>), then <C>GraphInverseSemigroup</C> returns the
 graph inverse semigroup <M>G(<A>E</A>)</M> where, roughly speaking, elements
 correspond to paths in the graph <A>E</A>. 

 Let us describe <A>E</A> as a digraph
 <M><A>E</A> = (E ^ 0, E ^ 1, r, s)</M>, where <M>E^0</M> is the set of
 vertices, <M>E^1</M> is the set of edges, and <M>r</M> and <M>s</M> are
 functions <M>E^1 \to E^0</M> giving the <E>range</E> and <E>source</E> of an
 edge, respectively.
 The <E>graph inverse semigroup <M>G(<A>E</A>)</M> of <M>E</M></E> is the
 semigroup-with-zero generated by the sets <M><A>E</A> ^ 0</M> and
 <M><A>E</A> ^ 1</M>, together with a set of variables
 <M>\{e ^ {-1} \mid e \in <A>E</A> ^ 1\}</M>, satisfying the following
 relations for all <M>v, w \in <A>E</A> ^ 0</M> and
 <M>e, f \in <A>E</A> ^ 1</M>:

 <List>
 (V)
 <Item><M>vw = \delta_{v,w} \cdot v</M>,</Item>

 (E1)
 <Item><M>s(e) \cdot e=e \cdot r(e)=e</M>,</Item>

 (E2)
 <Item><M>r(e) \cdot e^{-1} = e^{-1} \cdot s(e) =e^{-1}</M>,</Item>

 (CK1)
 <Item><M>e^{-1} \cdot f = \delta_{e,f} \cdot r(e)</M>.</Item>
 </List>

 (Here <M>\delta</M> is the Kronecker delta.) We define <M>v^{-1}=v</M> for
 each <M>v \in E^0</M>, and for any path <M>y=e_1\dots e_n</M> (<M>e_1\dots
 e_n \in E^1</M>) we let <M>y^{-1} = e_n^{-1} \dots e_1^{-1}</M>. With this
 notation, every nonzero element of <M>G(E)</M> can be written uniquely as
 <M>xy^{-1}</M> for some paths <M>x, y</M> in <M>E</M>, by the CK1 relation.
 

 For a more complete description, see <Cite Key = "Mesyan2016"/>. 

 <Example><![CDATA[
gap> gr := Digraph([[2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10], [1],
> [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10],
> [1, 4], [1, 5, 9], [1, 2, 7, 8], [3, 5]]);
<immutable digraph with 10 vertices, 37 edges>
gap> S := GraphInverseSemigroup(gr);
<infinite graph inverse semigroup with 10 vertices, 37 edges>
gap> GeneratorsOfInverseSemigroup(S);
[ e_1, e_2, e_3, e_4, e_5, e_6, e_7, e_8, e_9, e_10, e_11, e_12,
 e_13, e_14, e_15, e_16, e_17, e_18, e_19, e_20, e_21, e_22, e_23,
 e_24, e_25, e_26, e_27, e_28, e_29, e_30, e_31, e_32, e_33, e_34,
 e_35, e_36, e_37, v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8, v_9, v_10
]
gap> AssignGeneratorVariables(S);
gap> e_1 * e_1 ^ -1;
e_1e_1^-1
gap> e_1 ^ -1 * e_1 ^ -1;
0
gap> e_1 ^ -1 * e_1;
v_2]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="RangeSourceGraphInverseSemigroupElement">
<ManSection>
 <Attr Name="Range" Arg="x" Label="for a graph inverse semigroup element"/>
 <Attr Name="Source" Arg="x" Label="for a graph inverse semigroup element"/>
 <Returns>A graph inverse semigroup element.</Returns>
 <Description>
 If <A>x</A> is an element of a graph inverse semigroup
 (i.e. it satisfies <Ref Filt = "IsGraphInverseSemigroupElement"/>), then
 <C>Range</C> and <C>Source</C> give, respectively, the start and end
 vertices of <A>x</A> when viewed as a path in the digraph over which the
 semigroup is defined. 

 For a fuller description, see <Ref Oper="GraphInverseSemigroup"/>. 

 <Example><![CDATA[
gap> gr := Digraph([[], [1], [3]]);;
gap> S := GraphInverseSemigroup(gr);;
gap> e := S.1;
e_1
gap> Source(e);
v_2
gap> Range(e);
v_1
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsVertex">
<ManSection>
 <Oper Name="IsVertex" Arg="x" Label="for a graph inverse semigroup element"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 If <A>x</A> is an element of a graph inverse semigroup (i.e. it satisfies
 <Ref Filt = "IsGraphInverseSemigroupElement"/>), then this attribute returns
 <K>true</K> if <A>x</A> corresponds to a vertex in the digraph over which
 the semigroup is defined, and <K>false</K> otherwise.
 

 For a fuller description, see <Ref Oper="GraphInverseSemigroup"/>. 

 <Example><![CDATA[
gap> gr := Digraph([[], [1], [3]]);;
gap> S := GraphInverseSemigroup(gr);;
gap> e := S.1;
e_1
gap> IsVertex(e);
false
gap> v := S.3;
v_1
gap> IsVertex(v);
true
gap> z := v * e;
0
gap> IsVertex(z);
false
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsGraphInverseSemigroup">
<ManSection>
 <Filt Name="IsGraphInverseSemigroup" Arg="x"/>
 <Filt Name="IsGraphInverseSemigroupElement" Arg="x"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 The category <C>IsGraphInverseSemigroup</C> contains any semigroup defined
 over a digraph using the <Ref Oper="GraphInverseSemigroup"/> operation.
 The category <C>IsGraphInverseSemigroupElement</C> contains any element
 contained in such a semigroup. 

