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#W ncinterface.gd
#W NMO: Interface routines Randall Cone
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## User interface to lower-level functions
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## (C) 2010 Mathematics Dept., Virginia Polytechnic Institute, USA
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## <#GAPDoc Label="NCLessThanByOrdering">
## <ManSection>
## <Oper Name="NCLessThanByOrdering"
## Arg="<NoncommutativeMonomialOrdering>, <a>, <b>"/>
##
## <Description>
## Given a <C><NoncommutativeMonomialOrdering></C> on an algebra <M>A</M>
## and <M>a,b \in A</M>,
## <C>NCLessThanByOrdering</C> returns the (boolean) result of
## <M>a < b</M>, where <M><</M> represents the comparison operator
## determined by <C><NoncommutativeMonomialOrdering></C>.
## <P/>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("NCLessThanByOrdering",
[ IsNoncommutativeMonomialOrdering,
IsElementOfFreeAssociativeRing,
IsElementOfFreeAssociativeRing ]
);
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##
## <#GAPDoc Label="NCGreaterThanByOrdering">
## <ManSection>
## <Oper Name="NCGreaterThanByOrdering"
## Arg="<NoncommutativeMonomialOrdering>, <a>, <b>"/>
##
## <Description>
## Given a <C><NoncommutativeMonomialOrdering></C> on an algebra <M>A</M>
## and <M>a,b \in A</M>,
## <C>NCLessThanByOrdering</C> returns the (boolean) result of
## <M>a > b</M>, where <M>></M> represents the comparison operator
## determined by <C><NoncommutativeMonomialOrdering></C>.
## <P/>
## <P/>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("NCGreaterThanByOrdering",
[ IsNoncommutativeMonomialOrdering,
IsElementOfFreeAssociativeRing,
IsElementOfFreeAssociativeRing ]
);
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##
## <#GAPDoc Label="NCEquivalentByOrdering">
## <ManSection>
## <Oper Name="NCEquivalentByOrdering"
## Arg="<NoncommutativeMonomialOrdering>, <a>, <b>"/>
##
## <Description>
## Given a <C><NoncommutativeMonomialOrdering></C> on an algebra <M>A</M>
## and <M>a,b \in A</M>,
## <C>NCLessThanByOrdering</C> returns the (boolean) result of
## <M>a = b</M>, where <M>=</M> represents the comparison operator
## determined by <C><NoncommutativeMonomialOrdering></C>.
## <P/>
##
## Some examples of these methods in use:
## <Example>
## gap> A := FreeAssociativeAlgebraWithOne(Rationals,"x","y","z");
## <algebra-with-one over Rationals, with 3 generators>
## gap> x := A.x;; y := A.y;; z := A.z;; id := One(A);;
## gap> w1 := x*x*y;; w2 := x*y*x;; w3 := z*x;;
##
## gap> ml := NCMonomialLeftLengthLexOrdering(A);
## NCMonomialLeftLengthLexicographicOrdering([ (1)*x, (1)*y, (1)*z ])
##
## gap> ml2 := NCMonomialLengthOrdering(A);
## NCMonomialLengthOrdering([ (1)*x, (1)*y, (1)*z ])
##
## gap> ml7 := NCMonomialWeightOrdering(A,[1,2,3],[1,1,2]);
## NCMonomialWeightOrdering([ (1)*x, (1)*y, (1)*z ])
##
## gap> ml8 := NCMonomialWeightOrdering(A,[2,3,1],[1,1,2]);
## NCMonomialWeightOrdering([ (1)*y, (1)*z, (1)*x ])
##
## gap> # Left length-lex ordering, x<y<z:
## gap> NCEquivalentByOrdering(ml,w1,w2);
## false
## gap> # Length ordering:
## gap> NCEquivalentByOrdering(ml2,w1,w2);
## true
## gap> NCEquivalentByOrdering(ml2,w3,w2);
## false
## gap> # Weight ordering ( z=2, x=y=1 ):
## gap> NCEquivalentByOrdering(ml7,w1,w2);
## true
## gap> NCEquivalentByOrdering(ml7,w3,w2);
## true
## gap> # Weight ordering ( z=2, x=y=1 ), different lex:
## gap> NCEquivalentByOrdering(ml8,w1,w2);
## true
## gap> NCEquivalentByOrdering(ml8,w3,w2);
## true
## </Example>
## <P/>
##
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("NCEquivalentByOrdering",
[ IsNoncommutativeMonomialOrdering,
IsElementOfFreeAssociativeRing,
IsElementOfFreeAssociativeRing ]
);
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##
## <#GAPDoc Label="NCSortNP">
## <ManSection>
## <Oper Name="NCSortNP"
## Arg="<NoncommutativeMonomialOrdering>, <list>, <function>"/>
##
## <Description>
## Given a <C><list></C> of NP `list' representations for monomials from
## a noncommutative algebra, and an NP comparison (ordering) function
## <C><function></C>,
## <C>NCSortNP</C> returns a sorted version of <C><list></C>
## (with respect to the NP comparison function <C><function></C>).
## The sort used here is an insertion sort, per the recommendation
## from <Cite Key="gN02"/>.
## <P/>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("NCSortNP",
[ IsList, IsFunction ]
);
##
#E
