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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2Mathematica can do almost anything that you might want to do symbolically using your own custom rules and many built-in functions. – Somos May 19 '19 at 08:13
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awesome. thanks! – TheLast Cipher May 19 '19 at 08:15
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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.
Wuestenfux
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what about imposing non-commutativity and non-associativity and other rules? – TheLast Cipher May 19 '19 at 08:11
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1Non-commutativity holds in the first place. You have to specify $xy=yx$ to establish commutativity. Same for associativity. – Wuestenfux May 19 '19 at 08:14