In this section, we describe the operations in &SEMIGROUPS; that can be
used to create transformation semigroups belonging to several standard
classes of example. See <Ref Chap = "Transformations" BookName = "ref" />
for more information about transformations.
<#Include Label = "CatalanMonoid">
<#Include Label = "EndomorphismsPartition">
<#Include Label = "PartialTransformationMonoid">
<#Include Label = "SingularTransformationSemigroup">
<#Include Label = "OrderPreserving">
<#Include Label = "EndomorphismMonoid">
<Section Label = "Semigroups of partial permutations">
<Heading>
Semigroups of partial permutations
</Heading>
In this section, we describe the operations in &SEMIGROUPS; that can be used
to create semigroups of partial permutations belonging to several standard
classes of example. See <Ref Chap = "Partial permutations" BookName = "ref"
/> for more information about partial permutations.
<#Include Label = "MunnSemigroup">
<#Include Label = "RookMonoid">
<#Include Label = "POIPODIPOPIPORI">
</Section>
<Section Label = "Semigroups of bipartitions">
<Heading>
Semigroups of bipartitions
</Heading>
In this section, we describe the operations in &SEMIGROUPS; that can be
used to create bipartition semigroups belonging to several standard
classes of example. See Chapter <Ref Chap = "Bipartitions and blocks"/> for
more information about bipartitions.
<#Include Label = "PartitionMonoid">
<#Include Label = "BrauerMonoid">
<#Include Label = "JonesMonoid">
<#Include Label = "PartialJonesMonoid">
<#Include Label = "AnnularJonesMonoid">
<#Include Label = "MotzkinMonoid">
<#Include Label = "DualSymmetricInverseSemigroup">
<#Include Label = "UniformBlockBijectionMonoid">
<#Include Label = "PlanarPartitionMonoid">
<#Include Label = "ModularPartitionMonoid">
<#Include Label = "ApsisMonoid">
</Section>
In this section, we describe the operations in &SEMIGROUPS; that can be used
to create standard examples of semigroups of partitioned binary relations
(PBRs). See Chapter <Ref Chap = "Partitioned binary relations (PBRs)"/> for
more information about PBRs.
<#Include Label = "FullPBRMonoid">
</Section>
<Section Label = "Semigroups of matrices of a finite field">
<Heading>
Semigroups of matrices over a finite field
</Heading>
In this section, we describe the operations in &SEMIGROUPS; that can be used
to create semigroups of matrices over a finite field that belonging to
several standard classes of example. See the section
<Ref Sect = "Matrices over finite fields" Style="Text" /> for more
information about matrices over a finite field.
<#Include Label = "GeneralLinearMonoid">
<#Include Label = "SpecialLinearMonoid"> <!-- TODO(later) move to a more appropriate place -->
<#Include Label = "IsFullMatrixMonoid">
</Section>
<Section Label = "Semigroups of boolean matrices">
<Heading>
Semigroups of boolean matrices
</Heading>
In this section, we describe the operations in &SEMIGROUPS; that can be used
to create semigroups of boolean matrices belonging to several standard
classes of example. See the section <Ref Sect = "Boolean matrices"
Style="Text" /> for more information about boolean matrices.
<#Include Label = "FullBooleanMatMonoid">
<#Include Label = "RegularBooleanMatMonoid">
<#Include Label = "ReflexiveBooleanMatMonoid">
<#Include Label = "HallMonoid">
<#Include Label = "GossipMonoid">
<#Include Label = "TriangularBooleanMatMonoid">
</Section>
<Section Label = "Semigroups of matrices over a semiring">
<Heading>
Semigroups of matrices over a semiring
</Heading>
In this section, we describe the operations in &SEMIGROUPS; that can be used
to create semigroups of matices over a semiring that belong to several
standard classes of example. See Chapter
<Ref Chap = "Matrices over semirings" /> for more information about matrices
over a semiring.
<#Include Label = "FullTropicalMaxPlusMonoid"/>
<#Include Label = "FullTropicalMinPlusMonoid"/>
</Section>
<Section Label = "Examples in various representations">
<Heading>
Examples in various representations
</Heading>
In this section, we describe the functions in &SEMIGROUPS; that can be used
to create standard semigroups in various representations. For all of these
examples, the default representation is as a semigroup of transformations.
In general, these functions do not return a representation of minimal
degree.
This chapter describes the functions in &SEMIGROUPS; for dealing with free
bands. This part of the manual and the functions described herein were
originally written by Julius Jonušas, with later additions by Reinis
Cirpons, Tom Conti-Leslie, and Murray Whyte<P/>
A semigroup <M>B</M> is a <E>free band</E> on a non-empty set <M>X</M> if
<M>B</M> is a band with a map <M> f </M> from <M> B </M> to <M>X</M> such that
for every band <M> S </M> and every map <M> g </M> from <M>X</M> to <M> B </M>
there exists a unique homomorphism <M> g' from B to S such
that <M>fg' = g. The free band on a set X is unique up to
isomorphism. Moreover, by the universal property, every band can be expressed
as a quotient of a free band.<P/>
For an alternative description of a free band. Suppose that <M> X </M> is a
non-empty set and <M> X ^ + </M> a free semigroup on <M> X </M>. Also suppose
that <M> b </M> is the smallest congurance on <M> X ^ + </M> containing the set
<Display> \{(w ^ 2, w) : w \in X ^ + \}. </Display> Then the free band on
<M> X </M> is isomorphic to the quotient of <M> X ^ + </M> by <M> b </M>.
See Section 4.5 of <Cite Key = "Howie1995aa" /> for more information on
free bands.
<Subsection>
<Heading>Operators</Heading>
The following operators are also included for free band elements:
<List>
<Mark><C><A>u</A> * <A>v</A></C></Mark>
<Item>
returns the product of two free band elements <A>u</A> and
<A>v</A>.
