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#W semiex.xml
#Y Copyright (C) 2011-17 James D. Mitchell
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## Licensing information can be found in the README file of this package.
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<#GAPDoc Label="EndomorphismsPartition">
<ManSection>
<Oper Name = "EndomorphismsPartition" Arg = "list"/>
<Returns>A transformation monoid.</Returns>
<Description>
If <A>list</A> is a list of positive integers, then
<C>EndomorphismsPartition</C> returns a monoid of endomorphisms preserving
a partition of <C>[1 .. Sum(<A>list</A>)]</C> with
a part of length <C><A>list</A>[i]</C> for
every <C>i</C>. For example, if <C><A>list</A> = [1, 2, 3]</C>, then
<C>EndomorphismsPartition</C> returns the monoid of endomorphisms of the
partition <C>[[1], [2, 3], [4, 5, 6]]</C>. <P/>
If <C>f</C> is a transformation of <C>[1 .. n]</C>, then it is an
<B>endomorphism</B> of a partition <C>P</C> on <C>[1 .. n]</C> if
<C>(i, j)</C> in <C>P</C> implies that <C>(i ^ f, j ^ f)</C> is in
<C>P</C>.
<P/>
<C>EndomorphismsPartition</C> returns a monoid with a minimal size
generating set, as described in <Cite Key = "Araujo2015aa"/>.
<Example><![CDATA[
gap> S := EndomorphismsPartition([3, 3, 3]);
<transformation semigroup of degree 9 with 4 generators>
gap> Size(S);
531441]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="AnnularJonesMonoid">
<ManSection>
<Oper Name = "AnnularJonesMonoid" Arg = "n"/>
<Returns>A bipartition monoid.</Returns>
<Description>
If <A>n</A> is a non-negative integer, then <C>AnnularJonesMonoid</C>
returns the annular Jones monoid of degree <A>n</A>. The <E>annular Jones
monoid</E> is the subsemigroup of the partition monoid consisting of all
annular bipartitions whose blocks have size 2 (annular bipartitions are
defined in Chapter <Ref Chap="Bipartitions and blocks"/>). See <Ref
Oper="BrauerMonoid" />.
<Example><![CDATA[
gap> S := AnnularJonesMonoid(4);
<regular bipartition *-monoid of degree 4 with 2 generators>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="PartialJonesMonoid">
<ManSection>
<Oper Name = "PartialJonesMonoid" Arg = "n"/>
<Returns>A bipartition monoid.</Returns>
<Description>
If <A>n</A> is a non-negative integer, then <C>PartialJonesMonoid</C>
returns the partial Jones monoid of degree <A>n</A>. The <E>partial Jones
monoid</E> is a subsemigroup of the partial Brauer monoid. An element of
the partial Brauer monoid is contained in the partial Jones monoid if the
partition that it defines is finer than the partition defined by some element of the Jones monoid. In other words, an element of the partial
Jones monoid can be formed from some element <C>x</C> of the Jones monoid
by replacing some blocks <C>[a, b]</C> of <C>x</C> by singleton blocks
<C>[a], [b]</C>. <P/>
Note that, in general, the partial Jones monoid of degree <A>n</A> is
strictly contained in the Motzkin monoid of the same degree. <P/>
See <Ref Oper="PartialBrauerMonoid"/>, <Ref Oper="JonesMonoid" />, and
<Ref Oper="MotzkinMonoid"/>.
<Example><![CDATA[
gap> S := PartialJonesMonoid(4);
<regular bipartition *-monoid of degree 4 with 7 generators>
gap> T := JonesMonoid(4);
<regular bipartition *-monoid of degree 4 with 3 generators>
gap> U := MotzkinMonoid(4);
<regular bipartition *-monoid of degree 4 with 8 generators>
gap> IsSubsemigroup(U, S);
true
gap> IsSubsemigroup(S, T);
true
gap> Size(U);
323
gap> Size(S);
143
gap> Size(T);
14]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="JonesMonoid">
<ManSection>
<Oper Name = "JonesMonoid" Arg = "n"/>
<Oper Name = "TemperleyLiebMonoid" Arg = "n"/>
<Oper Name = "SingularJonesMonoid" Arg = "n"/>
<Returns>A bipartition monoid.</Returns>
<Description>
If <A>n</A> is a non-negative integer, then this operation returns the
Jones monoid of degree <A>n</A>. The <E>Jones monoid</E> is the
subsemigroup of the Brauer monoid consisting of those bipartitions that
are planar; see <Ref Oper="PlanarPartitionMonoid" />. The Jones monoid is
sometimes referred to as the <B>Temperley-Lieb monoid</B>. <P/>
<C>SingularJonesMonoid</C> returns the ideal of the Jones monoid
consisting of the non-invertible elements (i.e. those not in the group of
units), when <A>n</A> is at least 2.
