#############################################################################
##
#W congrms.xml
#Y Copyright (C) 2015 Michael Young
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
<#GAPDoc Label="IsRZMSCongruenceByLinkedTriple">
<ManSection>
<Filt Name = "IsRMSCongruenceByLinkedTriple" Arg = "obj"Type = "category"/>
<Filt Name = "IsRZMSCongruenceByLinkedTriple" Arg = "obj"Type = "category"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
These categories describe a type of semigroup congruence over a Rees
matrix or 0-matrix semigroup. Externally, an object of this type may be
used in the same way as any other object in the category <Ref
Prop = "IsSemigroupCongruence" BookName = "ref"/> but it is represented
internally by its <E>linked triple</E>, and certain functions may take
advantage of this information to reduce computation times.<P/>
An object of this type may be constructed with
<C>RMSCongruenceByLinkedTriple</C> or <C>RZMSCongruenceByLinkedTriple</C>,
or this representation may be selected automatically by <Ref
Func = "SemigroupCongruence"/>.
<#GAPDoc Label="IsRZMSCongruenceClassByLinkedTriple">
<ManSection>
<Filt Name = "IsRMSCongruenceClassByLinkedTriple" Arg = "obj" Type = "category"/>
<Filt Name = "IsRZMSCongruenceClassByLinkedTriple" Arg = "obj" Type = "category"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
These categories contain the congruence classes of a semigroup congruence
of the categories <Ref Filt = "IsRMSCongruenceByLinkedTriple"/> and <Ref
Filt = "IsRZMSCongruenceByLinkedTriple"/> respectively. <P/>
An object of one of these types may be used in the same way as any other
object in the category <Ref Filt = "IsCongruenceClass"/>,
but the class is represented internally by information related
to the congruence's linked triple, and certain functions may take
advantage of this information to reduce computation times. <P/>
<#GAPDoc Label="RZMSCongruenceByLinkedTriple">
<ManSection>
<Func Name = "RMSCongruenceByLinkedTriple"
Arg = "S, N, colBlocks, rowBlocks"/>
<Func Name = "RZMSCongruenceByLinkedTriple"
Arg = "S, N, colBlocks, rowBlocks"/>
<Returns>
A Rees matrix or 0-matrix semigroup congruence by linked triple.
</Returns>
<Description>
This function returns a semigroup congruence over the Rees matrix or
0-matrix semigroup <A>S</A> corresponding to the linked triple (<A>N</A>,
<A>colBlocks</A>, <A>rowBlocks</A>). The argument <A>N</A> should be a
normal subgroup of the underlying semigroup of <A>S</A>; <A>colBlocks</A>
should be a partition of the columns of the matrix of <A>S</A>; and
<A>rowBlocks</A> should be a partition of the rows of the matrix of
<A>S</A>. For example, if the matrix has 5 rows, then a possibility for
<A>rowBlocks</A> might be <C>[[1, 3], [2, 5], [4]]</C>.<P/>
If the arguments describe a valid linked triple on <A>S</A>, then an
object in the category <C>IsRZMSCongruenceByLinkedTriple</C> is returned.
This object can be used like any other semigroup congruence in
&GAP;.<P/>
If the arguments describe a triple which is not <E>linked</E> in the sense
described above, then this function returns an error.
<Example><![CDATA[
gap> G := Group([(1, 4, 5), (1, 5, 3, 4)]);;
gap> mat := [[0, 0, (1, 4, 5), 0, 0, (1, 4, 3, 5)],
> [0, (), 0, 0, (3, 5), 0],
> [(), 0, 0, (3, 5), 0, 0]];;
gap> S := ReesZeroMatrixSemigroup(G, mat);;
gap> N := Group([(1, 4)(3, 5), (1, 5)(3, 4)]);;
gap> colBlocks := [[1], [2, 5], [3, 6], [4]];;
gap> rowBlocks := [[1], [2], [3]];;
gap> cong := RZMSCongruenceByLinkedTriple(S, N, colBlocks, rowBlocks);;
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="RZMSCongruenceClassByLinkedTriple">
<ManSection>
<Oper Name = "RMSCongruenceClassByLinkedTriple"
Arg = "cong, nCoset, colClass, rowClass"/>
<Oper Name = "RZMSCongruenceClassByLinkedTriple"
Arg = "cong, nCoset, colClass, rowClass"/>
<Returns>
A Rees matrix or 0-matrix semigroup congruence class by linked
triple.
</Returns>
<Description>
This operation returns one congruence class of the congruence
<A>cong</A>, as defined by the other three parameters.<P/>
The argument <A>cong</A> must be a Rees matrix or 0-matrix semigroup
congruence by linked triple. If the linked triple consists of the three
parameters <C>N</C>, <C>colBlocks</C> and <C>rowBlocks</C>, then
<A>nCoset</A> must be a right coset of <C>N</C>, <A>colClass</A> must be
a positive integer corresponding to a position in the list
<C>colBlocks</C>, and <A>rowClass</A> must be a positive integer
corresponding to a position in the list <C>rowBlocks</C>.<P/>
If the arguments are valid, an <C>IsRMSCongruenceClassByLinkedTriple</C>
or <C>IsRZMSCongruenceClassByLinkedTriple</C> object is returned, which
can be used like any other equivalence class in &GAP;. Otherwise, an
error is returned.
<Example><![CDATA[
gap> G := Group([(1, 4, 5), (1, 5, 3, 4)]);;
gap> mat := [[0, 0, (1, 4, 5), 0, 0, (1, 4, 3, 5)],
> [0, (), 0, 0, (3, 5), 0],
> [(), 0, 0, (3, 5), 0, 0]];;
gap> S := ReesZeroMatrixSemigroup(G, mat);;
gap> N := Group([(1, 4)(3, 5), (1, 5)(3, 4)]);;
gap> colBlocks := [[1], [2, 5], [3, 6], [4]];;
gap> rowBlocks := [[1], [2], [3]];;
gap> cong := RZMSCongruenceByLinkedTriple(S, N, colBlocks, rowBlocks);;
gap> class := RZMSCongruenceClassByLinkedTriple(cong,
> RightCoset(N, (1, 5)), 2, 3);
<2-sided congruence class of (2,(3,4),3)>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="IsLinkedTriple">
<ManSection>
<Oper Name = "IsLinkedTriple" Arg = "S, N, colBlocks, rowBlocks"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
This operation returns true if and only if the arguments (<A>N</A>,
<A>colBlocks</A>, <A>rowBlocks</A>) describe a linked triple of
the Rees matrix or 0-matrix semigroup <A>S</A>, as described above.
<Example><![CDATA[
gap> G := Group([(1, 4, 5), (1, 5, 3, 4)]);;
gap> mat := [[0, 0, (1, 4, 5), 0, 0, (1, 4, 3, 5)],
> [0, (), 0, 0, (3, 5), 0],
> [(), 0, 0, (3, 5), 0, 0]];;
gap> S := ReesZeroMatrixSemigroup(G, mat);;
gap> N := Group([(1, 4)(3, 5), (1, 5)(3, 4)]);;
gap> colBlocks := [[1], [2, 5], [3, 6], [4]];;
gap> rowBlocks := [[1], [2], [3]];;
gap> IsLinkedTriple(S, N, colBlocks, rowBlocks);
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
¤ Dauer der Verarbeitung: 0.28 Sekunden
(vorverarbeitet)
¤
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.
