2021/01/01 by Jack, Trevor
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2101.00324
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 AC0: (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.