#############################################################################
##
#W pbr.xml
#Y Copyright (C) 2015 James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
<#GAPDoc Label="IsPBR">
<ManSection>
<Filt Name = "IsPBR" Arg = "obj"Type = "Category"/>
<Returns>
<K>true</K> or <K>false</K>.
</Returns>
<Description>
Every PBR in &GAP; belongs to the category <C>IsPBR</C>. Basic operations
for PBRs are
<Ref Attr = "DegreeOfPBR"/>,
<Ref Oper = "ExtRepOfObj" Label = "for a PBR"/>,
<Ref Oper = "PBRNumber"/>,
<Ref Oper = "NumberPBR"/>,
<Ref Oper = "StarOp" Label = "for a PBR"/>,
and multiplication of two PBRs of equal degree is via <K>*</K>.
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="IsPBRCollection">
<ManSection>
<Filt Name = "IsPBRCollection" Arg = "obj"Type = "Category"/>
<Filt Name = "IsPBRCollColl" Arg = "obj"Type = "Category"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
Every collection of PBRs belongs to the category
<C>IsPBRCollection</C>. For example, PBR semigroups
belong to <C>IsPBRCollection</C>.
<P/>
Every collection of collections of PBRs belongs to
<C>IsPBRCollColl</C>. For example, a list of PBR
semigroups belongs to <C>IsPBRCollColl</C>.
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="PBR">
<ManSection>
<Oper Name = "PBR" Arg = "left, right"/>
<Returns>A PBR.</Returns>
<Description>
The arguments <A>left</A> and <A>right</A> of this function should each
be a list of length <C>n</C> whose entries are lists of integers in the
ranges <C>[-n .. -1]</C> and <C>[1 .. n]</C> for some <C>n</C> greater
than 0.
<P/>
Given such an argument, <C>PBR</C> returns the PBR <C>x</C> where:
<List>
<Item>
for each <C>i</C> in the range <C>[1 .. n]</C> there is an edge from
<C>i</C> to every <C>j</C> in <A>left[i]</A>;
</Item>
<Item>
for each <C>i</C> in the range <C>[-n .. -1]</C> there is an edge from
<C>i</C> to every <C>j</C> in <A>right[-i]</A>;
</Item>
</List>
<#GAPDoc Label="RandomPBR">
<ManSection>
<Oper Name = "RandomPBR" Arg = "n[, p]"/>
<Returns>A PBR.</Returns>
<Description>
If <A>n</A> is a positive integer and <A>p</A> is an float between
<C>0</C> and <C>1</C>, then <C>RandomPBR</C> returns a random PBR of
degree <A>n</A> where the probability of there being an edge from
<C>i</C> to <C>j</C> is approximately <C>p</C>.
<P/>
<#GAPDoc Label="EmptyPBR">
<ManSection>
<Oper Name = "EmptyPBR" Arg = "n"/>
<Returns>A PBR.</Returns>
<Description>
If <A>n</A> is a positive integer, then <C>EmptyPBR</C>
returns the PBR of degree <A>n</A> with no edges.
<Example><![CDATA[
gap> x := EmptyPBR(3);
PBR([ [ ], [ ], [ ] ], [ [ ], [ ], [ ] ])
gap> IsEmptyPBR(x);
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="IdentityPBR">
<ManSection>
<Oper Name = "IdentityPBR" Arg = "n"/>
<Returns>A PBR.</Returns>
<Description>
If <A>n</A> is a positive integer, then <C>IdentityPBR</C> returns the
identity PBR of degree <A>n</A>.
This PBR has <C>2</C><A>n</A> edges: specifically, for each <C>i</C> in
the ranges <C>[1 .. n]</C> and <C>[-n .. -1]</C>, the identity PBR has an
edge from <C>i</C> to <C>-i</C>.
<Example><![CDATA[
gap> x := IdentityPBR(3);
PBR([ [ -1 ], [ -2 ], [ -3 ] ], [ [ 1 ], [ 2 ], [ 3 ] ])
gap> IsIdentityPBR(x);
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="UniversalPBR">
<ManSection>
<Oper Name = "UniversalPBR" Arg = "n"/>
<Returns>A PBR.</Returns>
<Description>
If <A>n</A> is a positive integer, then <C>UniversalPBR</C> returns the
PBR of degree <A>n</A> with <C>4 * n ^ 2</C> edges, i.e. every possible
edge.
