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#W conglatt.xml
#Y Copyright (C) 2016-2022 Michael Young
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## Licensing information can be found in the README file of this package.
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<#GAPDoc Label="IsCongruencePoset">
<ManSection>
<Filt Name = "IsCongruencePoset" Arg = "poset"Type = "Category"/>
<Filt Name = "IsCayleyDigraphOfCongruences" Arg = "poset"Type = "Category"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
This category contains all congruence posets. A <E>congruence poset</E>
is a partially ordered set of congruences over a specific semigroup, where
the ordering is defined by containment according
to <Ref Oper = "IsSubrelation"/>: given two congruences <C>cong1</C> and
<C>cong2</C>, we say that <C>cong1</C> < <C>cong2</C> if and only if
<C>cong1</C> is a subrelation (a refinement) of <C>cong2</C>. The
congruences in a congruence poset can be left, right, or two-sided. <P/>
A congruence poset is a digraph (see <Ref Filt = "IsDigraph" BookName = "Digraphs"/>) with a vertex for each congruence, and an edge from
vertex <C>i</C> to vertex <C>j</C> only if the congruence numbered
<C>i</C> is a subrelation of the congruence numbered <C>j</C>. To avoid
using an unnecessarily large amount of memory in some cases,
a congruence poset does not necessarily belong to
<Ref Prop="IsPartialOrderDigraph" BookName="Digraphs"/>. In other words,
although every congruence poset represents a partial order it is not
necessarily the case that there is an edge from vertex <C>i</C> to vertex
<C>j</C> if and only if the congruence numbered <C>i</C> is a subrelation
of the congruence numbered <C>j</C>. <P/>
The list of congruences can be obtained using <Ref Attr = "CongruencesOfPoset"/>; and the underlying semigroup of the poset can
be obtained using <Ref Attr="UnderlyingSemigroupOfCongruencePoset"/>.<P/>
Congruence posets can be created using any of:
<List>
<Item>
<Ref Oper = "PosetOfCongruences"/>,
</Item>
<Item>
<Ref Oper = "JoinSemilatticeOfCongruences"/>
</Item>
<Item>
<Ref Attr = "LatticeOfCongruences" Label="for a semigroup"/>,
<Ref Attr = "LatticeOfLeftCongruences" Label="for a semigroup"/>, or
<Ref Attr = "LatticeOfRightCongruences" Label="for a semigroup"/>
</Item>
<Item>
<Ref Attr = "CayleyDigraphOfCongruences" Label="for a semigroup"/>,
<Ref Attr = "CayleyDigraphOfLeftCongruences" Label="for a semigroup"/>, or
<Ref Attr = "CayleyDigraphOfRightCongruences" Label="for a semigroup"/>.
</Item>
</List>
<C>IsCayleyDigraphOfCongruences</C> only applies to the output of
<Ref Oper = "JoinSemilatticeOfCongruences"/>,
<Ref Attr = "CayleyDigraphOfCongruences" Label="for a semigroup"/>,
<Ref Attr = "CayleyDigraphOfLeftCongruences" Label="for a semigroup"/>, and
<Ref Attr = "CayleyDigraphOfRightCongruences" Label="for a semigroup"/>.
The congruences used as the generating set for these operations can be obtained using
<Ref Attr="GeneratingCongruencesOfJoinSemilattice"/>.
