Quelle conginv.xml Sprache: XML

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#W conginv.xml
#Y Copyright (C) 2015 Michael Young
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## Licensing information can be found in the README file of this package.
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<#GAPDoc Label="TraceOfSemigroupCongruence">
 <ManSection>
 <Attr Name="TraceOfSemigroupCongruence" Arg="cong"/>
 <Returns>A list of lists.</Returns>
 <Description>
 If <A>cong</A> is an inverse semigroup congruence by kernel and trace,
 then this attribute returns the restriction of <A>cong</A> to the
 idempotents of the semigroup. This is in block form: each idempotent will
 appear in precisely one list, and two idempotents will be in the same list
 if and only if they are related by <A>cong</A>.

 <Example><![CDATA[
gap> I := InverseSemigroup([
> PartialPerm([2, 3]), PartialPerm([2, 0, 3])]);;
gap> cong := SemigroupCongruence(I,
> [[PartialPerm([0, 1, 3]), PartialPerm([0, 1])],
> [PartialPerm([]), PartialPerm([1, 2])]]);
<2-sided semigroup congruence over <inverse partial perm semigroup
of size 19, rank 3 with 2 generators> with 2 generating pairs>
gap> TraceOfSemigroupCongruence(cong);
[ [ <empty partial perm>, <identity partial perm on [ 1 ]>,
 <identity partial perm on [ 2 ]>,
 <identity partial perm on [ 1, 2 ]>,
 <identity partial perm on [ 3 ]>,
 <identity partial perm on [ 2, 3 ]>,
 <identity partial perm on [ 1, 3 ]> ] ]]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="KernelOfSemigroupCongruence">
 <ManSection>
 <Attr Name = "KernelOfSemigroupCongruence" Arg = "cong"/>
 <Returns>
 An inverse semigroup.
 </Returns>
 <Description>
 If <A>cong</A> is a congruence over a semigroup with inverse op,
 then this attribute returns the <E>kernel</E> of that congruence; that
 is, the inverse subsemigroup consisting of all elements which are related
 to an idempotent by <A>cong</A>. 
 <Example><![CDATA[
gap> I := InverseSemigroup([
> PartialPerm([2, 3]), PartialPerm([2, 0, 3])]);;
gap> cong := SemigroupCongruence(I,
> [[PartialPerm([0, 1, 3]), PartialPerm([0, 1])],
> [PartialPerm([]), PartialPerm([1, 2])]]);
<2-sided semigroup congruence over <inverse partial perm semigroup
of size 19, rank 3 with 2 generators> with 2 generating pairs>
gap> KernelOfSemigroupCongruence(cong);
<inverse partial perm semigroup of size 19, rank 3 with 5 generators>
]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="AsInverseSemigroupCongruenceByKernelTrace">
 <ManSection>
 <Attr Name = "AsInverseSemigroupCongruenceByKernelTrace" Arg = "cong"/>
 <Returns>
 An inverse semigroup congruence by kernel and trace.
 </Returns>
 <Description>
 If <A>cong</A> is a semigroup congruence over an inverse semigroup, then
 this attribute returns an object which describes the same congruence, but
 with an internal representation defined by that congruence's kernel and
 trace. 

 See <Cite Key = "Howie1995aa"/> section 5.3 for more details.

 <Example><![CDATA[
gap> I := InverseSemigroup([
> PartialPerm([2, 3]), PartialPerm([2, 0, 3])]);;
gap> cong := SemigroupCongruenceByGeneratingPairs(I,
> [[PartialPerm([0, 1, 3]), PartialPerm([0, 1])],
> [PartialPerm([]), PartialPerm([1, 2])]]);
<2-sided semigroup congruence over <inverse partial perm semigroup of
rank 3 with 2 generators> with 2 generating pairs>
gap> cong2 := AsInverseSemigroupCongruenceByKernelTrace(cong);
<semigroup congruence over <inverse partial perm semigroup
of size 19, rank 3 with 2 generators> with congruence pair (19,1)>]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsInverseSemigroupCongruenceByKernelTrace">
 <ManSection>
 <Filt Name = "IsInverseSemigroupCongruenceByKernelTrace" Arg = "cong"
 Type = "Category"/>
 <Returns>
 <K>true</K> or <K>false</K>.</Returns>
 <Description>
 This category contains any inverse semigroup congruence <A>cong</A> which
 is represented internally by its kernel and trace. The <Ref
 Func = "SemigroupCongruence"/> function may create an
 object of this category if its first argument <A>S</A> is an inverse
 semigroup and has sufficiently large size. It can be treated
 like any other semigroup congruence object.
 