 <Example><![CDATA[
gap> gr := Digraph([[], [1], [3]]);;
gap> S := GraphInverseSemigroup(gr);
<infinite graph inverse semigroup with 3 vertices, 2 edges>
gap> IsGraphInverseSemigroup(S);
true
gap> x := GeneratorsOfSemigroup(S)[1];
e_1
gap> IsGraphInverseSemigroupElement(x);
true
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="GraphOfGraphInverseSemigroup">
<ManSection>
 <Attr Name="GraphOfGraphInverseSemigroup" Arg="S"/>
 <Returns>A digraph.</Returns>
 <Description>
 If <A>S</A> is a graph inverse semigroup (i.e. it satisfies
 <Ref Filt="IsGraphInverseSemigroup"/>), then this attribute returns the
 original digraph over which <A>S</A> was defined (most likely the argument
 given to <Ref Oper="GraphInverseSemigroup"/> to create <A>S</A>). 

 <Example><![CDATA[
gap> gr := Digraph([[], [1], [3]]);
<immutable digraph with 3 vertices, 2 edges>
gap> S := GraphInverseSemigroup(gr);;
gap> GraphOfGraphInverseSemigroup(S);
<immutable digraph with 3 vertices, 2 edges>
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsGraphInverseSemigroupElementCollection">
 <ManSection>
 <Filt Name="IsGraphInverseSemigroupElementCollection" Type="Category"/>
 <Description>
 Every collection of elements of a graph inverse semigroup belongs to the
 category <C>IsGraphInverseSemigroupElementCollection</C>. For example,
 every graph inverse semigroup belongs to
 <C>IsGraphInverseSemigroupElementCollection</C>.
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsGraphInverseSubsemigroup">
<ManSection>
 <Filt Name = "IsGraphInverseSubsemigroup"/>
 <Description>
 <C>IsGraphInverseSubsemigroup</C> is a synonym for
 <C>IsSemigroup</C> and <C>IsInverseSemigroup</C> and
 <C>IsGraphInverseSemigroupElementCollection</C>. 

 See <Ref Filt = "IsGraphInverseSemigroupElementCollection"/> and
 <Ref Prop = "IsInverseSemigroup" BookName = "ref"/>.

<Example><![CDATA[
gap> gr := Digraph([[], [1], [2]]);
<immutable digraph with 3 vertices, 2 edges>
gap> S := GraphInverseSemigroup(gr);
<finite graph inverse semigroup with 3 vertices, 2 edges>
gap> Elements(S);
[ e_2^-1, e_1^-1, e_1^-1e_2^-1, 0, e_1, e_1e_1^-1, e_1e_1^-1e_2^-1,
 e_2, e_2e_2^-1, e_2e_1, e_2e_1e_1^-1, e_2e_1e_1^-1e_2^-1, v_1, v_2,
 v_3 ]
gap> T := InverseSemigroup(Elements(S){[3, 5]});;
gap> IsGraphInverseSubsemigroup(T);
true]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="VerticesOfGraphInverseSemigroup">
<ManSection>
 <Attr Name="VerticesOfGraphInverseSemigroup" Arg="S"/>
 <Returns>A list.</Returns>
 <Description>
 If <A>S</A> is a graph inverse semigroup (i.e. it satisfies
 <Ref Filt="IsGraphInverseSemigroup"/>), then this attribute returns the
 list of vertices of <A>S</A>.

 <Example><![CDATA[
gap> D := Digraph([[3, 4], [3, 4], [4], []]);
<immutable digraph with 4 vertices, 5 edges>
gap> S := GraphInverseSemigroup(D);
<finite graph inverse semigroup with 4 vertices, 5 edges>
gap> VerticesOfGraphInverseSemigroup(S);
[ v_1, v_2, v_3, v_4 ]
gap> D := ChainDigraph(12);
<immutable chain digraph with 12 vertices>
gap> S := GraphInverseSemigroup(D);
<finite graph inverse semigroup with 12 vertices, 11 edges>
gap> VerticesOfGraphInverseSemigroup(S);
[ v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8, v_9, v_10, v_11, v_12 ]
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="IndexOfVertexOfGraphInverseSemigroup">
<ManSection>
 <Attr Name="IndexOfVertexOfGraphInverseSemigroup" Arg="v"/>
 <Returns>A positive integer.</Returns>
 <Description>
 If <A>v</A> is a vertex of a graph inverse semigroup (i.e. it satisfies
 <Ref Filt="IsGraphInverseSemigroup"/>), then this attribute returns the
 index of this vertex in <A>S</A>.

 <Example><![CDATA[
gap> D := Digraph([[3, 4], [3, 4], [4], []]);
<immutable digraph with 4 vertices, 5 edges>
gap> S := GraphInverseSemigroup(D);
<finite graph inverse semigroup with 4 vertices, 5 edges>
gap> IndexOfVertexOfGraphInverseSemigroup(v_1);
1
gap> IndexOfVertexOfGraphInverseSemigroup(v_3);
3]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

quality90%

¤ Dauer der Verarbeitung: 0.26 Sekunden (vorverarbeitet) ¤

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.