<Example><![CDATA[
gap> S := SymmetricInverseMonoid(2);;
gap> poset := LatticeOfCongruences(S);
<lattice of 4 two-sided congruences over
<symmetric inverse monoid of degree 2>>
gap> IsCongruencePoset(poset);
true
gap> IsDigraph(poset);
true
gap> IsIsomorphicDigraph(poset,
> Digraph([[1, 2, 3, 4], [2], [2, 3], [2, 3, 4]]));
true
gap> T := FullTransformationMonoid(3);;
gap> congs := PrincipalCongruencesOfSemigroup(T);;
gap> poset := JoinSemilatticeOfCongruences(PosetOfCongruences(congs));
<lattice of 6 two-sided congruences over
<full transformation monoid of degree 3>>
gap> IsCayleyDigraphOfCongruences(poset);
false
gap> IsCongruencePoset(poset);
true
gap> DigraphNrVertices(poset);
6
gap> poset := CayleyDigraphOfCongruences(T);
<poset of 7 two-sided congruences over
<full transformation monoid of degree 3>>
gap> IsCayleyDigraphOfCongruences(poset);
true
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="LatticeOfCongruences">
<ManSection>
<Attr Name = "LatticeOfCongruences" Arg = "S"
Label = "for a semigroup"/>
<Attr Name = "LatticeOfLeftCongruences" Arg = "S"
Label = "for a semigroup"/>
<Attr Name = "LatticeOfRightCongruences" Arg = "S"
Label = "for a semigroup"/>
<Oper Name = "LatticeOfCongruences" Arg = "S, restriction"
Label = "for a semigroup and a list or collection"/>
<Oper Name = "LatticeOfLeftCongruences" Arg = "S, restriction"
Label = "for a semigroup and a list or collection"/>
<Oper Name = "LatticeOfRightCongruences" Arg = "S, restriction"
Label = "for a semigroup and a list or collection"/>
<Returns>A lattice digraph.</Returns>
<Description>
If <A>S</A> is a semigroup, then <C>LatticeOfCongruences</C> returns a
congruence poset object containing all the congruences of <A>S</A> and
information about how they are contained in each other.
See <Ref Filt = "IsCongruencePoset"/> for more details. <P/>
<C>LatticeOfLeftCongruences</C> and <C>LatticeOfRightCongruences</C> do
the same thing for left and right congruences, respectively. <P/>
If <A>restriction</A> is specified and is a collection of elements from
<A>S</A>, then the result will only include congruences generated by pairs
of elements from <A>restriction</A>. Otherwise, all congruences will be
calculated.<P/>
See <Ref Attr = "CongruencesOfSemigroup" Label = "for a semigroup"/>. <P/>
<Example><![CDATA[
gap> S := OrderEndomorphisms(2);;
gap> LatticeOfCongruences(S);
<lattice of 3 two-sided congruences over <regular transformation
monoid of size 3, degree 2 with 2 generators>>
gap> LatticeOfLeftCongruences(S);
<lattice of 3 left congruences over <regular transformation monoid
of size 3, degree 2 with 2 generators>>
gap> LatticeOfRightCongruences(S);
<lattice of 5 right congruences over <regular transformation monoid
of size 3, degree 2 with 2 generators>>
gap> IsIsomorphicDigraph(LatticeOfRightCongruences(S),
> Digraph([[1, 2, 3, 4, 5], [2], [2, 3], [2, 4], [2, 5]]));
true
gap> S := FullTransformationMonoid(4);;
gap> restriction := [Transformation([1, 1, 1, 1]),
> Transformation([1, 1, 1, 2]),
> Transformation([1, 1, 1, 3])];;
gap> latt := LatticeOfCongruences(S, Combinations(restriction, 2));
<lattice of 2 two-sided congruences over
<full transformation monoid of degree 4>>
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="CayleyDigraphOfCongruences">
<ManSection>
<Attr Name = "CayleyDigraphOfCongruences"
Arg = "S"
Label = "for a semigroup"/>
<Attr Name = "CayleyDigraphOfLeftCongruences"
Arg = "S"
Label = "for a semigroup"/>
<Attr Name = "CayleyDigraphOfRightCongruences"
Arg = "S"
Label = "for a semigroup"/>
<Oper Name = "CayleyDigraphOfCongruences"
Arg = "S, restriction"
Label = "for a semigroup and a list or collection"/>
<Oper Name = "CayleyDigraphOfLeftCongruences"
Arg = "S, restriction"
Label = "for a semigroup and a list or collection"/>
<Oper Name = "CayleyDigraphOfRightCongruences"
Arg = "S, restriction"
Label = "for a semigroup and a list or collection"/>
<Returns>A digraph.</Returns>
<Description>
If <A>S</A> is a semigroup, then <C>CayleyDigraphOfCongruences</C>
returns the right Cayley graph of the semilattice of congruences of
<A>S</A> with respect to the generating set consisting of the principal
congruences congruence poset.
