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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.

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    $\begingroup$ Mathematica can do almost anything that you might want to do symbolically using your own custom rules and many built-in functions. $\endgroup$ – Somos May 19 at 8:13
  • $\begingroup$ awesome. thanks! $\endgroup$ – TheLast Cipher May 19 at 8:15
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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.

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  • $\begingroup$ what about imposing non-commutativity and non-associativity and other rules? $\endgroup$ – TheLast Cipher May 19 at 8:11
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    $\begingroup$ Non-commutativity holds in the first place. You have to specify $xy=yx$ to establish commutativity. Same for associativity. $\endgroup$ – Wuestenfux May 19 at 8:14

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