<Example><![CDATA[
gap> x := UniversalPBR(2);
PBR([ [ -2, -1, 1, 2 ], [ -2, -1, 1, 2 ] ],
[ [ -2, -1, 1, 2 ], [ -2, -1, 1, 2 ] ])
gap> IsUniversalPBR(x);
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="DegreeOfPBR">
<ManSection>
<Attr Name = "DegreeOfPBR" Arg = "x"/>
<Attr Name = "DegreeOfPBRCollection" Arg = "x"/>
<Returns>A positive integer.</Returns>
<Description>
The degree of a PBR is, roughly speaking, the number of points
where it is defined. More precisely, if <A>x</A> is a PBR defined
on <C>2 * n</C> points, then the degree of <A>x</A> is <C>n</C>.
<P/>
The degree of a collection <A>coll</A> of PBRs of equal degree is
just the degree of any (and every) PBR in <A>coll</A>. The degree
of collection of PBRs of unequal degrees is not defined.
<#GAPDoc Label="ExtRepOfObjPBR">
<ManSection>
<Oper Name = "ExtRepOfObj" Arg = "x" Label = "for a PBR"/>
<Returns>A pair of lists of lists of integers.</Returns>
<Description>
If <C>n</C> is the degree of the PBR <A>x</A>, then <C>ExtRepOfObj</C>
returns the argument required by <Ref Oper = "PBR"/> to create a PBR
equal to <A>x</A>, i.e. <C>PBR(ExtRepOfObj(<A>x</A>))</C>
returns a PBR equal to <A>x</A>.
<#GAPDoc Label="NumberPBR">
<ManSection>
<Oper Name = "PBRNumber" Arg = "m, n"/>
<Oper Name = "NumberPBR" Arg = "mat"/>
<Returns>A PBR, or a positive integer.</Returns>
<Description>
These functions implement a bijection from the set of all PBRs of
degree <A>n</A> and the numbers <C>[1 .. 2 ^ (4 * <A>n</A> ^ 2)]</C>.
<P/>
More precisely, if <A>m</A> and <A>n</A> are positive integers such that
<A>m</A> is at most <C>2 ^ (4 * <A>n</A> ^ 2)</C>, then <C>PBRNumber</C>
returns the <A>m</A>th PBR of degree <A>n</A>.
<P/>
If <A>mat</A> is a PBR of degree <A>n</A>, then <C>NumberPBR</C> returns
the number in <C>[1 .. 2 ^ (4 * <A>n</A> ^ 2)]</C> that corresponds to
<A>mat</A>.
<#GAPDoc Label="IsEmptyPBR">
<ManSection>
<Prop Name="IsEmptyPBR" Arg="x"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
A PBR is <B>empty</B> if it has no edges. <C>IsEmptyPBR</C> returns
<K>true</K> if the PBR <A>x</A> is empty and <K>false</K> if it is not.
<Example><![CDATA[
gap> x := PBR([[]], [[]]);;
gap> IsEmptyPBR(x);
true
gap> x := PBR([[-2, 1], [2]], [[-1], [-2, 1]]);
PBR([ [ -2, 1 ], [ 2 ] ], [ [ -1 ], [ -2, 1 ] ])
gap> IsEmptyPBR(x);
false]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="IsIdentityPBR">
<ManSection>
<Prop Name="IsIdentityPBR" Arg="x"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
A PBR of degree <C>n</C> is the <B>identity</B> PBR of degree <C>n</C>
if it is the identity of the full PBR monoid of degree <C>n</C>.
The identity PBR of degree <C>n</C> has <C>2n</C> edges. Specifically,
for each <C>i</C> in the ranges <C>[1 .. n]</C> and <C>[-n .. -1]</C>,
the identity PBR has an edge from <C>i</C> to <C>-i</C>. <P/>
<C>IsIdentityPBR</C> returns <K>true</K> is the PBR <A>x</A> is
an identity PBR and <K>false</K> if it is not.
<Example><![CDATA[
gap> x := PBR([[-2], [-1]], [[1], [2]]);
PBR([ [ -2 ], [ -1 ] ], [ [ 1 ], [ 2 ] ])
gap> IsIdentityPBR(x);
false
gap> x := PBR([[-1]], [[1]]);
PBR([ [ -1 ] ], [ [ 1 ] ])
gap> IsIdentityPBR(x);
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="IsUniversalPBR">
<ManSection>
<Prop Name = "IsUniversalPBR" Arg = "x"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
A PBR of degree <C>n</C> is <B>universal</B> if it has
<C>4 * n ^ 2</C> edges, i.e. every possible edge.