See <Ref Filt = "IsCayleyDigraphOfCongruences"/> for more details. <P/>
<C>CayleyDigraphOfLeftCongruences</C> and
<C>CayleyDigraphOfRightCongruences</C> do
the same thing for left and right congruences, respectively. <P/>
If <A>restriction</A> is specified and is a collection of elements from
<A>S</A>, then the result will only include congruences generated by pairs
of elements from <A>restriction</A>. Otherwise, all congruences will be
calculated.<P/>
Note that <Ref Attr="LatticeOfCongruences" Label="for a semigroup"/>, and
its analogues for right and left congruences, return the reflexive
transitive closure of the digraph returned by this function (with any
multiple edges removed). If there are a large number of congruences, then
it might be the case that forming the reflexive transitive closure takes
a significant amount of time, and so it might be desirable to use this
function instead. <P/>
See <Ref Attr = "CongruencesOfSemigroup" Label = "for a semigroup"/>. <P/>
<Example><![CDATA[
gap> S := OrderEndomorphisms(2);;
gap> CayleyDigraphOfCongruences(S);
<poset of 3 two-sided congruences over <regular transformation monoid
of size 3, degree 2 with 2 generators>>
gap> CayleyDigraphOfLeftCongruences(S);
<poset of 3 left congruences over <regular transformation monoid
of size 3, degree 2 with 2 generators>>
gap> CayleyDigraphOfRightCongruences(S);
<poset of 5 right congruences over <regular transformation monoid
of size 3, degree 2 with 2 generators>>
gap> IsIsomorphicDigraph(CayleyDigraphOfRightCongruences(S),
> Digraph([[2, 3, 4], [2, 5, 5], [5, 3, 5], [5, 5, 4], [5, 5, 5]]));
true
gap> S := FullTransformationMonoid(4);;
gap> restriction := [Transformation([1, 1, 1, 1]),
> Transformation([1, 1, 1, 2]),
> Transformation([1, 1, 1, 3])];;
gap> CayleyDigraphOfCongruences(S, Combinations(restriction, 2));
<poset of 2 two-sided congruences over
<full transformation monoid of degree 4>>
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="PosetOfPrincipalCongruences">
<ManSection>
<Attr Name = "PosetOfPrincipalCongruences" Arg = "S"
Label = "for a semigroup"/>
<Attr Name = "PosetOfPrincipalLeftCongruences" Arg = "S"
Label = "for a semigroup"/>
<Attr Name = "PosetOfPrincipalRightCongruences" Arg = "S"
Label = "for a semigroup"/>
<Oper Name = "PosetOfPrincipalCongruences" Arg = "S, restriction"
Label = "for a semigroup and a multiplicative element collection"/>
<Oper Name = "PosetOfPrincipalLeftCongruences" Arg = "S, restriction"
Label = "for a semigroup and a multiplicative element collection"/>
<Oper Name = "PosetOfPrincipalRightCongruences" Arg = "S, restriction"
Label = "for a semigroup and a multiplicative element collection"/>
<Returns>A congruence poset.</Returns>
<Description>
If <A>S</A> is a semigroup, then <C>PosetOfPrincipalCongruences</C>
returns a congruence poset object which contains all the principal
congruences of <A>S</A>, ordered by containment according to
<Ref Oper = "IsSubrelation"/>. A congruence is <E>principal</E> if it can
be defined by a single generating pair.
