The subpower membership problem for bands

TL;DR

This paper investigates the computational complexity of the subpower membership problem for bands, establishing a dichotomy between polynomial-time solvability and NP-completeness based on the band’s algebraic properties.

Contribution

It provides a classification of bands into tractable and NP-complete cases for SMP, identifies the maximal variety with tractable SMP, and presents the first example of two algebras with the same variety but different SMP complexities.

Findings

Dichotomy: SMP for bands is either in P or NP-complete.

Identified the largest variety of bands with tractable SMP.

Constructed the first example of algebras with same variety but different SMP complexities.

Abstract

Fix a finite semigroup $S$ and let $a_1,\ldots,a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) for $S$ asks whether $b$ can be generated by $a_1,\ldots,a_k$. For bands (idempotent semigroups), we provide a dichotomy result: if a band $S$ belongs to a certain quasivariety, then $SMP(S)$ is in P; otherwise it is NP-complete. Furthermore we determine the greatest variety of bands all of whose finite members induce a tractable SMP. Finally we present the first example of two finite algebras that generate the same variety and have tractable and NP-complete SMPs, respectively.