| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/semigroups/doc/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 29.7.2025 mit Größe 8 kB |
|
Quelle maximal.xml Sprache: XML |
|||||||||
|
#############################################################################
## #W maximal.xml #Y Copyright (C) 2013-16 James D. Mitchell ## Wilf A. Wilson ## ## Licensing information can be found in the README file of this package. ## ############################################################################# ## <#GAPDoc Label="MaximalSubsemigroups"> <ManSection> <Attr Name = "MaximalSubsemigroups" Arg = "S" Label="for a finite semigroup"/> <Oper Name = "MaximalSubsemigroups" Arg = "S, opts" Label="for a finite semigroup and a record"/> <Returns> The maximal subsemigroups of <A>S</A>. </Returns> <Description> If <A>S</A> is a finite semigroup, then the attribute <C>MaximalSubsemigroups</C> returns a list of the non-empty maximal subsemigroups of <A>S</A>. The methods used by <C>MaximalSubsemigroups</C> are based on <Cite Key="Graham1968aa"/>, and are described in <Cite Key="Donoven2018aa"/>. <P/> It is computationally expensive to search for the maximal subsemigroups of a semigroup, and so computations involving <C>MaximalSubsemigroups</C> may be very lengthy. A substantial amount of information on the progress of <C>MaximalSubsemigroups</C> is provided through the info class <Ref InfoClass="InfoSemigroups"/>, with increasingly detailed information given at levels 1, 2, and 3. <P/> The behaviour of <C>MaximalSubsemigroups</C> can be altered via the second argument <A>opts</A>, which should be a record. The optional components of <A>opts</A> are: <List> <Mark>gens (a boolean)</Mark> <Item> If <C><A>opts</A>.gens</C> is <K>false</K> or unspecified, then the maximal subsemigroups themselves are returned and not just generating sets for these subsemigroups. <P/> It can be more computationally expensive to return the generating sets for the maximal subsemigroups, than to return the maximal subsemigroups themselves. </Item> <Mark>contain (a list)</Mark> <Item> If <C><A>opts</A>.contain</C> is duplicate-free list of elements of <A>S</A>, then <C>MaximalSubsemigroups</C> will search for the maximal subsemigroups of <A>S</A> which contain those elements. </Item> <Mark>D (a &D;-class)</Mark> <Item> For a maximal subsemigroup <C>M</C> of a finite semigroup <A>S</A>, there exists a unique &D;-class which contains the complement of <C>M</C> in <A>S</A>. In other words, the elements of <A>S</A> which <C>M</C> lacks are contained in a unique &D;-class. <P/> If <C><A>opts</A>.D</C> is a &D;-class of <A>S</A>, then <C>MaximalSubsemigroups</C> will search exclusively for those maximal subsemigroups of <A>S</A> whose complement is contained in <C><A>opts</A>.D</C>. </Item> <Mark>types (a list)</Mark> <Item> <E>This option is relevant only if <A>S</A> is a regular Rees (0-)matrix semigroup over a group.</E> <P/> As described at the start of this subsection, <Ref Sect = "Computing maximal subsemigroups" />, a maximal subsemigroup of a regular Rees (0-)matrix semigroup over a group is one of 6 possible types. <P/> If <A>S</A> is a regular Rees (0-)matrix semigroup over a group and <C><A>opts</A>.types</C> is a subset of <C>[1 .. 6]</C>, then <C>MaximalSubsemigroups</C> will search for those maximal subsemigroups of <A>S</A> of the types enumerated by <C><A>opts</A>.types</C>. <P/> The default value for this option is <C>[1 .. 