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## #W factor.xml #Y Copyright (C) 2011-14 James D. Mitchell ## ## Licensing information can be found in the README file of this package. ## ############################################################################# ## <#GAPDoc Label="Factorization"> <ManSection> <Oper Name = "Factorization" Arg = "S, x"/> <Returns>A word in the generators.</Returns> <Description> <List> <Mark>for semigroups</Mark> <Item> When <A>S</A> is a semigroup and <A>x</A> belongs to <A>S</A>, <C>Factorization</C> return a word in the generators of <A>S</A> that is equal to <A>x</A>. In this case, a word is a list of positive integers where an entry <C>i</C> corresponds to <C>GeneratorsOfSemigroups(S)[i]</C>. More specifically, <Log>EvaluateWord(GeneratorsOfSemigroup(S), Factorization(S, x)) = x;</Log> </Item> <!-- FIXME(later) improve: it not clear to me whether this applies to all inverse semigroups or to ones with an inverse op WW --> <Mark>for inverse semigroups</Mark> <Item> When <A>S</A> is an inverse semigroup and <A>x</A> belongs to <A>S</A>, <C>Factorization</C> return a word in the generators of <A>S</A> that is equal to <A>x</A>. In this case, a word is a list of non-zero integers where an entry <C>i</C> corresponds to <C>GeneratorsOfSemigroup(S)[i]</C> and <C>-i</C> corresponds to <C>GeneratorsOfSemigroup(S)[i] ^ -1</C>. As in the previous case, <Log>EvaluateWord(GeneratorsOfSemigroup(S), Factorization(S, x)) = x;</Log> </Item> </List> Note that <C>Factorization</C> does not always return a word of minimum length; see <Ref Oper = "MinimalFactorization"/>. <P/> See also <Ref Oper = "EvaluateWord"/> and <Ref Func = "GeneratorsOfSemigroup" BookName = "ref"/>. <Example><![CDATA[ gap> gens := [Transformation([2, 2, 9, 7, 4, 9, 5, 5, 4, 8]), > Transformation([4, 10, 5, 6, 4, 1, 2, 7, 1, 2])];; gap> S := Semigroup(gens);; gap> x := Transformation([1, 10, 2, 10, 1, 2, 7, 10, 2, 7]);; gap> word := Factorization(S, x); [ 2, 2, 1, 2 ] gap> EvaluateWord(gens, word); Transformation( [ 1, 10, 2, 10, 1, 2, 7, 10, 2, 7 ] ) gap> S := SymmetricInverseMonoid(8); <symmetric inverse monoid of degree 8> gap> x := PartialPerm([1, 2, 3, 4, 5, 8], [7, 1, 4, 3, 2, 6]); [5,2,1,7][8,6](3,4) gap> word := Factorization(S, x); [ -2, -2, -2, -2, -2, -2, 2, 4, 4, 2, 3, 2, -3, -2, -2, 3, 2, -3, -2, -2, 4, -3, -4, 2, 2, 3, -2, -3, 4, -3, -4, 2, 2, 3, -2, -3, 2, 2, 3, -2, -3, 2, 2, 3, -2, -3, 4, -3, -4, 3, 2, -3, -2, -2, 3, 2, -3, -2, -2, 4, 3, -4, 3, 2, -3, -2, -2, 3, 2, -3, -2, -2, 3, 2, 2, 3, 2, 2, 2, 2 ] gap> EvaluateWord(GeneratorsOfSemigroup(S), word); [5,2,1,7][8,6](3,4) gap> S := DualSymmetricInverseMonoid(6);; gap> x := S.1 * S.2 * S.3 * S.2 * S.1; <block bijection: [ 1, 6, -4 ], [ 2, -2, -3 ], [ 3, -5 ], [ 4, -6 ], [ 5, -1 ]> gap> word := Factorization(S, x); [ -2, -2, -2, -2, -2, 4, 2 ] gap> EvaluateWord(GeneratorsOfSemigroup(S), word); <block bijection: [ 1, 6, -4 ], [ 2, -2, -3 ], [ 3, -5 ], [ 4, -6 ], [ 5, -1 ]>]]></Example> </Description> </ManSection> <#/GAPDoc> <#GAPDoc Label="MinimalFactorization"> <ManSection> <Oper Name = "MinimalFactorization" Arg = "S, x"/> <Returns>A minimal word in the generators.