<Chapter Label = "Bipartitions and blocks">
<Heading>
Bipartitions and blocks
</Heading>
In this chapter we describe the functions in &SEMIGROUPS; for creating and
manipulating bipartitions and semigroups of bipartitions. We begin by
describing what these objects are.
<P/>
A <E>partition</E> of a set &X; is a set of pairwise disjoint non-empty
subsets of &X; whose union is &X;. A partition of &X; is the
collection of equivalence classes of an equivalence relation on &X;, and
vice versa.
<P/>
Let <M>n\in</M>&bbN;, let &bfn;<M> = \{1, 2, \ldots, n\}</M>, and let
<M>-</M>&bfn;<M> = \{-1, -2, \ldots, -n\}</M>.
<P/>
The <E>partition monoid</E> of degree &n; is the set of all partitions of
&bfn;<M>\cup</M>-&bfn; with a multiplication we describe below. To avoid
conflict with other uses of the word "partition" in &GAP;, and to reflect
their definition, we have opted to refer to the elements of the partition
monoid as <E>bipartitions</E> of degree &n;; we will do so from this point on.
<P/>
Let &x; be any bipartition of degree &n;. Then &x; is a set of pairwise
disjoint non-empty subsets of &bfn;<M>\cup</M>-&bfn; whose union is
&bfn;<M>\cup</M>-&bfn;; these subsets are called the <E>blocks</E> of &x;. A
block containing elements of both &bfn; and -&bfn; is called a <E>transverse
block</E>. If &i;, &j;<M>\in</M>&bfn;<M>\cup</M>-&bfn; belong to the same
block of a bipartition &x;, then we write (&i;, &j;)<M>\in</M>&x;.
<P/>
Let &x; and &y; be bipartitions of degree &n;. Their product &x;&y; can be
described as follows. Define &bfn;' = \{1', 2', \ldots, n'\}</M>. From &x;,
create a partition &x;' of the set &bfn;\cup&bfn;' by replacing each
negative point -&i; in a block of &x; by the point &i;', and create from &y; a
partition &y;' of the set &bfn;'<M>\cup</M>-&bfn; by replacing each positive
point &i; in a block of &y; by the point &i;'. Then define a relation on the
set &bfn;<M>\cup</M>&bfn;'\cup-&bfn;, where &i; and &j; are related if
they are related in either &x;' or &y;', and let &p; be the transitive closure of
this relation. Finally, define &x;&y; to be the bipartition of degree &n;
defined by the restriction of the equivalence relation &p; to the set
&bfn;<M>\cup</M>-&bfn;.
<P/>
Equivalently, the product &x;&y; is defined to be the bipartition where
&i;,&j;<M>\in</M>&bfn;<M>\cup</M>-&bfn; (we assume without loss of generality
that &i;<M>\geq</M>&j;) belong to the same block of &x;&y; if either:
<List>
<Item>
&i;<C>=</C>&j;,
</Item>
<Item>
&i;, &j; <M>\in</M> &bfn; and <M>(</M>&i;,&j;<M>)</M><M>\in</M> &x;, or
</Item>
<Item>
&i;, &j; <M>\in</M> -&bfn; and <M>(</M>&i;,&j;<M>)</M><M>\in</M> &y;;
</Item>
</List>
or there exists
<M>r\in</M>&bbN; and
<Alt Not = "Text">
<M>k(1), k(2),\ldots, k(r)\in \mathbf{n}</M>
</Alt>
<Alt Only = "Text">
<C>k(1),k(2)</C>,<M>\ldots</M>,<C>k(r)</C><M>\in</M><E>n</E>
</Alt>,
and one of the following holds:
<List>
<Item>
<Alt Not = "Text">
<M>r=2s-1</M> for some <M>s\geq 1</M>
</Alt>
<Alt Only = "Text">
<C>r=2s-1</C> for some <C>s</C><M>\geq</M><C>1</C>
</Alt>,
&i;<M>\in</M>&bfn;, &j;<M>\in</M> -&bfn; and
<Alt Not = "Text">
<Display>(i,-k(1))\in x,\ (k(1),k(2))\in y,\ (-k(2),-k(3))\in x,\
\ldots,\qquad</Display>
