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##
#W semiboolmat.xml
#Y Copyright (C) 2017 Wilf A. Wilson
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## Licensing information can be found in the README file of this package.
##
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<#GAPDoc Label="FullBooleanMatMonoid">
<ManSection>
<Oper Name="FullBooleanMatMonoid" Arg="d"/>
<Returns>The monoid of all boolean matrices of dimension <A>d</A>.</Returns>
<Description>
If <A>d</A> is a positive integer less than or equal to <C>5</C>, then
this operation returns the full boolean matrix monoid of dimension
<A>d</A>. The <E>full boolean matrix monoid of dimension <A>d</A></E> is
the monoid consisting of all <A>d</A> by <A>d</A> boolean matrices, and
has <C>2 ^ (<A>n</A> ^ 2)</C> matrices. <P/>
<C>FullBooleanMatMonoid</C> returns a monoid with a generating set that is
minimal in size. These generating sets are pre-computed.
<Example><![CDATA[
gap> S := FullBooleanMatMonoid(3);
<monoid of 3x3 boolean matrices with 5 generators>
gap> Size(S);
512]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="RegularBooleanMatMonoid">
<ManSection>
<Oper Name="RegularBooleanMatMonoid" Arg="d"/>
<Returns>A monoid of boolean matrices.</Returns>
<Description>
If <A>d</A> is a positive integer, then
<C>RegularBooleanMatMonoid</C> returns the monoid generated by the
regular <A>d</A> by <A>d</A> boolean matrices. Note that this monoid is
<E>not</E> regular in general. <C>RegularBooleanMatMonoid(<A>d</A>)</C> is
generated by the four boolean matrices <C>A, B, C, D</C>, whose <K>true</K>
entries are:
<List>
<Item>
<C>A[i][i + 1]</C> and <C>A[n][1]</C>, for
<M>i \in \{1, \ldots, n - 1\}</M>;
</Item>
<Item>
<C>B[1][2]</C>, <C>B[2][1]</C>, and <C>B[i][i]</C> for <M>i \in \{3,
\ldots, n\}</M>;
</Item>
<Item>
<C>C[1][2]</C> and <C>C[i][i]</C>, for <M>i \in \{2, \ldots, n -
1\}</M>; and
</Item>
<Item>
<C>D[1][2]</C>, <C>D[i][i]</C>, for <M>i \in \{2, \ldots, n\}</M>,
and <C>D[n][1]</C>.
</Item>
</List>
This monoid has nearly <C>2 ^ (n ^ 2)</C> elements.
<Example><![CDATA[
gap> S := RegularBooleanMatMonoid(3);
<monoid of 3x3 boolean matrices with 4 generators>
gap> Size(S);
506]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="ReflexiveBooleanMatMonoid">
<ManSection>
<Oper Name="ReflexiveBooleanMatMonoid" Arg="d"/>
<Returns>A monoid of boolean matrices.</Returns>
<Description>
If <A>d</A> is a positive integer less than or equal to <C>5</C>, then
this operation returns the monoid consisting of all reflexive <A>d</A> by
<A>d</A> boolean matrices. A boolean matrix <C>mat</C> is <E>reflexive</E>
if each entry of its leading diagonal is <K>true</K>, i.e. if
<C>mat[i][i]</C> is <K>true</K> for all <M>i \in \{1, \ldots, d\}</M>.
<P/>
The generating sets for the monoids returned by
<C>ReflexiveBooleanMatMonoid</C> are pre-computed, and read from a file.
Small generating sets are not known for <M><A>d</A> \geq 6</M>.
<Example><![CDATA[
gap> S := ReflexiveBooleanMatMonoid(3);
<monoid of 3x3 boolean matrices with 8 generators>
gap> Size(S);
64]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="HallMonoid">
<ManSection>
<Oper Name="HallMonoid" Arg="d"/>
<Returns>A monoid of boolean matrices.</Returns>
<Description>
If <A>d</A> is a positive integer less than or equal to <C>5</C>, then
this operation returns the monoid consisting Hall matrices of degree
<A>d</A>. A <E>Hall matrix</E> is a boolean matrix in which every column
and every row contains at least one <K>true</K> entry. Equivalently, a
Hall matrix is a boolean matrix than contains a permutation. <P/>
A Hall matrix of dimension <A>d</A> corresponds to a solution to Hall's
Marriage Problem, when there are two collection of <A>d</A> people. Thus
the number of solutions to Hall's Marriage Problem in this instance is the
number of elements of <C>HallMonoid(<A>d</A>)</C>. <P/>
The operation <C>HallMonoid</C> returns a monoid with a generating set
that is minimal in size. These generating sets are pre-computed, and a
minimal generating set is not known for larger dimensions.
<Example><![CDATA[
gap> S := HallMonoid(3);
<monoid of 3x3 boolean matrices with 4 generators>
gap> Size(S);
247]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="GossipMonoid">
<ManSection>
<Oper Name="GossipMonoid" Arg="d"/>
<Returns>A monoid of boolean matrices.</Returns>
<Description>
If <A>d</A> is a positive integer, then this operation returns the
<A>d</A> by <A>d</A> gossip monoid. The <E>gossip monoid</E> is defined to
be the monoid generated by the collection of all <A>d</A> by <A>d</A>
boolean matrices that define an equivalence relation; see <Ref
Prop="IsEquivalenceBooleanMat" />. <P/>
For <M><A>d</A> \geq 2</M>, <C>GossipMonoid(<A>d</A>)</C> returns a monoid
with <M>{d \choose 2}</M> generators. The generating set is the
collection of boolean matrices that define an equivalence relation that
has one equivalence class of size <C>2</C>, and no other non-trivial
equivalence classes. Note that this generating set is strictly contained
within the collection of all equivalence relation boolean matrices. <P/>
The number of elements of <C>GossipMonoid(<A>d</A>)</C> is known for some
small values of <A>d</A> — see <Cite Key="Brouwer2015aa"/> for more
information about the gossip monoid, and its size for <M><A>d</A> \leq
9</M>.
<Example><![CDATA[
gap> S := GossipMonoid(3);
<monoid of 3x3 boolean matrices with 3 generators>
gap> Size(S);
11]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="TriangularBooleanMatMonoid">
<ManSection>
<Oper Name="TriangularBooleanMatMonoid" Arg="d"/>
<Oper Name="UnitriangularBooleanMatMonoid" Arg="d"/>
<Returns>A monoid of boolean matrices.</Returns>
<Description>
If <A>d</A> is a positive integer, then <C>TriangularBooleanMatMonoid</C>
returns the monoid consisting of the upper-triangular <A>d</A> by <A>d</A>
boolean matrices. A boolean matrix is <E>upper-triangular</E> if the entry
in row <C>i</C>, column <C>j</C> is <K>false</K> whenever <C>i > j</C>.
<P/>
<C>UnitriangularBooleanMatMonoid</C> returns the subsemigroup of the
<C>TriangularBooleanMatMonoid</C> that consists of reflexive
upper-triangular boolean matrices; see <Ref
Oper="ReflexiveBooleanMatMonoid" />.
<Example><![CDATA[
gap> S := TriangularBooleanMatMonoid(3);
<monoid of 3x3 boolean matrices with 6 generators>
gap> Size(S);
64
gap> T := UnitriangularBooleanMatMonoid(4);
<monoid of 4x4 boolean matrices with 6 generators>
gap> Size(T);
64]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
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