<Example><![CDATA[
gap> S := JonesMonoid(4);
<regular bipartition *-monoid of degree 4 with 3 generators>
gap> S = TemperleyLiebMonoid(4);
true
gap> SingularJonesMonoid(8);
<regular bipartition *-semigroup ideal of degree 8 with 1 generator>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="MotzkinMonoid">
<ManSection>
<Oper Name = "MotzkinMonoid" Arg = "n"/>
<Returns>A bipartition monoid.</Returns>
<Description>
If <A>n</A> is a non-negative integer, then this operation returns the
Motzkin monoid of degree <A>n</A>. The <E>Motzkin monoid</E> is the
subsemigroup of the partial Brauer monoid consisting of those bipartitions
that are planar (planar bipartitions are defined in Chapter <Ref
Chap="Bipartitions and blocks"/>). <P/>
Note that the Motzkin monoid of degree <A>n</A> contains the partial Jones
monoid of degree <A>n</A>, but in general, these monoids are not equal;
see <Ref Oper="PartialJonesMonoid"/>.
<Example><![CDATA[
gap> S := MotzkinMonoid(4);
<regular bipartition *-monoid of degree 4 with 8 generators>
gap> T := PartialJonesMonoid(4);
<regular bipartition *-monoid of degree 4 with 7 generators>
gap> IsSubsemigroup(S, T);
true
gap> Size(S);
323
gap> Size(T);
143]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="BrauerMonoid">
<ManSection>
<Oper Name="BrauerMonoid" Arg="n"/>
<Oper Name="PartialBrauerMonoid" Arg="n"/>
<Oper Name="SingularBrauerMonoid" Arg="n"/>
<Returns>A bipartition monoid.</Returns>
<Description>
If <A>n</A> is a non-negative integer, then this operation returns the
Brauer monoid of degree <A>n</A>. The <E>Brauer monoid</E> is the
submonoid of the partition monoid consisting of those bipartitions where
the size of every block is 2. <P/>
<C>PartialBrauerMonoid</C> returns the partial Brauer monoid, which is the
submonoid of the partition monoid consisting of those bipartitions where
the size of every block is <E>at most</E> 2. The partial Brauer monoid
contains the Brauer monoid as a submonoid. <P/>
<C>SingularBrauerMonoid</C> returns the ideal of the Brauer monoid
consisting of the non-invertible elements (i.e. those not in the group of
units), when <A>n</A> is at least 2.
<Example><![CDATA[
gap> S := BrauerMonoid(4);
<regular bipartition *-monoid of degree 4 with 3 generators>
gap> IsSubsemigroup(S, JonesMonoid(4));
true
gap> Size(S);
105
gap> SingularBrauerMonoid(8);
<regular bipartition *-semigroup ideal of degree 8 with 1 generator>
gap> S := PartialBrauerMonoid(3);
<regular bipartition *-monoid of degree 3 with 8 generators>
gap> IsSubsemigroup(S, BrauerMonoid(3));
true
gap> Size(S);
76]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="PartitionMonoid">
<ManSection>
<Oper Name="PartitionMonoid" Arg="n"/>
<Oper Name="RookPartitionMonoid" Arg="n"/>
<Oper Name="SingularPartitionMonoid" Arg="n"/>
<Returns>A bipartition monoid.</Returns>
<Description>
If <A>n</A> is a non-negative integer, then this operation returns the
partition monoid of degree <A>n</A>. The <E>partition monoid of degree
<A>n</A></E> is the monoid consisting of all the bipartitions of degree
<A>n</A>.
<P/>
<C>SingularPartitionMonoid</C> returns the ideal of the partition monoid
consisting of the non-invertible elements (i.e. those not in the group of
units), when <A>n</A> is positive.
<P/>
If <A>n</A> is positive, then <C>RookPartitionMonoid</C> returns submonoid
of the partition monoid of degree <C><A>n</A> + 1</C> consisting of those
bipartitions with <C><A>n</A> + 1</C> and <C>-<A>n</A> - 1</C> in the same
block; see <Cite Key="Halverson2005PartitionAlgebras"/>, <Cite
Key="Grood2006aa"/>, and <Cite Key="East2019aa"/>.
