Algebras from Congruences

TL;DR

This paper introduces a functorial algebra construction that relates congruence intervals to new algebras, enabling decidability results and efficient algorithms for problems in finite algebra varieties.

Contribution

It provides a novel functorial method to construct algebras from congruences, preserving key properties and enabling applications in decidability and computational complexity.

Findings

Supernilpotence is decidable for certain finite algebras.

Subpower membership problem reduces to simpler subdirect product questions.

Polynomial time algorithm for specific finite algebras with a cube term.

Abstract

We present a functorial construction which, starting from a congruence $\alpha$ 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 $\alpha$ 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.