On the complexity of inverse semigroup conjugacy

TL;DR

This paper explores the computational complexity of conjugacy problems in finite inverse semigroups, providing polynomial-time algorithms and complexity classifications for various decision problems related to semigroup properties.

Contribution

It introduces new polynomial-time algorithms for conjugacy checks and establishes complexity classifications for several decision problems in inverse semigroups.

Findings

Polynomial-time algorithms for conjugacy checks

Complexity classifications for nilpotency and R-triviality

Partition covering properties of conjugacy relations

Abstract

We investigate the computational complexity of various decision problems related to conjugacy in finite inverse semigroups. We describe polynomial-time algorithms for checking if two elements in such a semigroup are ~p conjugate and whether an inverse monoid is factorizable. We describe a connection between checking ~i conjugacy and checking membership in inverse semigroups. We prove that ~o and ~c are partition covering for any countable set and that ~p, ~p* , and ~tr are partition covering for any finite set. Finally, we prove that checking for nilpotency, R-triviality, and central idempotents in partial bijection semigroups are NL-complete problems and we extend several complexity results for partial bijection semigroups to inverse semigroups.