<Example><![CDATA[
gap> S := PartitionMonoid(4);
<regular bipartition *-monoid of size 4140, degree 4 with 4
generators>
gap> Size(S);
4140
gap> T := SingularPartitionMonoid(4);
<regular bipartition *-semigroup ideal of degree 4 with 1 generator>
gap> Size(S) - Size(T) = Factorial(4);
true
gap> S := RookPartitionMonoid(4);
<regular bipartition *-monoid of degree 5 with 5 generators>
gap> Size(S);
21147]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="PlanarPartitionMonoid">
<ManSection>
<Oper Name="PlanarPartitionMonoid" Arg="n"/>
<Oper Name="SingularPlanarPartitionMonoid" Arg="n"/>
<Returns>A bipartition monoid.</Returns>
<Description>
If <A>n</A> is a positive integer, then this operation returns the planar
partition monoid of degree <A>n</A> which is the monoid consisting of all
the planar bipartitions of degree <A>n</A> (planar bipartitions are defined
in Chapter <Ref Chap="Bipartitions and blocks"/>). <P/>
<C>SingularPlanarPartitionMonoid</C> returns the ideal of the planar
partition monoid consisting of the non-invertible elements
(i.e. those not in the group of units). <P/>
<Example><![CDATA[
gap> S := PlanarPartitionMonoid(3);
<regular bipartition *-monoid of degree 3 with 5 generators>
gap> Size(S);
132
gap> T := SingularPlanarPartitionMonoid(3);
<regular bipartition *-semigroup ideal of degree 3 with 1 generator>
gap> Size(T);
131
gap> Difference(S, T);
[ <block bijection: [ 1, -1 ], [ 2, -2 ], [ 3, -3 ]> ]
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="UniformBlockBijectionMonoid">
<ManSection>
<Oper Name="UniformBlockBijectionMonoid" Arg="n"/>
<Oper Name="FactorisableDualSymmetricInverseMonoid" Arg="n"/>
<Oper Name="SingularUniformBlockBijectionMonoid" Arg="n"/>
<Oper Name="PartialUniformBlockBijectionMonoid" Arg="n"/>
<Oper Name="SingularFactorisableDualSymmetricInverseMonoid" Arg="n"/>
<Oper Name="PlanarUniformBlockBijectionMonoid" Arg="n"/>
<Oper Name="SingularPlanarUniformBlockBijectionMonoid" Arg="n"/>
<Returns>An inverse bipartition monoid.</Returns>
<Description>
If <A>n</A> is a positive integer, then this operation returns the uniform
block bijection monoid of degree <A>n</A>. The <E>uniform block bijection
monoid</E> is the submonoid of the partition monoid consisting of the
block bijections of degree &n; where the number of positive integers in a
block equals the number of negative integers in that block.
The uniform block bijection monoid is also referred to as the
<E>factorisable dual symmetric inverse monoid</E>. <P/>
<C>SingularUniformBlockBijectionMonoid</C> returns the ideal of the
uniform block bijection monoid consisting of the non-invertible elements
(i.e. those not in the group of units), when <A>n</A> is at least 2. <P/>
<C>PlanarUniformBlockBijectionMonoid</C> returns the submonoid of the
uniform block bijection monoid consisting of the planar elements
(i.e. those in the planar partition monoid, see <Ref
Oper="PlanarPartitionMonoid"/>). <P/>
<C>SingularPlanarUniformBlockBijectionMonoid</C> returns the ideal of the
planar uniform block bijection monoid consisting of the non-invertible
elements (i.e. those not in the group of units), when <A>n</A> is
at least 2. <P/> <!-- TODO(later) reference -->
<C>PartialUniformBlockBijectionMonoid</C> returns the submonoid of the
uniform block bijection monoid of degree <C><A>n</A> + 1</C> consisting
of those uniform block bijection with <C><A>n</A> + 1</C> and
<C>-<A>n</A> - 1</C> in the same block. <P/>
See <Ref Prop="IsUniformBlockBijection"/>.
