On the Complexity of Properties of Partial Bijection Semigroups
Abstract
We examine the computational complexity of problems in which we are given generators for a partial bijection semigroup and asked to check properties of the generated semigroup. We prove that the following problems are in AC: (1) enumerating left and right identities and (2) checking if the semigroup is completely regular. We also describe a nondeterministic logspace algorithm for checking if an inverse semigroup given by generators satisfies a fixed semigroup identity that may involve a unary inverse operation. We conclude with an alternative proof that checking membership of a given idempotent in a partial bijection semigroup is a PSPACE-complete problem. The proof reduces from the well-known PSPACE-complete Rectangle Tiling Problem, thereby illustrating a connection between Wang tilings and partial bijection semigroups.
Cite
@article{arxiv.2101.00324,
title = {On the Complexity of Properties of Partial Bijection Semigroups},
author = {Trevor Jack},
journal= {arXiv preprint arXiv:2101.00324},
year = {2025}
}