<C>PosetOfPrincipalLeftCongruences</C> and
<C>PosetOfPrincipalRightCongruences</C> do the same thing for left and
right congruences respectively. <P/>
If <A>restriction</A> is specified and is a collection of elements from
<A>S</A>, then the result will only include principal congruences
generated by pairs of elements from <A>restriction</A>. Otherwise, all
principal congruences will be calculated.<P/>
See also <Ref Attr = "LatticeOfCongruences" Label = "for a semigroup"/>
and <Ref Attr = "PrincipalCongruencesOfSemigroup"
Label = "for a semigroup"/>. <P/>
<Example><![CDATA[
gap> S := Semigroup(Transformation([1, 3, 1]),
> Transformation([2, 3, 3]));;
gap> PosetOfPrincipalLeftCongruences(S);
<poset of 12 left congruences over <transformation semigroup
of size 11, degree 3 with 2 generators>>
gap> PosetOfPrincipalCongruences(S);
<lattice of 3 two-sided congruences over <transformation semigroup
of size 11, degree 3 with 2 generators>>
gap> restriction := [Transformation([3, 2, 3]),
> Transformation([3, 1, 3]),
> Transformation([2, 2, 2])];;
gap> poset := PosetOfPrincipalRightCongruences(S,
> Combinations(restriction, 2));
<poset of 3 right congruences over <transformation semigroup
of size 11, degree 3 with 2 generators>>
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="CongruencesOfPoset">
<ManSection>
<Attr Name = "CongruencesOfPoset" Arg = "poset"/>
<Returns>A list.</Returns>
<Description>
If <A>poset</A> is a congruence poset object, then this attribute returns
a list of all the congruence objects in the poset (these may be left,
right, or two-sided). The order of this list corresponds to the order of
the entries in the poset. <P/>
See also <Ref Attr = "LatticeOfCongruences" Label = "for a semigroup"/>
and <Ref Attr = "CongruencesOfSemigroup" Label = "for a semigroup"/>. <P/>
<Example><![CDATA[
gap> S := OrderEndomorphisms(2);;
gap> latt := LatticeOfRightCongruences(S);
<lattice of 5 right congruences over <regular transformation monoid
of size 3, degree 2 with 2 generators>>
gap> CongruencesOfPoset(latt);
[ <2-sided semigroup congruence over <regular transformation monoid
of size 3, degree 2 with 2 generators> with 0 generating pairs>,
<right semigroup congruence over <regular transformation monoid
of size 3, degree 2 with 2 generators> with 1 generating pairs>,
<right semigroup congruence over <regular transformation monoid
of size 3, degree 2 with 2 generators> with 1 generating pairs>,
<right semigroup congruence over <regular transformation monoid
of size 3, degree 2 with 2 generators> with 1 generating pairs>,
<right semigroup congruence over <regular transformation monoid
of size 3, degree 2 with 2 generators> with 2 generating pairs> ]
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="UnderlyingSemigroupOfCongruencePoset">
<ManSection>
<Attr Name = "UnderlyingSemigroupOfCongruencePoset" Arg = "poset"/>
<Returns>A semigroup.</Returns>
<Description>
If <A>poset</A> is a congruence poset object, then this attribute returns
the semigroup on which all its congruences are defined.
<Example><![CDATA[
gap> S := OrderEndomorphisms(2);
<regular transformation monoid of degree 2 with 2 generators>
gap> latt := LatticeOfRightCongruences(S);
<lattice of 5 right congruences over <regular transformation monoid
of size 3, degree 2 with 2 generators>>
gap> UnderlyingSemigroupOfCongruencePoset(latt) = S;
true
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="PosetOfCongruences">
<ManSection>
<Oper Name = "PosetOfCongruences" Arg = "coll"/>
<Returns>A congruence poset.</Returns>
<Description>
If <A>coll</A> is a list or collection of semigroup congruences (which may
be left, right, or two-sided) then this operation returns the congruence
poset formed by these congruences partially ordered by containment. <P/>
This operation does not create any new congruences or take any joins.
See also <Ref Oper = "JoinSemilatticeOfCongruences"/>,
<Ref Filt = "IsCongruencePoset"/>,
and <Ref Attr = "LatticeOfCongruences" Label = "for a semigroup"/>. <P/>
<#GAPDoc Label="JoinSemilatticeOfCongruences">
<ManSection>
<Attr Name = "JoinSemilatticeOfCongruences" Arg = "poset"/>
<Returns>A congruence poset.</Returns>
<Description>
If <A>poset</A> is a congruence poset (i.e. it satisfies
<Ref Filt = "IsCongruencePoset"/>), then
this function returns the congruence poset formed by these congruences
partially ordered by containment, along with all their joins.