<Example><![CDATA[
gap> S := UniformBlockBijectionMonoid(4);
<inverse block bijection monoid of degree 4 with 3 generators>
gap> Size(PlanarUniformBlockBijectionMonoid(8));
128
gap> S := DualSymmetricInverseMonoid(4);
<inverse block bijection monoid of degree 4 with 3 generators>
gap> IsFactorisableInverseMonoid(S);
false
gap> S := UniformBlockBijectionMonoid(4);
<inverse block bijection monoid of degree 4 with 3 generators>
gap> IsFactorisableInverseMonoid(S);
true
gap> S := AsSemigroup(IsBipartitionSemigroup,
> SymmetricInverseMonoid(5));
<inverse bipartition monoid of degree 5 with 3 generators>
gap> IsFactorisableInverseMonoid(S);
true
gap> S := PartialUniformBlockBijectionMonoid(5);
<inverse block bijection monoid of degree 6 with 4 generators>
gap> NrIdempotents(S);
203
gap> IsFactorisableInverseMonoid(S);
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="DualSymmetricInverseSemigroup">
<ManSection>
<Oper Name = "DualSymmetricInverseSemigroup" Arg = "n"/>
<Oper Name = "DualSymmetricInverseMonoid" Arg = "n"/>
<Oper Name = "SingularDualSymmetricInverseMonoid" Arg = "n"/>
<Oper Name = "PartialDualSymmetricInverseMonoid" Arg = "n"/>
<Returns>An inverse bipartition monoid.</Returns>
<Description>
If <A>n</A> is a positive integer, then the operations
<C>DualSymmetricInverseSemigroup</C> and <C>DualSymmetricInverseMonoid</C>
return the dual symmetric inverse monoid of degree <A>n</A>, which is the
subsemigroup of the partition monoid consisting of the block bijections of
degree <A>n</A>.<P/>
<C>SingularDualSymmetricInverseMonoid</C> returns the ideal of the
dual symmetric inverse monoid consisting of the non-invertible elements
(i.e. those not in the group of units), when <A>n</A> is at least 2.<P/>
<C>PartialDualSymmetricInverseMonoid</C> returns the submonoid of the
dual symmetric inverse monoid of degree <C><A>n</A> + 1</C> consisting of
those block bijections with <C><A>n</A> + 1</C> and <C>-<A>n</A> - 1</C> in
the same block; see <Cite Key="Kudryavtseva2011aa"/> and
<Cite Key="Kudryavtseva2015aa"/>. <P/>
See <Ref Prop = "IsBlockBijection"/>.
<Example><![CDATA[
gap> Number(PartitionMonoid(3), IsBlockBijection);
25
gap> S := DualSymmetricInverseSemigroup(3);
<inverse block bijection monoid of degree 3 with 3 generators>
gap> Size(S);
25
gap> S := PartialDualSymmetricInverseMonoid(5);
<inverse block bijection monoid of degree 6 with 4 generators>
gap> Size(S);
29072]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="ApsisMonoid">
<ManSection>
<Oper Name="ApsisMonoid" Arg="m, n"/>
<Oper Name="SingularApsisMonoid" Arg="m, n"/>
<Oper Name="CrossedApsisMonoid" Arg="m, n"/>
<Oper Name="SingularCrossedApsisMonoid" Arg="m, n"/>
<Returns>A bipartition monoid.</Returns>
<Description>
If <A>m</A> and <A>n</A> are positive integers, then this operation
returns the <A>m</A>-apsis monoid of degree <A>n</A>. The
<A>m</A><E>-apsis monoid</E> is the monoid of bipartitions generated when
the diapses in generators of the Jones monoid are replaced
with <A>m</A>-apses. Note that an <A>m</A><E>-apsis</E> is a block that
contains precisely <A>m</A> consecutive integers. <P/>
<C>SingularApsisMonoid</C> returns the ideal of the
apsis monoid consisting of the non-invertible elements
(i.e. those not in the group of units), when <A>m</A> <M>\leq</M>
<A>n</A>. <P/>
<C>CrossedApsisGeneratedMonoid</C> returns the semigroup generated by the
symmetric group of degree <A>n</A> and the <A>m</A>-apsis monoid of
degree <A>n</A>. <P/>
<C>SingularCrossedApsisMonoid</C> returns the ideal of the
crossed apsis monoid consisting of the non-invertible elements
(i.e. those not in the group of units), when <A>m <= n</A>. <!-- TODO(later) reference and/or definition -->
<Example><![CDATA[
gap> S := ApsisMonoid(3, 7);
<regular bipartition *-monoid of degree 7 with 5 generators>
gap> Size(S);
320
gap> T := SingularApsisMonoid(3, 7);
<regular bipartition *-semigroup ideal of degree 7 with 1 generator>
gap> Difference(S, T) = [One(S)];
true
gap> Size(CrossedApsisMonoid(2, 5));
945
gap> SingularCrossedApsisMonoid(4, 6);
<regular bipartition *-semigroup ideal of degree 6 with 1 generator>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="PartialTransformationMonoid">
<ManSection>
<Oper Name = "PartialTransformationMonoid" Arg = "n"/>
<Returns>A transformation monoid.</Returns>
<Description>
If <A>n</A> is a positive integer, then this function returns a
