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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%A orders.msk GAP documentation Isabel Araujo
%%
%A @(#)$Id: orders.msk,v 1.6 2002/04/15 10:02:31 sal Exp $
%%
%Y (C) 2000 School Math and Comp. Sci., University of St. Andrews, Scotland
%Y Copyright (C) 2002 The GAP Group
%%
\Chapter{Orderings}
\FileHeader{orders}[1]
\Declaration{IsOrdering}
\Declaration{OrderingsFamily}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Building new orderings}
\Declaration{OrderingByLessThanFunctionNC}
\Declaration{OrderingByLessThanOrEqualFunctionNC}
\beginexample
gap> f := FreeSemigroup("a","b");;
gap> a := GeneratorsOfSemigroup(f)[1];;
gap> b := GeneratorsOfSemigroup(f)[2];;
gap> lt := function(x,y) return Length(x)<Length(y); end;
function( x, y ) ... end
gap> fam := FamilyObj(a);;
gap> ord := OrderingByLessThanFunctionNC(fam,lt);
Ordering
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Properties and basic functionality}
\Declaration{IsWellFoundedOrdering}
\Declaration{IsTotalOrdering}
\Declaration{IsIncomparableUnder}
\Declaration{FamilyForOrdering}
\Declaration{LessThanFunction}
\Declaration{LessThanOrEqualFunction}
\Declaration{IsLessThanUnder}
\Declaration{IsLessThanOrEqualUnder}
\beginexample
gap> IsLessThanUnder(ord,a,a*b);
true
gap> IsLessThanOrEqualUnder(ord,a*b,a*b);
true
gap> IsIncomparableUnder(ord,a,b);
true
gap> FamilyForOrdering(ord) = FamilyObj(a);
true
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Orderings on families of associative words}
\FileHeader{orders}[2]
\Declaration{IsOrderingOnFamilyOfAssocWords}
\FileHeader{orders}[3]
\Declaration{IsTranslationInvariantOrdering}
\Declaration{IsReductionOrdering}
\Declaration{OrderingOnGenerators}
\Declaration{LexicographicOrdering}
\beginexample
gap> f := FreeSemigroup(3);
<free semigroup on the generators [ s1, s2, s3 ]>
gap> lex := LexicographicOrdering(f,[2,3,1]);
Ordering
gap> IsLessThanUnder(lex,f.2*f.3,f.3);
true
gap> IsLessThanUnder(lex,f.3,f.2);
false
\endexample
\Declaration{ShortLexOrdering}
\Declaration{IsShortLexOrdering}
\beginexample
gap> f := FreeSemigroup(3);
<free semigroup on the generators [ s1, s2, s3 ]>
gap> sl := ShortLexOrdering(f,[2,3,1]);
Ordering
gap> IsLessThanUnder(sl,f.1,f.2);
false
gap> IsLessThanUnder(sl,f.3,f.2);
false
gap> IsLessThanUnder(sl,f.3,f.1);
true
\endexample
\Declaration{WeightLexOrdering}
\Declaration{IsWeightLexOrdering}
\Declaration{WeightOfGenerators}
\beginexample
gap> f := FreeSemigroup(3);
<free semigroup on the generators [ s1, s2, s3 ]>
gap> wtlex := WeightLexOrdering(f,[f.2,f.3,f.1],[3,2,1]);
Ordering
gap> IsLessThanUnder(wtlex,f.1,f.2);
true
gap> IsLessThanUnder(wtlex,f.3,f.2);
true
gap> IsLessThanUnder(wtlex,f.3,f.1);
false
gap> OrderingOnGenerators(wtlex);
[ s2, s3, s1 ]
gap> WeightOfGenerators(wtlex);
[ 3, 2, 1 ]
\endexample
\Declaration{BasicWreathProductOrdering}
\Declaration{IsBasicWreathProductOrdering}
\beginexample
gap> f := FreeSemigroup(3);
<free semigroup on the generators [ s1, s2, s3 ]>
gap> basic := BasicWreathProductOrdering(f,[2,3,1]);
Ordering
gap> IsLessThanUnder(basic,f.3,f.1);
true
gap> IsLessThanUnder(basic,f.3*f.2,f.1);
true
gap> IsLessThanUnder(basic,f.3*f.2*f.1,f.1*f.3);
false
\endexample
\Declaration{WreathProductOrdering}
\Declaration{IsWreathProductOrdering}
\Declaration{LevelsOfGenerators}
\beginexample
gap> f := FreeSemigroup(3);
<free semigroup on the generators [ s1, s2, s3 ]>
gap> wrp := WreathProductOrdering(f,[1,2,3],[1,1,2,]);
Ordering
gap> IsLessThanUnder(wrp,f.3,f.1);
false
gap> IsLessThanUnder(wrp,f.3,f.2);
false
gap> IsLessThanUnder(wrp,f.1,f.2);
true
gap> LevelsOfGenerators(wrp);
[ 1, 1, 2 ]
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%E
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