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<Chapter Label="Chapter_Translations">
<Heading>Translations</Heading>
In this chapter we describe the functionality in &SEMIGROUPS; for working
with translations of semigroups. The notation used (as well as a number of
results) is based on <Cite Key="Petrich1970aa"/>. Translations are of
interest mainly due to their role in ideal extensions. A description of
this role can also be found in <Cite Key="Petrich1970aa"/>. The
implementation of translations in this package is only applicable to
finite semigroups satisfying <Ref Filt="CanUseFroidurePin"/>.
<P/>
For a semigroup <M>S</M>, a transformation <M>\lambda</M> of <M>S</M>
(written on the left) is a left translation if for all <M>s, t</M> in
<M>S</M>, <M>\lambda (s)t = \lambda (st) </M>.
A right translation <M>\rho</M> (written on the right) is defined
dually.
<P/>
The set <M>L</M> of left translations of <M>S</M> is a semigroup under
composition of functions, as is the set <M>R</M> of right translations.
The associativity of <M>S</M> guarantees that left [right] multiplication by
any element <M>s</M> of <M>S</M> represents a left [right] translation;
this is the <E>inner</E> left [right] translation <M>\lambda_s</M>
[<M>\rho_s</M>]. The inner left [right] translations form an ideal in
<M>L</M> [<M>R</M>].
<P/>
A left translation <M>\lambda</M> and right translation <M>\rho</M> are
<E>linked</E> if for all <C>s, t</C> in <M>S</M>,
<M>s\lambda(t) = (s)\rho t</M>.
A pair of linked translations is called a <E>bitranslation</E>. The set of
all bitranslations forms a semigroup <M>H</M> called the <E>translational
hull</E> of <M>S</M> where the operation is componentwise. If the
components are inner translations corresponding to multiplication by
the same element, then the bitranslation is <E>inner</E>. The inner
bitranslations form an ideal of the translational hull.
<P/>
<Heading>
Translations of normalised Rees matrix semigroups
</Heading>
Translations of a normalized Rees matrix semigroup
<M>T</M> (see <Ref Attr = "RMSNormalization"/>) over a group <M>G</M>
can be represented through certain tuples, which can be computed
very efficiently compared to arbitrary translations. For left translations
these tuples consist of a function from the row indices of <M>T</M> to G and
a transformation on the row indices of <M>T</M>; the same is true of right
translations and columns. More specifically, given a normalised Rees matrix
semigroup <M>S</M> over a group <M>G</M> with sandwich matrix <M>P</M>, rows
<M>I</M> and columns <M>J</M>, the left translations are
characterised by pairs <M>(\theta, \chi)</M> where <M>\theta</M> is a
function from <M>I</M> to <M>G</M> and <M>\chi</M> is a transformation of
<M>I</M>. The left translation <M>\lambda</M> defined by
<M>(\theta, \chi)</M> acts on <M>S</M> via
<P/>
<Display Mode="M">
\lambda((i, a, \mu)) = ((i)\chi, (i)\theta \cdot a, \mu),
</Display>
<P/>
where <M>i \in I</M>, <M>a \in G</M>, and <M>\mu \in J</M>
Dually, right translations <M>\rho</M> are characterised by
pairs <M>(\omega, \psi)</M> where <M>\omega</M> is a function from
<M>J</M> to <M>G</M> and <M>\psi</M> is a transformation of
<M>J</M>, with action given by
<P/>
<Display Mode="M">
((i, a, \mu))\rho = (i, a \cdot (\mu)\psi, (\mu)\psi).
</Display>
<P/>
Similarly, bitranslations <M>(\lambda, \rho)</M> of <M>S</M> can be
characterised by triples <M>(g, \chi, \psi)</M> such that <M>g \in G</M>,
<M>\chi</M> and <M>\psi</M> are transformations of <M>I, J</M> respectively,
and
<P/>
<Display Mode="M">
p_{\mu, (i)\chi} \cdot g \cdot p_{(1)\psi, i} =
p_{\mu, (1)\chi} \cdot g \cdot p_{(mu)\psi, i}.
</Display>
<P/>
The action of <M>\lambda</M> on <M>S</M> is then given by
<P/>
<Display Mode="M">
\lambda((i, a, \mu)) = ((i)\chi, b \cdot p_{(1)\psi, i} \cdot a, \mu),
</Display>
<P/>
and of <M>\rho</M> on <M>S</M> by
<P/>
<Display Mode="M">
((i, a, \mu))\rho = (i, a \cdot p_{\mu, (1)\chi} \cdot b, (\mu)\psi).