semigroup of transformations on <C><A>n</A> + 1</C> points which is
isomorphic to the semigroup consisting of all partial transformation on
<A>n</A> points. This monoid has <C>(<A>n</A> + 1) ^ <A>n</A></C> elements. <!--TODO(later) add generators-->
<Example><![CDATA[
gap> S := PartialTransformationMonoid(5);
<regular transformation monoid of degree 6 with 4 generators>
gap> Size(S);
7776]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="ModularPartitionMonoid">
<ManSection>
<Oper Name="ModularPartitionMonoid" Arg="m, n"/>
<Oper Name="SingularModularPartitionMonoid" Arg="m, n"/>
<Oper Name="PlanarModularPartitionMonoid" Arg="m, n"/>
<Oper Name="SingularPlanarModularPartitionMonoid" Arg="m, n"/>
<Returns>A bipartition monoid.</Returns>
<Description>
If <A>m</A> and <A>n</A> are positive integers, then this operation
returns the modular-<A>m</A> partition monoid of degree <A>n</A>. The
<E>modular-</E><A>m</A> <E>partition monoid</E> is the submonoid of the
partition monoid such that the numbers of positive and negative
integers contained in each block are congruent mod <A>m</A>. <P/>
<C>SingularModularPartitionMonoid</C> returns the ideal of the
modular partition monoid consisting of the non-invertible elements
(i.e. those not in the group of units), when either
<A>m = n = 1</A> or <A>m, n > 1</A>. <P/>
<C>PlanarModularPartitionMonoid</C> returns the submonoid of the
modular-<A>m</A> partition monoid consisting of the planar elements
(i.e. those in the planar partition monoid, see <Ref
Oper="PlanarPartitionMonoid" />). <P/>
<C>SingularPlanarModularPartitionMonoid</C> returns the ideal of the
planar modular partition monoid consisting of the non-invertible elements
(i.e. those not in the group of units), when either
<A>m = n = 1</A> or <A>m, n > 1</A>. <!-- TODO(later) reference -->
<Example><![CDATA[
gap> S := ModularPartitionMonoid(3, 6);
<regular bipartition *-monoid of degree 6 with 4 generators>
gap> Size(S);
36243
gap> S := SingularModularPartitionMonoid(1, 1);
<commutative inverse bipartition semigroup ideal of degree 1 with
1 generator>
gap> Size(SingularModularPartitionMonoid(2, 4));
355
gap> S := PlanarModularPartitionMonoid(4, 9);
<regular bipartition *-monoid of degree 9 with 14 generators>
gap> Size(S);
1795
gap> S := SingularPlanarModularPartitionMonoid(3, 5);
<regular bipartition *-semigroup ideal of degree 5 with 1 generator>
gap> Size(SingularPlanarModularPartitionMonoid(1, 2));
13]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="GeneralLinearMonoid">
<ManSection>
<Oper Name="FullMatrixMonoid" Arg="d, q"/>
<Oper Name="GeneralLinearMonoid" Arg="d, q"/>
<Oper Name="GLM" Arg="d, q"/>
<Returns>A matrix monoid.</Returns>
<Description>
These operations return the full matrix monoid of <A>d</A> by <A>d</A>
matrices over the field with <A>q</A> elements. The <E>full matrix
monoid</E>, also known as the <E>general linear monoid</E>, with these
parameters, is the monoid consisting of all <A>d</A> by <A>d</A> matrices
with entries from the field <C>GF(<A>q</A>)</C>. This
monoid has <C><A>q</A> ^ (<A>d</A> ^ 2)</C> elements.
<Example><![CDATA[
gap> S := FullMatrixMonoid(2, 4);
<general linear monoid 2x2 over GF(2^2)>
gap> Size(S);
256
gap> S = GeneralLinearMonoid(2, 4);
true
gap> GLM(2, 2);
<general linear monoid 2x2 over GF(2)>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="SpecialLinearMonoid">
<ManSection>
<Oper Name="SpecialLinearMonoid" Arg="d, q"/>
<Oper Name="SLM" Arg="d, q"/>
<Returns>A matrix monoid.</Returns>
<Description>
These operations return the special linear monoid of <A>d</A> by <A>d</A>
matrices over the field with <A>q</A> elements. The <E>special linear
monoid</E> is the monoid consisting of all <A>d</A> by <A>d</A> matrices
with entries from the field <C>GF(<A>q</A>)</C> that have determinant
<C>0</C> or <C>1</C>. In other words, the special linear monoid is formed
from the general linear monoid of the same parameters by replacing its
group of units (the general linear group) by the special linear group.
<Example><![CDATA[
gap> S := SpecialLinearMonoid(2, 4);
<regular monoid of 2x2 matrices over GF(2^2) with 3 generators>
gap> S = SLM(2, 4);
true
gap> Size(S);
136]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="RookMonoid">
<ManSection>
<Oper Name="RookMonoid" Arg="n"/>
<Returns>An inverse monoid of partial permutations.</Returns>
<Description>
<C>RookMonoid</C> is a synonym for <Ref Oper="SymmetricInverseMonoid"
BookName="ref" />.
