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Is there any software that is designed specifically for algebraic structures, that can do some of the basic constructions in algebra, like

  • Constructing the congruence lattice of a (finite) algebra, the lattices of sub-algebras, maximal sub-algebras, the sub-algebra generated by a set, quotients, ect.

    • Testing whether a set of term equations $E$ deduces another set of equations $E'$. Or finding a counter example, ect.

    • Basic constructions for (finite, commutative) rings, like the Jacobson radical ideal, or the nil-potent radical ideal, or testing whether a subring is integral.

    • Various constructions on groups, like the Jordon-Holder decomposition, the center, sylow subgroups, ect.

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    $\begingroup$ +1 for the soft-question pun. Also, have you looked at Sage, PARI, and GAP? (The first contains the other two IIRC. Certainly GAP.) $\endgroup$ – Dan Nov 11 '16 at 18:02
  • $\begingroup$ I havent, but Ill check them out. I have only seen prover 9, but that doesnt seem like it is what I describe above $\endgroup$ – Mike Nov 11 '16 at 18:05
  • $\begingroup$ Yeah you're looking more for a "computer algebra system" than an automated prover it sounds like. Take a look at Sage first. I've never attempted a few of the calculations you mentioned in a CAS (computing Jacobson and nil radicals, for instance), but if a halting algorithm is known for computing some kind of algebraic object, it is (or probably can be efficiently) implemented in Sage. As such you probably won't find generic <Claim about Objects> --> <Counterexample> functionality in a CAS, but you can still use them to find counterexamples by testing hypotheses on huge sample sets. $\endgroup$ – Dan Nov 11 '16 at 18:19
  • $\begingroup$ Here's a link to a tutorial on Sage's group theory capabilities: doc.sagemath.org/html/en/thematic_tutorials/group_theory.html $\endgroup$ – Dan Nov 11 '16 at 18:20
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    $\begingroup$ Also look at the universal algebra calculator: uacalc.org $\endgroup$ – Eran Nov 11 '16 at 18:26

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