</Display>
<P/>
Further details may be found in <Cite Key="Clifford1977aa"/>.
<P/>
<Section Label="Chapter_Translations_Section_Methods_for_translations">
<Heading>Methods for translations</Heading>
<ManSection Label="IsXTranslation">
<Heading>IsXTranslation</Heading>
<Filt Arg="arg" Name="IsSemigroupTranslation" Label="for IsAssociativeElement and IsMultiplicativeElementWithOne"/>
<Filt Arg="arg" Name="IsLeftTranslation" Label="for IsSemigroupTranslation"/>
<Filt Arg="arg" Name="IsRightTranslation" Label="for IsSemigroupTranslation"/>
<Returns><K>true</K> or <K>false</K>
</Returns>
<Description>
All, and only, left [right] translations belong to <C>IsLeftTranslation</C>
[<C>IsRightTranslation</C>]. These are both subcategories of
<C>IsSemigroupTranslation</C>, which itself is a subcategory of
<C>IsAssociativeElement</C>.
<Example><![CDATA[
gap> S := RectangularBand(3, 4);;
gap> l := Representative(LeftTranslations(S));;
gap> IsSemigroupTranslation(l);
true
gap> IsLeftTranslation(l);
true
gap> IsRightTranslation(l);
false
gap> l = One(LeftTranslations(S));
true
gap> l * l = l;
true
]]></Example>
<P/>
</Description>
</ManSection>
<ManSection>
<Filt Arg="arg" Name="IsBitranslation" Label="for IsAssociativeElement and IsMultiplicativeElementWithOne"/>
<Returns><K>true</K> or <K>false</K>
</Returns>
<Description>
All, and only, bitranslations belong to <C>IsBitranslation</C>. This is a
subcategory of <Ref Filt="IsAssociativeElement" BookName="ref"/>.
<Example><![CDATA[
gap> G := Group((1, 2), (3, 4));;
gap> A := AsList(G);;
gap> mat := [[A[1], 0, A[1]],
> [A[2], A[2], A[4]]];;
gap> S := ReesZeroMatrixSemigroup(G, mat);;
gap> L := LeftTranslations(S);;
gap> R := RightTranslations(S);;
gap> l := OneOp(Representative(L));;
gap> r := OneOp(Representative(R));;
gap> H := TranslationalHull(S);;
gap> x := Bitranslation(H, l, r);;
gap> IsBitranslation(x);
true
gap> IsSemigroupTranslation(x);
false
]]></Example>
</Description>
</ManSection>
<ManSection Label="IsXTranslationCollection">
<Heading>IsXTranslationCollection</Heading>
<Filt Arg="obj" Name="IsSemigroupTranslationCollection" />
<Filt Arg="obj" Name="IsLeftTranslationCollection" />
<Filt Arg="obj" Name="IsRightTranslationCollection" />
<Filt Arg="obj" Name="IsBitranslationCollection" />
<Returns><K>true</K> or <K>false</K>
</Returns>
<Description>
Every collection of left-, right-, or bi-translations belongs to the
respective category <Ref Filt="IsXTranslationCollection"/>.
<P/>
</Description>
</ManSection>
<ManSection Label="XPartOfBitranslation">
<Heading>XPartOfBitranslation</Heading>
<Func Arg="h" Name="LeftPartOfBitranslation" />
<Func Arg="arg" Name="RightPartOfBitranslation" />
<Returns>a left or right translation
</Returns>
<Description>
For a Bitranslation <A>h</A> consisting of a linked pair <M>(l, r)</M>,
of left and right translations, <Code>LeftPartOfBitranslation(<A>b</A>)</Code> returns
the left translation <Code>l</Code>, and <Code>RightPartOfBitranslation(<A>b</A>)</Code> returns
the right translation <Code>r</Code>.