<Example><![CDATA[
gap> S := RookMonoid(4);
<symmetric inverse monoid of degree 4>
gap> S = SymmetricInverseMonoid(4);
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="MunnSemigroup">
<ManSection>
<Attr Name = "MunnSemigroup" Arg = "S"/>
<Returns>The Munn semigroup of a semilattice.</Returns>
<Description>
If <A>S</A> is a semilattice, then <C>MunnSemigroup</C> returns the
inverse semigroup of partial permutations of isomorphisms of principal
ideals of <A>S</A>; called the <E>Munn semigroup</E> of <A>S</A>.<P/>
This function was written jointly by J. D. Mitchell, Yann Péresse
(St Andrews), Yanhui Wang (York).
<Example><![CDATA[
gap> S := InverseSemigroup([
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 10], [4, 6, 7, 3, 8, 2, 9, 5]),
> PartialPerm([1, 2, 7, 9], [5, 6, 4, 3])]);
<inverse partial perm semigroup of rank 10 with 2 generators>
gap> T := IdempotentGeneratedSubsemigroup(S);;
gap> M := MunnSemigroup(T);
<inverse partial perm semigroup of rank 60 with 7 generators>
gap> NrIdempotents(M);
60
gap> NrIdempotents(S);
60]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="OrderPreserving">
<ManSection>
<Heading>Semigroups of order-preserving transformations</Heading>
<Oper Name = "OrderEndomorphisms" Arg = "n"
Label = "monoid of order preserving transformations"/>
<Oper Name = "SingularOrderEndomorphisms" Arg = "n"/>
<Oper Name = "OrderAntiEndomorphisms" Arg = "n"/>
<Oper Name = "PartialOrderEndomorphisms" Arg = "n"/>
<Oper Name = "PartialOrderAntiEndomorphisms" Arg = "n"/>
<Returns>
A semigroup of transformations related to a linear order.
</Returns>
<Description>
<List>
<Mark><C>OrderEndomorphisms(<A>n</A>)</C></Mark>
<Item>
<C>OrderEndomorphisms(<A>n</A>)</C> returns the monoid of
transformations that preserve the usual order on <M>\{1, 2, \ldots,
n\}</M>, where <A>n</A> is a positive integer.
<Alt Not = "Text">
<C>OrderEndomorphisms(<A>n</A>)</C> is generated by the
<M><A>n + 1</A></M> transformations:
<Display>
\left(
\begin{array}{ccccccccc}
1&2&3&\cdots&n-1& n\\
1&1&2&\cdots&n-2&n-1
\end{array}\right), \qquad
\left(
\begin{array}{ccccccccc}
1&2&\cdots&i-1& i& i+1&i+2&\cdots
&n\\
1&2&\cdots&i-1& i+1&i+1&i+2&\cdots
&n\\
\end{array}\right)
</Display>
where <M>i=0,\ldots,n-1</M>, and has <M>{2n-1\choose n-1}</M>
elements.
</Alt>
</Item>
<Mark><C>SingularOrderEndomorphisms(<A>n</A>)</C></Mark>
<Item>
<C>SingularOrderEndomorphisms(<A>n</A>)</C> returns the ideal of
<C>OrderEndomorphisms(<A>n</A>)</C> consisting of the non-invertible
elements, when <A>n</A> is at least <C>2</C>. The only invertible element in <C>OrderEndomorphisms(<A>n</A>)</C> is the identity
transformation.
<Alt Not="Text">
Therefore <C>SingularOrderEndomorphisms(<A>n</A>)</C> has
<M>{2n-1\choose n-1} - 1</M> elements.
</Alt>
</Item>
<Mark><C>OrderAntiEndomorphisms(<A>n</A>)</C></Mark>
<Item>
<C>OrderAntiEndomorphisms(<A>n</A>)</C> returns the monoid of
transformations that preserve or reverse the usual order on <M>\{1, 2,
\ldots, n\}</M>, where <A>n</A> is a positive integer.
<C>OrderAntiEndomorphisms(<A>n</A>)</C> is generated by the generators
of <C>OrderEndomorphisms(<A>n</A>)</C> along with the bijective
transformation that reverses the order on <M>\{1, 2, \ldots, n\}</M>.
<Alt Not = "text">
The monoid <C>OrderAntiEndomorphisms(<A>n</A>)</C> has <M>{2n-1\choose
n-1} - n</M> elements.
</Alt>
</Item>
<Mark><C>PartialOrderEndomorphisms(<A>n</A>)</C></Mark>
<Item>
<C>PartialOrderEndomorphisms(<A>n</A>)</C> returns a monoid of
transformations on <C><A>n</A> + 1</C> points that is isomorphic to the
monoid consisting of all partial transformations that preserve the usual
order on <M>\{1, 2, \ldots, n\}</M>.
