The subpower membership problem of 2-nilpotent algebras

TL;DR

This paper investigates the computational complexity of the subpower membership problem for 2-nilpotent Mal'tsev algebras, showing it is solvable in polynomial time for this class and developing tools for broader classes.

Contribution

It proves polynomial-time solvability of SMP for a large class of 2-nilpotent Mal'tsev algebras and introduces new tools for future research on nilpotent Mal'tsev algebras.

Findings

SMP(A) is polynomial-time solvable for 2-nilpotent Mal'tsev algebras.

Develops new algebraic tools for analyzing subpower membership.

Provides a foundation for future work on general nilpotent Mal'tsev algebras.

Abstract

The subpower membership problem SMP(A) of a finite algebraic structure A asks whether a given partial function from A^k to A can be interpolated by a term operation of A, or not. While this problem can be EXPTIME-complete in general, Willard asked whether it is always solvable in polynomial time if A is a Mal'tsev algebras. In particular, this includes many important structures studied in abstract algebra, such as groups, quasigroups, rings, Boolean algebras. In this paper we give an affirmative answer to Willard's question for a big class of 2-nilpotent Mal'tsev algebras. We furthermore develop tools that might be essential in answering the question for general nilpotent Mal'tsev algebras in the future.