<P/>
</Description>
</ManSection>
<ManSection Label="XTranslation">
<Heading>XTranslation</Heading>
<Oper Arg="T, x[, y]" Name="LeftTranslation" Label="for IsLeftTranslationsSemigroup, IsGeneralMapping"/>
<Oper Arg="arg1,arg2" Name="RightTranslation" Label="for IsRightTranslationsSemigroup, IsGeneralMapping"/>
<Returns>a left or right translation
</Returns>
<Description>
For the semigroup <A>T</A> of left or right translations of a semigroup <M>
S</M> and <A>x</A> one of:
<List>
<Item>
a mapping on the underlying semigroup; note that in this case only the
values of the mapping on the <Code>UnderlyingRepresentatives</Code> of
<A>T</A> are checked and used, so mappings which do not define translations
can be used to create translations if they are valid on that subset of S;
</Item>
<Item>
a list of indices representing the images of the
<Code>UnderlyingRepresentatives</Code> of <A>T</A>, where the ordering
is that of <Ref Oper="PositionCanonical"/> on <A>S</A>;
</Item>
<Item>
(for <Code>LeftTranslation</Code>) a list of length <Code>Length(Rows(S))</Code>
containing elements of <Code>UnderlyingSemigroup(S)</Code>; in this case
<A>S</A> must be a normalised Rees matrix semigroup and <Code>y</Code> must be
a Transformation of <Code>Rows(S)</Code>;
</Item>
<Item>
(for <Code>RightTranslation</Code>) a list of length <Code>Length(Columns(S))</Code>
containing elements of <Code>UnderlyingSemigroup(S)</Code>; in this case
<A>S</A> must be a normalised Rees matrix semigroup and <Code>y</Code> must be
a Transformation of <Code>Columns(S)</Code>;
</Item>
</List>
<Code>LeftTranslation</Code> and <Code>RightTranslation</Code> return the corresponding
translations.
<Example><![CDATA[
gap> S := RectangularBand(3, 4);;
gap> L := LeftTranslations(S);;
gap> s := AsList(S)[1];;
gap> map := MappingByFunction(S, S, x -> s * x);;
gap> l := LeftTranslation(L, map);
<left translation on <regular transformation semigroup of size 12,
degree 8 with 4 generators>>
gap> s ^ l;
Transformation( [ 1, 2, 1, 1, 5, 5, 5, 5 ] )
]]></Example>
<P/>
</Description>
</ManSection>
<ManSection>
<Oper Arg="H, l, r" Name="Bitranslation" Label="for IsBitranslationsSemigroup, IsLeftTranslation, IsRightTranslation"/>
<Returns>a bitranslation
</Returns>
<Description>
If <A>H</A> is a translational hull over a semigroup <M>S</M>, and <A>l</A>
and <A>r</A> are linked left and right translations respectively over
<M>S</M>, then this function returns the bitranslation
<M>(<A>l</A>, <A>r</A>)</M>. If <A>l</A> and <A>r</A> are not linked, then
an error is produced.
<Example><![CDATA[
gap> G := Group((1, 2), (3, 4));;
gap> A := AsList(G);;
gap> mat := [[A[1], 0],
> [A[2], A[2]]];;
gap> S := ReesZeroMatrixSemigroup(G, mat);;
gap> L := LeftTranslations(S);;
gap> R := RightTranslations(S);;
gap> l := LeftTranslation(L, MappingByFunction(S, S, s -> S.1 * s));;
gap> r := RightTranslation(R, MappingByFunction(S, S, s -> s * S.1));;
gap> H := TranslationalHull(S);;
gap> x := Bitranslation(H, l, r);
<bitranslation on <regular semigroup of size 17, with 4 generators>>
]]></Example>
</Description>
</ManSection>
<ManSection Label="UnderlyingSemigroup">
<Heading>UnderlyingSemigroup</Heading>
<Attr Arg="S" Name="UnderlyingSemigroup" Label="for IsTranslationsSemigroup"/>
<Attr Arg="arg" Name="UnderlyingSemigroup" Label="for IsBitranslationsSemigroup"/>
<Returns>a semigroup
</Returns>
<Description>
Given a semigroup of translations or bitranslations, returns the
semigroup on which these translations act.
<P/>
</Description>
</ManSection>
<ManSection Label="XTranslationsSemigroupOfFamily">
<Heading>XTranslationsSemigroupOfFamily</Heading>
<Attr Arg="fam" Name="LeftTranslationsSemigroupOfFamily" Label="for IsFamily"/>
<Attr Arg="arg" Name="RightTranslationsSemigroupOfFamily" Label="for IsFamily"/>
<Attr Arg="arg" Name="TranslationalHullOfFamily" Label="for IsFamily"/>
<Returns> the semigroup of left or right translations, or the translational hull
</Returns>
<Description>
Given a family <A>fam</A> of left-, right- or bi-translations, returns
the translations semigroup or translational hull to which they belong.