This includes the empty join which equals the trivial congruence.
<P/>
The digraph returned by this function represents the Cayley graph of
the semilattice generated by <Ref Attr="CongruencesOfPoset"/> with
identity adjoined. The reflexive transitive closure of this digraph is a
join semillatice in the sense of
<Ref Prop="IsJoinSemilatticeDigraph" BookName="Digraphs"/>.
<P/>
See also <Ref Filt = "IsCongruencePoset"/>
and <Ref Oper = "PosetOfCongruences"/>.
<#GAPDoc Label="MinimalCongruences">
<ManSection>
<Attr Name = "MinimalCongruences" Arg = "coll"
Label = "for a list or collection"/>
<Attr Name = "MinimalCongruences" Arg = "poset"
Label = "for a congruence poset"/>
<Returns>A list.</Returns>
<Description>
If <A>coll</A> is a list or collection of semigroup congruences (which may
be left, right, or two-sided) then this attribute returns a list of all
the congruences from <A>coll</A> which do not contain any of the others as
subrelations. <P/>
Alternatively, a congruence poset <A>poset</A> can be specified; in this
case, the congruences contained in <A>poset</A> will be used in place of
<A>coll</A>, and information already known about their containments will
be used. <P/>
This function should not be confused with
<Ref Attr = "MinimalCongruencesOfSemigroup" Label = "for a semigroup"/>.
See also <Ref Filt = "IsCongruencePoset"/>
and <Ref Oper = "PosetOfCongruences"/>. <P/>
<Example><![CDATA[
gap> S := SymmetricInverseMonoid(2);;
gap> pair1 := [PartialPerm([1], [1]), PartialPerm([2], [1])];;
gap> pair2 := [PartialPerm([1], [1]), PartialPerm([1, 2], [1, 2])];;
gap> pair3 := [PartialPerm([1, 2], [1, 2]),
> PartialPerm([1, 2], [2, 1])];;
gap> coll := [RightSemigroupCongruence(S, pair1),
> RightSemigroupCongruence(S, pair2),
> RightSemigroupCongruence(S, pair3)];;
gap> MinimalCongruences(PosetOfCongruences(coll));
[ <right semigroup congruence over <symmetric inverse monoid of degree\
2> with 1 generating pairs>,
<right semigroup congruence over <symmetric inverse monoid of degree\
2> with 1 generating pairs> ]
gap> poset := LatticeOfCongruences(S);
<lattice of 4 two-sided congruences over
<symmetric inverse monoid of degree 2>>
gap> MinimalCongruences(poset);
[ <2-sided semigroup congruence over <symmetric inverse monoid of degr\
ee 2> with 0 generating pairs> ]
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="GeneratingCongruencesOfJoinSemilattice">
<ManSection>
<Attr Name = "GeneratingCongruencesOfJoinSemilattice" Arg="poset"/>
<Returns>A list of congruences.</Returns>
<Description>
If <A>poset</A> satisfies <Ref Filt="IsCayleyDigraphOfCongruences"/>,
then this attribute holds the generating set for the semilattice of
congruences (where the operation is join).
<Example><![CDATA[
gap> S := OrderEndomorphisms(3);;
gap> D := CayleyDigraphOfCongruences(S);
<poset of 4 two-sided congruences over <regular transformation monoid
of size 10, degree 3 with 3 generators>>
gap> GeneratingCongruencesOfJoinSemilattice(D);
[ <universal semigroup congruence over <regular transformation monoid
of size 10, degree 3 with 3 generators>>,
<2-sided semigroup congruence over <regular transformation monoid
of size 10, degree 3 with 3 generators> with 1 generating pairs>,
<2-sided semigroup congruence over <regular transformation monoid
of size 10, degree 3 with 3 generators> with 1 generating pairs>
]
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
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