</Returns> <Description> This operation returns a minimal length word in the generators of the semigroup <A>S</A> that equals the element <A>x</A>. In this case, a word is a list of positive integers where an entry <C>i</C> corresponds to <C>GeneratorsOfSemigroups(<A>S</A>)[i]</C>. More specifically, <Log>EvaluateWord(GeneratorsOfSemigroup(S), MinimalFactorization(S, x)) = x;</Log> <P/> <C>MinimalFactorization</C> involves exhaustively enumerating <A>S</A> until the element <A>x</A> is found, and so <C>MinimalFactorization</C> may be less efficient than <Ref Oper="Factorization"/> for some semigroups. <P/> Unlike <Ref Oper = "Factorization"/> this operation does not distinguish between semigroups and inverse semigroups. See also <Ref Oper = "EvaluateWord"/> and <Ref Func = "GeneratorsOfSemigroup" BookName="ref"/>. <Example><![CDATA[ gap> S := Semigroup(Transformation([2, 2, 9, 7, 4, 9, 5, 5, 4, 8]), > Transformation([4, 10, 5, 6, 4, 1, 2, 7, 1, 2])); <transformation semigroup of degree 10 with 2 generators> gap> x := Transformation([8, 8, 2, 2, 9, 2, 8, 8, 9, 9]); Transformation( [ 8, 8, 2, 2, 9, 2, 8, 8, 9, 9 ] ) gap> Factorization(S, x); [ 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1 ] gap> MinimalFactorization(S, x); [ 1, 2, 1, 1, 1, 1, 2, 2, 1 ]]]></Example> </Description> </ManSection> <#/GAPDoc> <#GAPDoc Label="NonTrivialFactorization"> <ManSection> <Oper Name = "NonTrivialFactorization" Arg = "S, x"/> <Returns>A non-trivial word in the generators, or <K>fail</K>.</Returns> <Description> When <A>S</A> is a semigroup and <A>x</A> belongs to <A>S</A>, this operation returns a non-trivial word in the generators of the semigroup <A>S</A> that equals <A>x</A>, if one exists. The definition of a word in the generators is the same as given in <Ref Oper="Factorization" /> for semigroups and inverse semigroups. A word is non-trivial if it has length two or more. <P/> If no non-trivial word for <A>x</A> exists, then <A>x</A> is an indecomposable element of <A>S</A> and this operation returns <K>fail</K>; see <Ref Attr="IndecomposableElements" />. <P/> When <A>x</A> does not belong to <C>GeneratorsOfSemigroup(<A>S</A>)</C>, any factorization of <A>x</A> is non-trivial. In this case, <C>NonTrivialFactorization</C> returns the same word as <Ref Oper="Factorization" />. <P/> See also <Ref Func = "EvaluateWord"/> and <Ref Func = "GeneratorsOfSemigroup" BookName="ref"/>. <Example><![CDATA[ gap> x := Transformation([5, 4, 2, 1, 3]);; gap> y := Transformation([4, 4, 2, 4, 1]);; gap> S := Semigroup([x, y]); <transformation semigroup of degree 5 with 2 generators> gap> NonTrivialFactorization(S, x * y); [ 1, 2 ] gap> Factorization(S, x); [ 1 ] gap> NonTrivialFactorization(S, x); [ 1, 1, 1, 1, 1, 1 ] gap> Factorization(S, y); [ 2 ] gap> NonTrivialFactorization(S, y); [ 2, 1, 1, 1, 1, 1 ] gap> z := PartialPerm([2]);; gap> S := Semigroup(z); <commutative partial perm semigroup of rank 1 with 1 generator> gap> NonTrivialFactorization(S, z); fail]]></Example> </Description> </ManSection> <#/GAPDoc> ¤ Dauer der Verarbeitung: 0.1 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28