<Example><![CDATA[
gap> S := RectangularBand(3, 3);;
gap> L := LeftTranslations(S);;
gap> l := Representative(L);;
gap> LeftTranslationsSemigroupOfFamily(FamilyObj(l)) = L;
true
gap> H := TranslationalHull(S);;
gap> h := Representative(H);;
gap> TranslationalHullOfFamily(FamilyObj(h)) = H;
true
]]></Example>
<P/>
</Description>
</ManSection>
<ManSection Label="TypeXTranslationSemigroupElements">
<Heading>TypeXTranslationSemigroupElements</Heading>
<Attr Arg="arg" Name="TypeLeftTranslationsSemigroupElements" Label="for IsLeftTranslationsSemigroup"/>
<Attr Arg="arg" Name="TypeRightTranslationsSemigroupElements" Label="for IsRightTranslationsSemigroup"/>
<Attr Arg="arg" Name="TypeBitranslations" Label="for IsBitranslationsSemigroup"/>
<Returns>a type
</Returns>
<Description>
Given a (bi)translations semigroup, returns the type of the elements that
it contains.
<P/>
</Description>
</ManSection>
<ManSection Label="XTranslations">
<Heading>XTranslations</Heading>
<Attr Arg="S" Name="LeftTranslations" Label="for IsSemigroup and CanUseFroidurePin and IsFinite"/>
<Attr Arg="arg" Name="RightTranslations" Label="for IsSemigroup and CanUseFroidurePin and IsFinite"/>
<Returns>the semigroup of left or right translations
</Returns>
<Description>
Given a finite semigroup <A>S</A> satisfying <Ref Filt="CanUseFroidurePin"/>,
returns the semigroup of all left or right translations of <A>S</A>.
<Example><![CDATA[
gap> S := Semigroup([Transformation([1, 4, 3, 3, 6, 5]),
> Transformation([3, 4, 1, 1, 4, 2])]);;
gap> L := LeftTranslations(S);
<the semigroup of left translations of <transformation semigroup of
degree 6 with 2 generators>>
gap> Size(L);
361
]]></Example>
<P/>
</Description>
</ManSection>
<ManSection>
<Attr Arg="S" Name="TranslationalHull" Label="for IsSemigroup and CanUseFroidurePin and IsFinite"/>
<Returns>the translational hull
</Returns>
<Description>
Given a finite semigroup <A>S</A> satisfying <Ref Filt="CanUseFroidurePin"/>,
returns the translational hull of <A>S</A>.
<Example><![CDATA[
gap> S := Semigroup([Transformation([1, 4, 3, 3, 6, 5]),
> Transformation([3, 4, 1, 1, 4, 2])]);;
gap> H := TranslationalHull(S);
<translational hull over <transformation semigroup of degree 6 with 2
generators>>
gap> Size(H);
38
]]></Example>
</Description>
</ManSection>
<ManSection Label="NrXTranslations">
<Heading>NrXTranslations</Heading>
<Attr Arg="S" Name="NrLeftTranslations" Label="for IsSemigroup and CanUseFroidurePin and IsFinite"/>
<Attr Arg="arg" Name="NrRightTranslations" Label="for IsSemigroup and CanUseFroidurePin and IsFinite"/>
<Attr Arg="arg" Name="NrBitranslations" Label="for IsSemigroup and CanUseFroidurePin and IsFinite"/>
<Returns>the number of left-, right-, or bi-translations
</Returns>
<Description>
Given a finite semigroup <A>S</A> satisfying <Ref Filt="CanUseFroidurePin"/>,
returns the number of left-, right-, or bi-translations of <A>S</A>. This
is typically more efficient than calling <Code>Size(LeftTranslations(<A>S</A>))</Code>,
as the [bi]translations may not be produced.
<Example><![CDATA[
gap> S := Semigroup([Transformation([1, 4, 3, 3, 6, 5, 2]),
> Transformation([3, 4, 1, 1, 4, 2])]);;
gap> NrLeftTranslations(S);
1444
gap> NrRightTranslations(S);
398
gap> NrBitranslations(S);
69
]]></Example>
<P/>
</Description>
</ManSection>
<ManSection Label="InnerXTranslations">
<Heading>InnerXTranslations</Heading>
<Attr Arg="S" Name="InnerLeftTranslations" Label="for IsSemigroup and CanUseFroidurePin and IsFinite"/>
<Attr Arg="arg" Name="InnerRightTranslations" Label="for IsSemigroup and CanUseFroidurePin and IsFinite"/>
<Returns>the monoid of inner left or right translations
</Returns>
<Description>
For a finite semigroup <A>S</A> satisfying <Ref Filt="CanUseFroidurePin"/>,
<C>InnerLeftTranslations</C>(<A>S</A>)
returns the inner left translations of S (i.e. the translations
defined by left multiplication by a fixed element of <A>S</A>), and
<C>InnerRightTranslations</C>(<A>S</A>) returns the inner right translations
of <A>S</A> (i.e. the translations defined by right multiplication by
a fixed element of <A>S</A>).
