I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task:

I have two graded (generally noncommutative!) algebras $A$ and $B$ given by generators and homogeneous relations. Given two graded algebra homomorphisms $f,g:A\to B$ which are given by their values on the generators. I want to be able, for every fixed $n\in\mathbb N$, to know whether the kernel of $A$ in degree $n$ is contained in the kernel of $B$ in degree $n$.

This is a completely elementary linear-algebraic problem, in the sense that no Gröbner bases are required (because everything is graded), so it shouldn't be hard to do for a computer. I would like to be able to do this and many similar tasks with the least possible amount of work. In particular, I don't want to have to write scripts that translate my problem into linear algebra in order to feed them to the software (that should be done by the software), and I don't want to have to install experimental and undocumented extensions/libraries. I am not sure whether Singular does all I want in the commutative case, but it's certainly the right direction. Plural (the noncommutative algebra module of Singular) is not the right direction.

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    $\begingroup$ Have you looked into Sage? $\endgroup$ – Alexander Woo Jan 24 '12 at 3:32
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    $\begingroup$ I think GAP can be useful here. gap-system.org $\endgroup$ – Lyosha Jan 24 '12 at 5:36
  • $\begingroup$ Let me mention: mathoverflow.net/questions/85400/… $\endgroup$ – Alexander Chervov Jan 24 '12 at 5:52
  • $\begingroup$ Thanks, I'll look into both (but I can't say I want to have to do anything with GAP again...) - which libraries are closest to what I want? $\endgroup$ – darij grinberg Jan 24 '12 at 8:31
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    $\begingroup$ I think Magma algebra system is useful for your question. Also it has a combinatorics and linear algebra packages. $\endgroup$ – Shahrooz Janbaz Jan 24 '12 at 11:04

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