</Item>
<Mark><C><A>u</A> = <A>v</A> </C></Mark>
<Item>
checks if two free band elements are equal.
</Item>
<Mark><C><A>u</A> < <A>v</A> </C></Mark>
<Item>
compares the sizes of the internal representations of two free
band elements.
</Item>
</List>
</Subsection>
</Section>
In this chapter we describe a class of semigroups arising from directed
graphs. <P/>
The functionality in &SEMIGROUPS; for graph inverse semigroups was written
jointly by Zak Mesyan (UCCS) and J. D. Mitchell (St Andrews). The
functionality for graph inverse semigroup congruences was written by Marina
Anagnostopoulou-Merkouri (St Andrews).
<P/>
This chapter describes the functions in &SEMIGROUPS; for dealing with free
inverse semigroups. This part of the manual and the functions described
herein were written by Julius Jonušas.<P/>
An inverse semigroup <M>F</M> is said to be <E>free</E> on a non-empty set
<M>X</M> if there is a map <M>f</M> from <M>F</M> to <M>X</M> such that for
every inverse semigroup <M>S</M> and a map <M>g</M> from <M>X</M> to
<M>S</M> there exists a unique homomorphism <M>g' from F to
<M>S</M> such that <M>fg' = g. Moreover, by this universal property,
every inverse semigroup can be expressed as a quotient of a free inverse
semigroup.
<P/>
The internal representation of an element of a free inverse semigroup
uses a Munn tree. A <E>Munn tree</E> is a directed tree with distinguished
start and terminal vertices and where the edges are labeled by generators so
that two edges labeled by the same generator are only incident to the same
vertex if one of the edges is coming in and the other is leaving the vertex.
For more information regarding free inverse semigroups and the Munn
representations see Section 5.10 of <Cite Key = "Howie1995aa"/>.
<P/>
See also <Ref Chap = "Inverse semigroups and monoids" BookName = "ref"/>,
<Ref Chap = "Partial permutations" BookName = "ref"/> and <Ref Sect = "Free
Groups, Monoids and Semigroups" BookName = "ref" />.
<P/>
An element of a free inverse semigroup in &SEMIGROUPS; is displayed, by
default, as a shortest word corresponding to the element. However, there
might be more than one word of the minimum length. For example, if <M>x</M>
and <M>y</M> are generators of a free inverse semigroups, then <Display>xyy
^ {-1}xx ^ {-1}x ^ {-1} = xxx ^ {-1}yy ^ {-1}x ^ {-1}.</Display> See <Ref
Attr = "MinimalWord" Label = "for free inverse semigroup element"/>.
Therefore we provide a another method for printing elements of a free
inverse semigroup: a unique canonical form. Suppose an element of a free
inverse semigroup is given as a Munn tree. Let <M>L</M> be the set of words
corresponding to the shortest paths from the start vertex to the leaves of
the tree. Also let <M>w</M> be the word corresponding to the shortest path
from the start vertex to the terminal vertex. The word <M>vv ^ {-1}</M> is an
idempotent for every <M>v</M> in <M>L</M>.
The canonical form is given by multiplying
these idempotents, in shortlex order, and then postmultiplying by <M>w</M>.
For example, consider the word <M>xyy ^ {-1}xx ^ {-1}x ^ {-1}</M> again.
The words corresponding to the paths to the leaves are in this case
<M>xx</M> and <M>xy</M>. And <M>w</M> is an empty word since start and
terminal vertices are the same. Therefore, the canonical form is
<Display>xxx ^ {-1}x ^ {-1}xyy ^ {-1}x ^ {-1}.</Display> See <Ref Oper = "CanonicalForm" Label = "for a free inverse semigroup element"/>.
<#Include Label = "FreeInverseSemigroup">
<#Include Label = "IsFreeInverseSemigroupCategory">
<#Include Label = "IsFreeInverseSemigroup">
<#Include Label = "IsFreeInverseSemigroupElement">
<#Include Label = "IsFreeInverseSemigroupElementCollection">
<#Include Label = "CanonicalForm">
<#Include Label = "MinimalWord">
<Subsection>
<Heading>Displaying free inverse semigroup elements </Heading>
<Heading>Displaying free inverse semigroup elements</Heading>
There is a way to change how &GAP; displays free inverse semigroup
elements using the user preference <C>FreeInverseSemigroupElementDisplay</C>.
See <Ref Func = "UserPreference" BookName = "ref"/> for more information
about user preferences.<P/>
There are two possible values for <C>FreeInverseSemigroupElementDisplay</C>:
<List>
<Mark>minimal </Mark>
<Item> With this option selected, &GAP; will display a shortest word
corresponding to the free inverse semigroup element. However,
this shortest word is not unique. This is a default setting.
</Item>
<Mark>canonical</Mark>
<Item> With this option selected, &GAP; will display a free inverse
semigroup element in the canonical form.
</Item>
</List>
<Subsection
Label="Operators and operations for free inverse semigroup elements">
<Heading>Operators for free inverse semigroup elements
</Heading>
<List>
<Mark><C><A>w</A> ^ -1</C></Mark>
<Item>
returns the semigroup inverse of the free inverse semigroup element
<A>w</A>.
</Item>
<Mark><C><A>u</A> * <A>v</A></C></Mark>
<Item>
returns the product of two free inverse semigroup elements <A>u</A>
and <A>v</A>.
</Item>
<Mark><C><A>u</A> = <A>v</A> </C></Mark>
<Item>
checks if two free inverse semigroup elements are equal, by comparing
their canonical forms.
</Item>
</List>
</Subsection>
</Section>
</Chapter>
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