[Axiom-math] How to make symbolic computations?

[Fabio S.]

Hi,
I am facing a problem that I can't solve and I can't find any help on the 
book.

The problem is the following: assume you have a field k (let's say k=Q, 
but I would like to more general if possible) and take two variable x and 
y which do not commute.

I would like to build the non-commutative algebra h=k[x,y] and then I 
would like to make computations in h using some predefined rules for x and 
y. As an example, take the three equations

x*y*x=y*x*y
x*x=a*x+b
y*y=a*y+b

where a and b are (generic, if possible) elements of k.

Then, I would like to be able to reduce polynomials in x and y according 
to the previous rules. For example,

(x+y)^2 (=x^2+x*y+y*x+y^2)

should reduce to

a*(x+y)+2*b+x*y+y*x

and


(x+y)^3

(
=(x+y)(x+y)^2
=(x+y)(a(x+y)+2b+xy+yx)
=axx+axy+ayx+ayy+2bx+2by+xxy+xyx+yxy+yyx
=a^2x+ab+axy+ayx+a^2y+ab+2bx+2by+(ax+b)y+xyx+yxy+(ay+b)x
=a^2x+ab+axy+ayx+a^2y+ab+2bx+2by+axy+by+xyx+yxy+ayx+bx
=xyx+yxy+axy+axy+ayx+ayx+a^2x+2bx+bx+a^2y+2by+by+ab+ab
)

should reduce to

2*x*y*x+2*a*x*y+2*a*y*x+(a^2+3*b)*x+(a^2+3*b)*y+2*a*b

Is this possible? I think it should, but I couldn't find how... :-((

Thanks for your help

Fabio