############################################################################
##
## congsemigraph.xml
## Copyright (C) 2022 Marina Anagnostopoulou-Merkouri
## James Mitchell
##
## Licensing information can be found in the README file of this package.
##
############################################################################
<#GAPDoc Label="IsCongruenceByWangPair">
<ManSection>
<Prop Name="IsCongruenceByWangPair" Arg="cong"/>
<Description>
A congruence by Wang pair <C>cong</C> is a congruence of a graph inverse
semigroup <C>S</C> which is expressed in terms of two sets <A>H</A> and
<A>W</A> of vertices of the corresponding graph of <A>S</A>. The set
<A>H</A> must be a hereditary subset (closed under reachability) and all
vertices in <A>W</A> must have all but one of their out-neighbours in
<A>H</A>. For more information on Wang pairs see <Cite Key="Wang2019aa"/>
and <Cite Key="Anagnostopoulou-Merkouri2021aa"/>.<P/>
<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> cong := CongruenceByWangPair(S, [3, 4], []);
<graph inverse semigroup congruence with H = [ 3, 4 ] and W = [ ]>
gap> IsCongruenceByWangPair(cong);
true
gap> cong := CongruenceByWangPair(S, [4], [2]);
<graph inverse semigroup congruence with H = [ 4 ] and W = [ 2 ]>
gap> IsCongruenceByWangPair(cong);
true
gap> e_1 := S.1;
e_1
gap> e_3 := S.3;
e_3
gap> cong := SemigroupCongruence(S, [[e_1, e_3]]);
<2-sided semigroup congruence over <finite graph inverse semigroup wit\
h 4 vertices, 5 edges> with 1 generating pairs>
gap> IsCongruenceByWangPair(cong);
false
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="CongruenceByWangPair">
<ManSection>
<Func Name = "CongruenceByWangPair" Arg = "S, H, W"/>
<Returns>A semigroup congruence.</Returns>
<Description>
This function returns a semigroup congruence over the graph inverse
semigroup <A>S</A> in the form of a Wang pair.<P/>
If <A>S</A> is a finite graph inverse semigroup <A>H</A> and <A>W</A> are
two lists of vertices in the graph of <A>S</A> representing a valid
hereditary subset and a W-set respectively, then this function will return
the semigroup congruence defined by this Wang pair. For the definition of
Wang pair <Ref Attr="IsCongruenceByWangPair"/>.<P/>
<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> cong := CongruenceByWangPair(S, [3, 4], []);
<graph inverse semigroup congruence with H = [ 3, 4 ] and W = [ ]>
gap> cong := CongruenceByWangPair(S, [4], [2]);
<graph inverse semigroup congruence with H = [ 4 ] and W = [ 2 ]>
gap> cong := CongruenceByWangPair(S, [3, 4], []);
<graph inverse semigroup congruence with H = [ 3, 4 ] and W = [ ]>
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="AsCongruenceByWangPair">
<ManSection>
<Oper Name = "AsCongruenceByWangPair" Arg = "cong"/>
<Returns>A congruence by Wang pair.</Returns>
<Description>
This operation takes <A>cong</A>, a finite graph inverse semigroup
congruence, and returns an object representing the same congruence, but
described as a congruence by Wang pairs: a pair of sets <A>H</A> and
<A>W</A> of the corresponding graph of <A>S</A> that are a hereditary
subset and a W-set of the graph of <A>S</A> respectively. For more
information about Wang pairs see <Cite Key="Wang2019aa"/> and <Cite
Key="Anagnostopoulou-Merkouri2021aa"/>. <P/>
<Example><![CDATA[
gap> D := Digraph([[2, 3], [3], [4], []]);
<immutable digraph with 4 vertices, 4 edges>
gap> S := GraphInverseSemigroup(D);
<finite graph inverse semigroup with 4 vertices, 4 edges>
gap> CongruenceByWangPair(S, [4], [2]);
<graph inverse semigroup congruence with H = [ 4 ] and W = [ 2 ]>
gap> cong := AsSemigroupCongruenceByGeneratingPairs(last);
<2-sided semigroup congruence over <finite graph inverse semigroup wit\
h 4 vertices, 4 edges> with 2 generating pairs>
gap> AsCongruenceByWangPair(cong);
<graph inverse semigroup congruence with H = [ 4 ] and W = [ 2 ]>
gap> CongruenceByWangPair(S, [3, 4], [1]);
<graph inverse semigroup congruence with H = [ 3, 4 ] and W = [ 1 ]>
gap> cong := AsSemigroupCongruenceByGeneratingPairs(last);
<2-sided semigroup congruence over <finite graph inverse semigroup wit\
h 4 vertices, 4 edges> with 3 generating pairs>
gap> AsCongruenceByWangPair(cong);
<graph inverse semigroup congruence with H = [ 3, 4 ] and W = [ 1 ]>
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="GeneratingCongruencesOfLattice">
<ManSection>
<Attr Name = "GeneratingCongruencesOfLattice" Arg = "S"/>
<Returns>A semigroup.</Returns>
<Description>
This attribute takes a finite graph inverse semigroup <A>S</A> and returns a
minimal generating set for the lattice of congruences of <A>S</A>, as described
in <Cite Key="Anagnostopoulou-Merkouri2021aa"/>. This operation works only if
the corresponding digraph of the graph inverse semigroup is simple. If there
are multiple edges, an error is returned.<P/>
<Example><![CDATA[
gap> D := Digraph([[2, 3], [3], [4], []]);
<immutable digraph with 4 vertices, 4 edges>
gap> S := GraphInverseSemigroup(D);
<finite graph inverse semigroup with 4 vertices, 4 edges>
gap> CongruenceByWangPair(S, [4], [2]);
<graph inverse semigroup congruence with H = [ 4 ] and W = [ 2 ]>
gap> cong := AsSemigroupCongruenceByGeneratingPairs(last);
<2-sided semigroup congruence over <finite graph inverse semigroup wit\
h 4 vertices, 4 edges> with 2 generating pairs>
gap> AsCongruenceByWangPair(cong);
<graph inverse semigroup congruence with H = [ 4 ] and W = [ 2 ]>
gap> CongruenceByWangPair(S, [3, 4], [1]);
<graph inverse semigroup congruence with H = [ 3, 4 ] and W = [ 1 ]>
gap> cong := AsSemigroupCongruenceByGeneratingPairs(last);
<2-sided semigroup congruence over <finite graph inverse semigroup wit\
h 4 vertices, 4 edges> with 3 generating pairs>
gap> AsCongruenceByWangPair(cong);
<graph inverse semigroup congruence with H = [ 3, 4 ] and W = [ 1 ]>
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
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