<Display>\qquad\ldots,\ (-k(2s-2),-k(2s-1))\in x,\
(k(2s-1),j)\in y;</Display>
</Alt>
<Alt Only = "Text">
<C>(i,-k(1))</C><M>\in</M><C>x</C>, <C>(k(1),k(2))</C><M>\in</M><C>y</C>,
<C>(-k(2),-k(3))</C><M>\in</M><C>x</C>, <M>\ldots</M>,
<C>(-k(2s-2),-k(2s-1))</C><M>\in</M><C>x</C>,
<C>(k(2s-1),j)</C><M>\in</M><C>y</C>;
</Alt>
</Item>
<Item>
<Alt Not = "Text">
<M>r=2s</M> for some <M>s\geq 1</M>
</Alt>
<Alt Only = "Text">
<C>r=2s</C> for some <C>s</C><M>\geq</M><C>1</C>
</Alt>,
and either
&i;,&j;<M>\in</M>&bfn;, and
<Alt Not = "Text">
<Display>(i,-k(1))\in x,\ (k(1),k(2))\in y,\ (-k(2),-k(3))\in x,\ \ldots,
(k(2s-1), k(2s))\in y,\ (-k(2s), j)\in x,</Display>
</Alt>
<Alt Only = "Text">
<P/>
<C>(i,-k(1))</C><M>\in</M><C>x</C>,
<C>(k(1),k(2))</C><M>\in</M><C>y</C>,
<C>(-k(2),-k(3))</C><M>\in</M><C>x</C>,
<M>\ldots</M>,
<C>(k(2s-1),k(2s))</C><M>\in</M><C>x</C>,
<C>(-k(2s),j)</C><M>\in</M><C>y</C>,<P/>
</Alt>
or &i;,&j;<M>\in</M>-&bfn;, and
<Alt Not = "Text">
<Display>(i,k(1))\in y,\ (-k(1),-k(2))\in x,\ (k(2),k(3))\in y,\ \ldots,
(-k(2s-1), -k(2s))\in x,\ (k(2s), j)\in y.</Display>
</Alt>
<Alt Only = "Text">
<P/>
<C>(i,k(1))</C><M>\in</M><C>y</C>,
<C>(-k(1),-k(2))</C><M>\in</M><C>x</C>,
<C>(k(2),k(3))</C><M>\in</M><C>y</C>,
<M>\ldots</M>,
<C>(-k(2s-1),-k(2s))</C><M>\in</M><C>y</C>,
<C>(k(2s),j)</C><M>\in</M><C>x</C>.<P/>
</Alt>
</Item>
</List>
This multiplication can be shown to be associative, and so the collection of
all bipartitions of any particular degree is a monoid; the identity element of
the partition monoid of degree <M>n</M> is the bipartition <Alt Not = "Text"><M>\left\{\{i,-i\}:i\in\mathbf{n}\right\}</M>.</Alt> <Alt Only = "Text">{{<M>i</M>,-<M>i</M>}:<M>i \in</M><E>n</E>}.</Alt> A bipartition is a
unit if and only if each block is of the form <M>\{</M>&i;,-&j;<M>\}</M> for
some &i;, &j;<M>\in</M>&bfn;. Hence the group of units is isomorphic to the
symmetric group on &bfn;.
<P/>
Let &x; be a bipartition of degree &n;. Then we define &x;<M>^*</M> to be the
bipartition obtained from &x; by replacing &i; by -&i; and -&i; by &i; in
every block of &x; for all &i;<M>\in</M>&bfn;. It is routine to verify that
if &x; and &y; are arbitrary bipartitions of equal degree, then
<Alt Not = "Text">
<Display>
(x^*)^*=x,\quad xx^*x=x,\quad x^*xx^*=x^*,\quad (xy)^*=y^*x^*.
</Display>
</Alt>
<Alt Only = "Text">
<P/>
<C>(x^*)^*=x, xx^*x=x, x^*xx^*=x^*, (xy)^*=y^*x^*.</C>
<P/>
</Alt>
In this way, the partition monoid is a <E>regular *-semigroup</E>. <P/>
A bipartition &x; of degree &n; is called <E>planar</E> if there do not exist
distinct blocks <M>A, U \in</M> &x;, along with <M>a, b \in A</M> and <M>u, v
\in U</M>, such that <M>a < u < b < v</M>. Define &p; to be the
bipartition of degree &n; with blocks
<Alt Not = "Text">
<M>\left\{\{i, -(i+1)\}:i\in\{1,\ldots,n-1\right\}\}</M> and
<M>\{n,-1\}</M>
</Alt>
<Alt Only = "Text">
{{<C>i</C>,-<C>(i+1)</C>}:<C>i</C><M>\in</M><E>n</E>,<C>i</C><<E>n</E>}
and {<C>n</C>,-<C>1</C>}
</Alt>. Note that &p; is a unit.
A bipartition &x; of degree &n; is called <E>annular</E> if
<Alt Not = "Text">
<M>x = p^{i} y p^{j}</M>
</Alt>
<Alt Only = "Text">
<C>x =(p^i)y(p^j)</C>
</Alt>
for some planar bipartition &y; of degree &n;, and some integers &i; and &j;.
