ON SEMIGROUPS WITH PSPACE-COMPLETE SUBPOWER MEMBERSHIP PROBLEM

Fix a finite semigroup S and let a(1), ..., 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), ..., a(k). For combinatorial Rees matrix semigroups we establish a dichotomy result: if the corresponding matrix is of a certain form, then the SMP is in P; otherwise it is NP-complete. For combinatorial Rees matrix semigroups with adjoined identity, we obtain a trichotomy: the SMP is either in P, NP-complete, or PSPACE-complete. This result yields various semigroups with PSPACE-complete SMP including the six-element Brandt monoid, the full transformation semigroup on three or more letters, and semigroups of all n by n matrices over a field for n >= 2.