<Example><![CDATA[
gap> S := Semigroup([Transformation([1, 4, 3, 3, 6, 5]),
> Transformation([3, 4, 1, 1, 4, 2])]);;
gap> I := InnerLeftTranslations(S);
<left translations semigroup over <transformation semigroup
of size 22, degree 6 with 2 generators>>
gap> Size(I) <= Size(S);
true
]]></Example>
<P/>
</Description>
</ManSection>
<ManSection>
<Attr Arg="S" Name="InnerTranslationalHull" Label="for IsSemigroup and CanUseFroidurePin and IsFinite"/>
<Returns>the inner translational hull
</Returns>
<Description>
Given a finite semigroup <A>S</A> satisfying <Ref Filt="CanUseFroidurePin"/>,
returns the inner translational hull of <A>S</A>, i.e. the bitranslations
whose left and right translation components are inner translations defined by
left and right multiplication by the same fixed element of <A>S</A>.
<Example><![CDATA[
gap> S := Semigroup([Transformation([1, 4, 3, 3, 6, 5]),
> Transformation([3, 4, 1, 1, 4, 2])]);;
gap> I := InnerTranslationalHull(S);
<semigroup of bitranslations over <transformation semigroup
of size 22, degree 6 with 2 generators>>
gap> L := LeftTranslations(S);;
gap> R := RightTranslations(S);;
gap> H := TranslationalHull(S);;
gap> inners := [];;
gap> for s in S do
> l := LeftTranslation(L, MappingByFunction(S, S, x -> s * x));
> r := RightTranslation(R, MappingByFunction(S, S, x -> x * s));
> AddSet(inners, Bitranslation(H, l, r));
> od;
gap> Set(I) = inners;
true
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Arg="T" Name="UnderlyingRepresentatives" Label="for IsTranslationsSemigroup"/>
<Returns>a set of representatives
</Returns>
<Description>
For efficiency, we typically store translations on a semigroup <M>S</M> as
their actions on a small subset of <M>S</M>. For left translations, this is a
set of representatives of the maximal &R;-classes of <M>S</M>; dually for
right translations we use representatives of the maximal &L;-classes. You can
use <Code>UnderlyingRepresentatives</Code> to access these representatives.
<Example><![CDATA[
gap> G := Range(IsomorphismPermGroup(SmallGroup(12, 1)));;
gap> mat := [[G.1, G.2], [G.1, G.1],
> [G.2, G.3], [G.1 * G.2, G.1 * G.3]];;
gap> S := ReesMatrixSemigroup(G, mat);;
gap> L := LeftTranslations(S);;
gap> R := RightTranslations(S);;
gap> UnderlyingRepresentatives(L);
[ (1,(),1), (2,(),2) ]
gap> UnderlyingRepresentatives(R);
[ (1,(),1), (2,(),2), (1,(),3), (1,(),4) ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Arg="t" Name="ImageSetOfTranslation" Label="for IsSemigroupTranslation"/>
<Returns>a set of elements
</Returns>
<Description>
Given a left or right translation <A>t</A> on a semigroup <M>S</M>, returns
the set of elements of <M>S</M> lying in the image of <A>t</A>.
<Example><![CDATA[
gap> S := Semigroup([Transformation([1, 3, 3, 4]),
> Transformation([3, 4, 1, 2])]);;
gap> t := Set(LeftTranslations(S))[4];
<left translation on <transformation semigroup of size 8, degree 4
with 2 generators>>
gap> ImageSetOfTranslation(t);
[ Transformation( [ 1, 2, 3, 1 ] ), Transformation( [ 1, 3, 3, 1 ] ),
Transformation( [ 3, 1, 1, 3 ] ), Transformation( [ 3, 4, 1, 3 ] ) ]
]]></Example>
</Description>
</ManSection>
</Section>
</Chapter>
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