[GAP Forum] Re:Extending Orderings, perhaps better posed

[rcone]

Alexander Hulpke wrote:

>What you could do is to form the semigroup algebra of the
>free semigroup (or free abelian semigroup), this way there
>is a connection between semigroup and algebra.

I've tried this, and gotten some unexpected results.  Perhaps you could 
explain where I've gone wrong:

gap> A := FreeAlgebra(Rationals,2);
<algebra over Rationals, with 2 generators>
gap> gensA := GeneratorsOfAlgebra(A);
[ (1)*x.1, (1)*x.2 ]
gap> a1 := gensA[1];
(1)*x.1
gap> a2 := gensA[2];
(1)*x.2
gap> a1*a2 in A;
true
gap> a1+a2 in A;
true
gap> G := FreeSemigroup(2);
<free semigroup on the generators [ s1, s2 ]>
gap> gensG := GeneratorsOfSemigroup(G);
[ s1, s2 ]
gap> g1 := gensG[1];
s1
gap> g2 := gensG[2];
s2
gap> g1*g2 in G;
true
gap> B := Algebra(Rationals,gensG,0);
<algebra over Rationals, with 2 generators>
gap> g1*g2 in B;
false
gap> gensB := GeneratorsOfAlgebra(B);
[ s1, s2 ]
gap> b1 := gensB[1];
s1
gap> b2 := gensB[2];
s2
gap> b1*b2 in B;
false
gap> b1*b2 in G;
true
gap> b1+b2 in B;
Error, no method found! For debug...


>The other option would be to write small functions that
>take a monomial and build a ``corresponding'' semigroup
>element, using these ``shadows'' for comparison.

My hope was to stay away from such machinery, and act on the objects 
(monomials of free algebras) directly as associative words.  Then, any 
improvements or extensions to the work on associative words would translate 
naturally to such objects.  For example, I've implemented the Horspool subword 
search algorithm for associative words.  Guess I'm outta luck, and should 
consider coding orderings, subword search, overlap, etc. for elements of free 
algebras?

Thanks for your time and response!
Randy Cone