Math software for symbolic computation with customizable rules.

Question

I am looking for software where we can define operations and rules of manipulation. For example, defining multiplication of $x,y \in A$ to be $x\cdot y:= xy-yx$ and $x^{2k}=1$ and $y^{3k}=1$, $\forall k \gt 0$. Also, we should be able to impose commutativity or non-commutativity, etc... I have heard of mathematica, maple, sagemath,.. , but I don't know which of those can accomplish what I want. Please give me some recommendations.

Answer 1

Well, I have some experience with Maple. It allows to define functions such as

$f := (x,y)\mapsto xy-yx$

and to manipulate and simplify expressions

simplify(f,{x^2=xy,y^2=yx}$

These belong to the standard capabilities of computer algebra systems.