<Example><![CDATA[
gap> x := PBR([[]], [[]]);
PBR([ [ ] ], [ [ ] ])
gap> IsUniversalPBR(x);
false
gap> x := PBR([[-2, 1], [2]], [[-1], [-2, 1]]);
PBR([ [ -2, 1 ], [ 2 ] ], [ [ -1 ], [ -2, 1 ] ])
gap> IsUniversalPBR(x);
false
gap> x := PBR([[-1, 1]], [[-1, 1]]);
PBR([ [ -1, 1 ] ], [ [ -1, 1 ] ])
gap> IsUniversalPBR(x);
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="IsBipartitionPBR">
<ManSection>
<Prop Name = "IsBipartitionPBR" Arg = "x"/>
<Prop Name = "IsBlockBijectionPBR" Arg = "x"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
If the PBR <A>x</A> defines a bipartition, then <C>IsBipartitionPBR</C>
returns <K>true</K>, and if not, then it returns <K>false</K>.
<P/>
A PBR <A>x</A> defines a bipartition if and only if when considered as a
boolean matrix it is an equivalence.<P/>
If <A>x</A> satisfies <C>IsBipartitionPBR</C> and when considered as a
bipartition it is a block bijection, then <C>IsBlockBijectionPBR</C>
returns <K>true</K>.
<#GAPDoc Label="IsTransformationPBR">
<ManSection>
<Prop Name = "IsTransformationPBR" Arg = "x"/>
<Returns>
<K>true</K> or <K>false</K>.
</Returns>
<Description>
If the PBR <A>x</A> defines a transformation, then
<C>IsTransformationPBR</C> returns <K>true</K>, and if not, then
<K>false</K> is returned.
<P/>
A PBR <A>x</A> defines a transformation if and only if
it satisfies <Ref Prop = "IsBipartitionPBR"/> and
when it is considered as a bipartition it satisfies
<Ref Prop = "IsTransBipartition"/>.
<P/>
With this definition, <Ref Oper = "AsPBR"/> and
<Ref Attr = "AsTransformation" Label = "for a PBR"/> define mutually
inverse isomorphisms from the full transformation monoid of degree <C>n</C>
to the submonoid of the full PBR monoid of degree <C>n</C> consisting of
all the elements satisfying <C>IsTransformationPBR</C>.
<#GAPDoc Label="IsDualTransformationPBR">
<ManSection>
<Prop Name = "IsDualTransformationPBR" Arg = "x"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
If the PBR <A>x</A> defines a dual transformation, then
<C>IsDualTransformationPBR</C> returns <K>true</K>, and if not, then
<K>false</K> is returned.
<P/>
A PBR <A>x</A> defines a dual transformation if and only if
<C>Star(<A>x</A>)</C> satisfies <Ref Prop = "IsTransformationPBR"/>.
<P/>
<#GAPDoc Label="IsPartialPermPBR">
<ManSection>
<Prop Name="IsPartialPermPBR" Arg="x"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
If the PBR <A>x</A> defines a partial permutation, then
<C>IsPartialPermPBR</C> returns <K>true</K>, and if not, then
<K>false</K> is returned.
<P/>
A PBR <A>x</A> defines a partial perm if and only if
it satisfies <Ref Prop = "IsBipartitionPBR"/> and
and when it is considered as a bipartition it satisfies
<Ref Prop = "IsPartialPermBipartition"/>.
<P/>
With this definition, <Ref Oper = "AsPBR"/> and
<Ref Oper = "AsPartialPerm" Label = "for a PBR"/> define mutually inverse
isomorphisms from the symmetric inverse monoid of degree <C>n</C> to the
submonoid of the full PBR monoid of degree <C>n</C> consisting of all the
elements satisfying <C>IsPartialPermPBR</C>.
<#GAPDoc Label="IsPermPBR">
<ManSection>
<Prop Name = "IsPermPBR" Arg = "x"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
If the PBR <A>x</A> defines a permutation, then
<C>IsPermPBR</C> returns <K>true</K>, and if not, then
<K>false</K> is returned.
<P/>
A PBR <A>x</A> defines a permutation if and only if
it satisfies <Ref Prop = "IsBipartitionPBR"/> and
and when it is considered as a bipartition it satisfies
<Ref Prop = "IsPermBipartition"/>.