<P/>
From a graphical perspective, as on Page 873 in <Cite
Key="Halverson2005PartitionAlgebras"/>, a bipartition of degree &n; is
planar if it can be represented as a graph without edges crossing inside of
the rectangle formed by its vertices &bfn;<M>\cup</M>-&bfn;. Similarly, as
shown in Figure 2 in <Cite Key="auinger2012krohn"/>, a bipartition of degree
&n; is annular if it can be represented as a graph without edges crossing
inside an annulus.
<Section><Heading>The family and categories of bipartitions</Heading>
<#Include Label = "IsBipartition">
<#Include Label = "IsBipartitionCollection">
</Section>
<Section Label = "creating-bipartitions">
<Heading>Creating bipartitions</Heading>
There are several ways of creating bipartitions in &GAP;, which are
described in this section. The maximum degree of a bipartition is set as
2 ^ 29 - 1. In reality, it is unlikely to be possible to create bipartitions
of degrees as small as 2 ^ 24 because they require too much memory.
<#Include Label = "Bipartition">
<#Include Label = "BipartitionByIntRep">
<#Include Label = "IdentityBipartition">
<#Include Label = "LeftProjection">
<#Include Label = "RightProjection">
<#Include Label = "StarOp">
<#Include Label = "RandomBipartition">
</Section>
<Section Label = "changing-rep-bipartitions">
<Heading>Changing the representation of a bipartition</Heading>
It is possible that a bipartition can be represented as another type of object, or that another type of &GAP; object can be represented as a
bipartition. In this section, we describe the functions in the &SEMIGROUPS;
package for changing the representation of bipartition, or for changing the
representation of another type of object to that of a bipartition.<P/>
The operations
<Ref Attr = "AsPermutation" Label = "for a bipartition"/>,
<Ref Oper = "AsPartialPerm" Label = "for a bipartition"/>,
<Ref Attr = "AsTransformation" Label = "for a bipartition"/> can be
used to convert bipartitions into permutations, partial permutations, or
transformations where appropriate.
<#Include Label = "AsBipartition"/>
<#Include Label = "AsBlockBijection"/>
<#Include Label = "AsTransformation"/>
<#Include Label = "AsPartialPerm"/>
<#Include Label = "AsPermutation"/>
</Section>
<Section Label = "operators-bipartitions">
<Heading>Operators for bipartitions</Heading>
<List>
<Mark><C><A>f</A> * <A>g</A></C></Mark>
<Item>
<Index Key = "*"><C>*</C> (for bipartitions)</Index>
returns the composition of <A>f</A> and <A>g</A> when <A>f</A> and
<A>g</A> are bipartitions.
</Item>
<Mark><C><A>f</A> < <A>g</A></C></Mark>
<Item>
<Index Key = "<"><C><</C> (for bipartitions)</Index>
returns <K>true</K> if the internal representation of <A>f</A> is
lexicographically less than the internal representation of <A>g</A> and
<K>false</K> if it is not.
</Item>
<Mark><C><A>f</A> = <A>g</A></C></Mark>
<Item>
<Index Key = "="><C>=</C> (for bipartitions)</Index>
returns <K>true</K> if the bipartition <A>f</A> equals the
bipartition <A>g</A> and returns <K>false</K> if it does not.
</Item>
</List>
<Section Label = "section-blocks">
<Heading>Creating blocks and their attributes</Heading>
As described above the left and right blocks of a bipartition characterise
Green's &R;- and &L;-relation of the partition monoid; see
<Ref Attr = "LeftBlocks"/> and <Ref Attr = "RightBlocks"/>.
The left or right blocks of a bipartition are &GAP; objects in their own
right.
<P/>
In this section, we describe the functions in the &SEMIGROUPS; package for
creating and manipulating the left or right blocks of a bipartition.
Bipartitions act on left and right blocks in several ways, which are
described in this section.
<#Include Label = "OnRightBlocks">
<#Include Label = "OnLeftBlocks">
</Section>
<Section>
<Heading>
Semigroups of bipartitions
</Heading>
Semigroups and monoids of bipartitions can be created in the usual way in
&GAP; using the functions <Ref Func = "Semigroup" BookName = "ref"/> and
<Ref Func = "Monoid" BookName = "ref"/>; see Chapter
<Ref Chap = "Semigroups and monoids defined by generating sets"/> for more details. <P/>
It is possible to create inverse semigroups and monoids of bipartitions
using <Ref Func = "InverseSemigroup" BookName = "ref"/> and <Ref Func = "InverseMonoid" BookName = "ref"/> when the argument is a collection of
block bijections or partial perm bipartions; see <Ref Prop = "IsBlockBijection"/> and <Ref Prop = "IsPartialPermBipartition"/>.
Note that every bipartition semigroup in &SEMIGROUPS; is finite.
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