</Item>
<Mark><C>PartialOrderAntiEndomorphisms(<A>n</A>)</C></Mark>
<Item>
<C>PartialAntiOrderEndomorphisms(<A>n</A>)</C> returns a monoid of
transformations on <C><A>n</A> + 1</C> points that is isomorphic to the
monoid consisting of all partial transformations that preserve or
reverse the usual order on <M>\{1, 2, \ldots, n\}</M>.
</Item>
</List>
<Example><![CDATA[
gap> S := OrderEndomorphisms(5);
<regular transformation monoid of degree 5 with 5 generators>
gap> IsIdempotentGenerated(S);
true
gap> Size(S) = Binomial(2 * 5 - 1, 5 - 1);
true
gap> Difference(S, SingularOrderEndomorphisms(5));
[ IdentityTransformation ]
gap> SingularOrderEndomorphisms(10);
<regular transformation semigroup ideal of degree 10 with 1 generator>
gap> T := OrderAntiEndomorphisms(4);
<regular transformation monoid of degree 4 with 5 generators>
gap> Transformation([4, 2, 2, 1]) in T;
true
gap> U := PartialOrderEndomorphisms(6);
<regular transformation monoid of degree 7 with 12 generators>
gap> V := PartialOrderAntiEndomorphisms(6);
<regular transformation monoid of degree 7 with 13 generators>
gap> IsSubsemigroup(V, U);
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="POIPODIPOPIPORI">
<ManSection>
<Heading>Inverse monoids of order-preserving partial permutations</Heading>
<Oper Name = "POI" Arg = "n"
Label = "monoid of order preserving partial perms"/>
<Oper Name = "PODI" Arg = "n"
Label = "monoid of order preserving or reversing partial perms"/>
<Oper Name = "POPI" Arg = "n"
Label = "monoid of orientation preserving partial perms"/>
<Oper Name = "PORI" Arg = "n"
Label = "monoid of orientation preserving or reversing partial perms"/>
<Returns>
An inverse monoid of partial permutations related to a linear order.
</Returns>
<Description>
<List>
<Mark><C>POI(<A>n</A>)</C></Mark>
<Item>
<C>POI(<A>n</A>)</C> returns the inverse
monoid of partial permutations that preserve the usual order
on <M>\{1, 2, \ldots, n\}</M>, where <A>n</A> is a positive integer.
<Alt Not="Text">
<C>POI(<A>n</A>)</C> is generated by the <M><A>n</A></M>
partial permutations:
<Display>
\left(
\begin{array}{ccccc}
1&2&3&\cdots&n\\
-&1&2&\cdots&n-1
\end{array}\right), \qquad
\left(
\begin{array}{ccccccccc}
1&2&\cdots&i-1& i& i+1&i+2&\cdots
&n\\
1&2&\cdots&i-1& i+1&-&i+2&\cdots&n\\
\end{array}\right)
</Display>
where <M>i = 1, \ldots, n - 1</M>,
and has <M>{2n\choose n}</M> elements. </Alt>
</Item>
<Mark><C>PODI(<A>n</A>)</C></Mark>
<Item>
<C>PODI(<A>n</A>)</C> returns the inverse monoid of partial permutations
that preserve or reverse the usual order on <M>\{1, 2, \ldots, n\}</M>,
where <A>n</A> is a positive integer. <C>PODI(<A>n</A>)</C> is generated
by the generators of <C>POI(<A>n</A>)</C>, along with the
permutation that reverses the usual order on <M>\{1, 2, \ldots, n\}</M>.
<Alt Not="Text">
<C>PODI(<A>n</A>)</C> has <M>{2n\choose n} - n^{2} - 1</M> elements.
</Alt>
</Item>
<Mark><C>POPI(<A>n</A>)</C></Mark>
<Item>
<C>POPI(<A>n</A>)</C> returns the inverse monoid of partial permutations
that preserve the orientation of <M>\{1,2,\ldots, n\}</M>, where
<M>n</M> is a positive integer.
<Alt Not="Text">
<C>POPI(<A>n</A>)</C> is generated by the partial permutations:
<Display>
\left(
\begin{array}{ccccc}
1&2&\cdots&n-1&n\\
2&3&\cdots&n&1
\end{array}\right),\qquad
\left(
\begin{array}{cccccc}
1&2&\cdots&n-2&n-1&n\\
1&2&\cdots&n-2&n&-
\end{array}\right),
</Display>
and has <M>1+\frac{n}{2}{2n\choose n}</M> elements.
