Algebras from congruences

We present a functorial construction which, starting from a congruence α of finite index in an algebra 𝐀 , yields a new algebra 𝐂 with the following properties: the congruence lattice of 𝐂 is isomorphic to the interval of congruences between 0 and α on 𝐀 , this isomorphism preserves higher commutators and TCT types, and 𝐂 inherits all idempotent Maltsev conditions from 𝐀 . As applications of this construction, we first show that supernilpotence is decidable for congruences of finite algebras in varieties that omit type 1 . Secondly, we prove that the subpower membership problem for finite algebras with a cube term can be effectively reduced to membership questions in subdirect products of subdirectly irreducible algebras with central monoliths. As a consequence, we obtain a polynomial time algorithm for the subpower membership problem for finite algebras with a cube term in which the monolith of every subdirectly irreducible section has a supernilpotent centralizer.