 See <Cite Key = "Howie1995aa"/> Section 5.3 for more details. See also
 <Ref Func = "InverseSemigroupCongruenceByKernelTrace"/>.

 <Example><![CDATA[
gap> S := InverseSemigroup([
> PartialPerm([4, 3, 1, 2]),
> PartialPerm([1, 4, 2, 0, 3])],
> rec(cong_by_ker_trace_threshold := 0));;
gap> cong := SemigroupCongruence(S, []);
<semigroup congruence over <inverse partial perm semigroup
of size 351, rank 5 with 2 generators> with congruence pair (24,24)>
gap> IsInverseSemigroupCongruenceByKernelTrace(cong);
true]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsInverseSemigroupCongruenceClassByKernelTrace">
 <ManSection>
 <Filt Name = "IsInverseSemigroupCongruenceClassByKernelTrace" Arg = "obj"
 Type = "Category"/>
 <Returns>
 <K>true</K> or <K>false</K>.
 </Returns>
 <Description>
 This category contains any congruence class which belongs to a congruence
 which is represented internally by its kernel and trace. See <Ref
 Func = "InverseSemigroupCongruenceByKernelTrace"/>. 

 See <Cite Key = "Howie1995aa"/> Section 5.3 for more details.

 <Example><![CDATA[
gap> I := InverseSemigroup([
> PartialPerm([2, 3]), PartialPerm([2, 0, 3])],
> rec(cong_by_ker_trace_threshold := 0));;
gap> cong := SemigroupCongruence(I,
> [[PartialPerm([0, 1, 3]), PartialPerm([0, 1])],
> [PartialPerm([]), PartialPerm([1, 2])]]);;
gap> class := EquivalenceClassOfElement(cong,
> PartialPerm([1, 2], [2, 3]));;
gap> IsInverseSemigroupCongruenceClassByKernelTrace(class);
true]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="InverseSemigroupCongruenceByKernelTrace">
 <ManSection>
 <Func Name = "InverseSemigroupCongruenceByKernelTrace"
 Arg = "S, kernel, traceBlocks"/>
 <Returns>
 An inverse semigroup congruence by kernel and trace.
 </Returns>
 <Description>
 If <A>S</A> is an inverse semigroup, <A>kernel</A> is a subsemigroup of
 <A>S</A>, <A>traceBlocks</A> is a list of lists describing a congruence
 on the idempotents of <A>S</A>, and <M>(<A>kernel</A>, <A>trace</A>)</M>
 describes a valid congruence pair for <A>S</A> (see <Cite Key =
 "Howie1995aa"/> Section 5.3) then this function returns the semigroup
 congruence defined by that congruence pair. 

 See also <Ref Attr = "KernelOfSemigroupCongruence"/> and
 <Ref Attr = "TraceOfSemigroupCongruence"/>.

 <Example><![CDATA[
gap> S := InverseSemigroup([
> PartialPerm([2, 3]), PartialPerm([2, 0, 3])]);;
gap> kernel := InverseSemigroup([
> PartialPerm([1, 0, 3]), PartialPerm([0, 2, 3]),
> PartialPerm([1, 2]), PartialPerm([3]),
> PartialPerm([2])]);;
gap> trace := [
> [PartialPerm([0, 2, 3])],
> [PartialPerm([1, 2])],
> [PartialPerm([1, 0, 3])],
> [PartialPerm([0, 0, 3]), PartialPerm([0, 2]),
> PartialPerm([1]), PartialPerm([], [])]];;
gap> cong := InverseSemigroupCongruenceByKernelTrace(S, kernel, trace);
<semigroup congruence over <inverse partial perm semigroup of rank 3
with 2 generators> with congruence pair (13,4)>]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="MinimumGroupCongruence">
 <ManSection>
 <Attr Name = "MinimumGroupCongruence" Arg = "S"/>
 <Returns>
 An inverse semigroup congruence by kernel and trace.
 </Returns>
 <Description>
 If <A>S</A> is an inverse semigroup, then this function returns the
 least congruence on <A>S</A> whose quotient is a group. 

 <Example><![CDATA[
gap> S := InverseSemigroup([
> PartialPerm([5, 2, 0, 0, 1, 4]),
> PartialPerm([1, 4, 6, 3, 5, 0, 2])]);;
gap> cong := MinimumGroupCongruence(S);
<semigroup congruence over <inverse partial perm semigroup
of size 101, rank 7 with 2 generators> with congruence pair (59,1)>
gap> IsGroupAsSemigroup(S / cong);
true]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

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