</Alt>
</Item>
<Mark><C>PORI(<A>n</A>)</C></Mark>
<Item>
<C>PORI(<A>n</A>)</C> returns the inverse monoid of partial permutations
that preserve or reverse the orientation of <M>\{1, 2, \ldots, n\}</M>,
where <M>n</M> is a positive integer. <C>PORI(<A>n</A>)</C> is generated
by the generators of <C>POPI(<A>n</A>)</C>, along with the
permutation that reverses the usual order on <M>\{1, 2, \ldots, n\}</M>.
<Alt Not="Text">
<C>PORI(<A>n</A>)</C> has <M>\frac{n}{2}{2n\choose n} - n(n + 1)</M>
elements.
</Alt>
</Item>
</List>
<Example><![CDATA[
gap> S := PORI(10);
<inverse partial perm monoid of rank 10 with 3 generators>
gap> S := POPI(10);
<inverse partial perm monoid of rank 10 with 2 generators>
gap> Size(S) = 1 + 5 * Binomial(20, 10);
true
gap> S := PODI(10);
<inverse partial perm monoid of rank 10 with 11 generators>
gap> S := POI(10);
<inverse partial perm monoid of rank 10 with 10 generators>
gap> Size(S) = Binomial(20, 10);
true
gap> IsSubsemigroup(PORI(10), PODI(10))
> and IsSubsemigroup(PORI(10), POPI(10))
> and IsSubsemigroup(PODI(10), POI(10))
> and IsSubsemigroup(POPI(10), POI(10));
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="CatalanMonoid">
<ManSection>
<Oper Name = "CatalanMonoid" Arg = "n"/>
<Returns>A transformation monoid.</Returns>
<Description>
If <A>n</A> is a positive integer, then this operation returns the Catalan
monoid of degree <A>n</A>. The <E>Catalan monoid</E> is the semigroup of
the order-preserving and order-decreasing transformations of <C>[1 ..
n]</C> with the usual ordering. <P/>
<Alt Not = "Text">
The Catalan monoid is generated by the <M><A>n - 1</A></M>
transformations <M>f_i</M>:
<Display>
\left(
\begin{array}{cccccccccc}
1&2&3&\cdots& i &i + 1& i + 2 & \cdots & n\\
1&2&3&\cdots& i &i & i + 2 & \cdots & n
\end{array}\right), \qquad
</Display>
where <M>i = 1, \ldots, n - 1</M> and has size equal to the <M>n</M>th
Catalan number.
</Alt>
<Example><![CDATA[
gap> S := CatalanMonoid(6);
<transformation monoid of degree 6 with 5 generators>
gap> Size(S);
132]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="SingularTransformationSemigroup">
<ManSection>
<Oper Name="SingularTransformationSemigroup" Arg="n"/>
<Oper Name="SingularTransformationMonoid" Arg="n"/>
<Returns>The semigroup of non-invertible transformations.</Returns>
<Description>
If <A>n</A> is a integer greater than 1, then this function
returns the semigroup of non-invertible transformations, which is
generated by the <C><A>n</A>(<A>n</A> - 1)</C> idempotents of degree
<A>n</A> and rank <C><A>n</A> - 1</C> and has <M>n ^ n - n!</M> elements.
<Example><![CDATA[
gap> S := SingularTransformationSemigroup(4);
<regular transformation semigroup ideal of degree 4 with 1 generator>
gap> Size(S);
232]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="FullTropicalMaxPlusMonoid">
<ManSection>
<Oper Name="FullTropicalMaxPlusMonoid" Arg="d, t"/>
<Returns>A monoid of tropical max plus matrices.</Returns>
<Description>
If <C><A>d</A> = 2</C> and <A>t</A> is a positive integer, then
<C>FullTropicalMaxPlusMonoid</C> returns the monoid consisting of all
<A>d</A> by <A>d</A> matrices with entries from the tropical max-plus
semiring with threshold <A>t</A>. A small generating set for larger values
of <A>d</A> is not currently known. <P/>
<Example><![CDATA[
gap> S := FullTropicalMaxPlusMonoid(2, 5);
<monoid of 2x2 tropical max-plus matrices with 24 generators>
gap> Size(S);
2401
gap> (5 + 2) ^ (2 ^ 2);
2401]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="FullTropicalMinPlusMonoid">
<ManSection>
<Oper Name="FullTropicalMinPlusMonoid" Arg="d, t"/>
<Returns>A monoid of tropical min plus matrices.</Returns>
<Description>
If <A>d</A> is equal to <C>2</C> or <C>3</C>, and <A>t</A> is a positive
integer, then <C>FullTropicalMinPlusMonoid</C> returns the monoid
consisting of all <A>d</A> by <A>d</A> matrices with entries from the
tropical min-plus semiring with threshold <A>t</A>. A small generating
set for larger values of <A>d</A> is not currently known. <P/>
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