The equation solvability problem over nilpotent Mal'cev algebras

TL;DR

This paper extends polynomial-time solvability of the equation problem from finite nilpotent groups and rings to all finite supernilpotent Mal'cev algebras, but shows some nilpotent Mal'cev algebras have coNP-complete identity checking.

Contribution

It generalizes the polynomial-time solvability of the equation problem to finite supernilpotent Mal'cev algebras and provides a counterexample for non-supernilpotent cases.

Findings

Equation solvability over finite supernilpotent Mal'cev algebras is in P

Existence of a nilpotent, non-supernilpotent Mal'cev algebra with coNP-complete identity checking

Extension of known results from groups and rings to broader algebraic structures

Abstract

By a result of Horv\'ath the equation solvability problem over finite nilpotent groups and rings is in P. We generalize his result, showing that the equation solvability over every finite supernilpotent Mal'cev algebra is in P. We also give an example of a nilpotent, but not supernilpotent Mal'cev algebra, whose identity checking problem is coNP-complete.