<P/>
With this definition, <Ref Oper = "AsPBR"/> and
<Ref Attr = "AsPermutation" Label = "for a PBR"/> define mutually inverse
isomorphisms from the symmetric group of degree <C>n</C> to the
subgroup of the full PBR monoid of degree <C>n</C> consisting of all the
elements satisfying <C>IsPermPBR</C> (i.e. the <Ref Attr = "GroupOfUnits"/>
of the full PBR monoid of degree <C>n</C>).
<#GAPDoc Label="AsPBR">
<ManSection>
<Oper Name = "AsPBR" Arg = "x[, n]"/>
<Returns>A PBR.</Returns>
<Description>
<C>AsPBR</C> returns the boolean matrix, bipartition, transformation,
partial permutation, or permutation <A>x</A> as a PBR of degree <A>n</A>.
<P/>
There are several possible arguments for <C>AsPBR</C>:
<List>
<Mark>bipartitions</Mark>
<Item>
If <A>x</A> is a bipartition and <A>n</A> is a positive integer, then
<C>AsPBR</C> returns a PBR corresponding to <A>x</A> with degree
<A>n</A>. The resulting PBR has an edge from <C>i</C> to <C>j</C>
whenever <C>i</C> and <C>j</C> belong to the same block of
<A>x</A>.<P/>
If the optional second argument <A>n</A> is not specified, then
degree of the bipartition <A>x</A> is used by default.
</Item>
<Mark>boolean matrices</Mark>
<Item>
If <A>x</A> is a boolean matrix of even dimension <C>2 * m</C> and
<A>n</A> is a positive integer, then <C>AsPBR</C> returns a PBR
corresponding to <A>x</A> with degree <A>n</A>.
If the optional second argument <A>n</A> is not specified, then
dimension of the boolean matrix <A>x</A> is used by default.
</Item>
<Mark>transformations, partial perms, permutations</Mark>
<Item>
If <A>x</A> is a transformation, partial perm, or permutation and
<A>n</A> is a positive integer, then <C>AsPBR</C> is a synonym for
<C>AsPBR(AsBipartition(<A>x</A>, <A>n</A>))</C>.
If the optional second argument <A>n</A> is not specified, then
<C>AsPBR</C> is a synonym for <C>AsPBR(AsBipartition(<A>x</A>))</C>.
See <Ref Oper = "AsBipartition"/> for more details.
</Item>
</List>
<#GAPDoc Label="AsTransformationPBR">
<ManSection>
<Attr Name = "AsTransformation" Arg = "x" Label = "for a PBR"/>
<Returns>A transformation.</Returns>
<Description>
When the argument <A>x</A> is a PBR which satisfies <Ref Prop = "IsTransformationPBR"/>, then this attribute returns that
transformation.
<Example><![CDATA[
gap> x := PBR([[-3], [-3], [-2]], [[], [3], [1, 2]]);;
gap> IsTransformationPBR(x);
true
gap> AsTransformation(x);
Transformation( [ 3, 3, 2 ] )
gap> x := PBR([[1], [1, 2]], [[-2, -1], [-2, -1]]);;
gap> AsTransformation(x);
Error, the argument (a pbr) does not define a transformation]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="AsPartialPermPBR">
<ManSection>
<Oper Name = "AsPartialPerm" Arg = "x" Label = "for a PBR"/>
<Returns>A partial perm.</Returns>
<Description>
When the argument <A>x</A> is a PBR which satisfies
<Ref Prop = "IsPartialPermPBR"/>, then this function returns that partial
perm.
<#GAPDoc Label="AsPermutationPBR">
<ManSection>
<Attr Name = "AsPermutation" Arg = "x" Label = "for a PBR"/>
<Returns>A permutation.</Returns>
<Description>
When the argument <A>x</A> is a PBR which satisfies
<Ref Prop = "IsPermPBR"/>, then this attribute returns that permutation.
<#GAPDoc Label="StarPBR">
<ManSection>
<Oper Name = "StarOp" Arg = "x" Label = "for a PBR"/>
<Attr Name = "Star" Arg = "x" Label = "for a PBR"/>
<Returns>A PBR.</Returns>
<Description>
<C>StarOp</C> returns the unique PBR <C>y</C> obtained by exchanging the
positive and negative numbers in <A>x</A> (i.e. multiplying
<Ref Oper = "ExtRepOfObj" Label = "for a PBR"/> by <C>-1</C> and swapping its first and
second components).
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