6]</C> (i.e. no restriction). </Item> </List> <Example><![CDATA[ gap> S := FullTransformationSemigroup(3); <full transformation monoid of degree 3> gap> MaximalSubsemigroups(S); [ <transformation semigroup of degree 3 with 7 generators>, <transformation semigroup of degree 3 with 7 generators>, <transformation semigroup of degree 3 with 7 generators>, <transformation semigroup of degree 3 with 7 generators>, <transformation monoid of degree 3 with 5 generators> ] gap> MaximalSubsemigroups(S, > rec(gens := true, D := DClass(S, Transformation([2, 2, 3])))); [ [ Transformation( [ 1, 1, 1 ] ), Transformation( [ 3, 3, 3 ] ), Transformation( [ 2, 2, 2 ] ), IdentityTransformation, Transformation( [ 2, 3, 1 ] ), Transformation( [ 2, 1 ] ) ] ] gap> MaximalSubsemigroups(S, > rec(contain := [Transformation([2, 3, 1])])); [ <transformation semigroup of degree 3 with 7 generators>, <transformation monoid of degree 3 with 5 generators> ] gap> R := PrincipalFactor( > DClass(FullTransformationMonoid(4), Transformation([2, 2]))); <Rees 0-matrix semigroup 6x4 over Group([ (2,3,4), (2,4) ])> gap> MaximalSubsemigroups(R, rec(types := [5], > contain := [RMSElement(R, 1, (), 1), > RMSElement(R, 1, (2, 3), 2)])); [ <subsemigroup of 6x4 Rees 0-matrix semigroup with 10 generators>, <subsemigroup of 6x4 Rees 0-matrix semigroup with 10 generators>, <subsemigroup of 6x4 Rees 0-matrix semigroup with 10 generators>, <subsemigroup of 6x4 Rees 0-matrix semigroup with 10 generators> ]]]></Example> </Description> </ManSection> <#/GAPDoc> <#GAPDoc Label="NrMaximalSubsemigroups"> <ManSection> <Attr Name = "NrMaximalSubsemigroups" Arg = "S" /> <Returns> The number of maximal subsemigroups of <A>S</A>. </Returns> <Description> If <A>S</A> is a finite semigroup, then <C>NrMaximalSubsemigroups</C> returns the number of non-empty maximal subsemigroups of <A>S</A>. The methods used by <C>MaximalSubsemigroups</C> are based on <Cite Key="Graham1968aa"/>, and are described in <Cite Key="Donoven2018aa"/>. <P/> It can be significantly faster to find the number of maximal subsemigroups of a semigroup than to find the maximal subsemigroups themselves. <P/> Unless the maximal subsemigroups of <A>S</A> are already known, the command <C>NrMaximalSubsemigroups(<A>S</A>)</C> simply calls the command <C>MaximalSubsemigroups(<A>S</A>, rec(number := true))</C>. <P/> For more information about searching for maximal subsemigroups of a finite semigroup in the &SEMIGROUPS; package, and for information about the options available to alter the search, see <Ref Oper="MaximalSubsemigroups" Label="for a finite semigroup and a record"/>. By supplying the additional option <C><A>opts</A>.number :=</C> <K>true</K>, the number of maximal subsemigroups will be returned rather than the subsemigroups themselves. <Example><![CDATA[ gap> S := FullTransformationSemigroup(3); <full transformation monoid of degree 3> gap> NrMaximalSubsemigroups(S); 5 gap> S := RectangularBand(3, 4);; gap> NrMaximalSubsemigroups(S); 7 gap> R := PrincipalFactor( > DClass(FullTransformationMonoid(4), Transformation([2, 2]))); <Rees 0-matrix semigroup 6x4 over Group([ (2,3,4), (2,4) ])> gap> MaximalSubsemigroups(R, rec(number := true, types := [3, 4])); 10]]></Example> </Description> </ManSection> <#/GAPDoc> <#GAPDoc Label="IsMaximalSubsemigroup"> <ManSection> <Oper Name = "IsMaximalSubsemigroup" Arg = "S, T"/> <Returns> <K>true</K> or <K>false</K>. </Returns> <Description> If <A>S</A> and <A>T</A> are semigroups, then <C>IsMaximalSubsemigroup</C> returns <K>true</K> if and only if <A>T</A> is a maximal subsemigroup of <A>S</A>. <P/> A <E>maximal subsemigroup</E> of <A>S</A> is a proper subsemigroup of <A>S</A> which is contained in no other proper subsemigroup of <A>S</A>. <Example><![CDATA[ gap> S := ZeroSemigroup(2);; gap> IsMaximalSubsemigroup(S, Semigroup(MultiplicativeZero(S))); true gap> S := FullTransformationSemigroup(4); <full transformation monoid of degree 4> gap> T := Semigroup(Transformation([3, 4, 1, 2]), > Transformation([1, 4, 2, 3]), > Transformation([2, 1, 1, 3])); <transformation semigroup of degree 4 with 3 generators> gap> IsMaximalSubsemigroup(S, T); true gap> R := Semigroup(Transformation([3, 4, 1, 2]), > Transformation([1, 4, 2, 2]), > Transformation([2, 1, 1, 3])); <transformation semigroup of degree 4 with 3 generators> gap> IsMaximalSubsemigroup(S, R); false]]></Example> </Description> </ManSection> <#/GAPDoc> ¤ Dauer der Verarbeitung: 0.22 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
|
2026-03-28