Membership and Conjugacy in Inverse Semigroups

Abstract: The membership problem for an algebraic structure asks whether a given element is contained in some substructure, which is usually given by generators. In this work we study the membership problem, as well as the conjugacy problem, for finite inverse semigroups. The closely related membership problem for finite semigroups has been shown to be PSPACE-complete in the transformation model by Kozen (1977) and NL-complete in the Cayley table model by Jones, Lien, and Laaser (1976). More recently, both the membership and the conjugacy problem for finite inverse semigroups were shown to be PSPACE-complete in the partial bijection model by Jack (2023). Here we present a more detailed analysis of the complexity of the membership and conjugacy problems parametrized by varieties of finite inverse semigroups. We establish dichotomy theorems for the partial bijection model and for the Cayley table model. In the partial bijection model these problems are in NC (resp. NP for conjugacy) for strict inverse semigroups and PSPACE-complete otherwise. In the Cayley table model we obtain general LOGSPACE-algorithms as well as NPOLYLOGTIME upper bounds for Clifford semigroups and LOGSPACE-completeness otherwise. Furthermore, by applying our findings, we show the following: the intersection non-emptiness problem for inverse automata is PSPACE-complete even for automata with only two states; the subpower membership problem is in NC for every strict inverse semi-group and PSPACE-complete otherwise; the minimum generating set and the equation satisfiability problems are in NP for varieties of finite strict inverse semigroups and PSPACE-complete otherwise.