Abstract. 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) asks whether b can be generated by a 1 , . . . , a k . If S is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in nk. For semigroups this problem always lies in PSPACE. We show that the SMP for a full transformation semigroup on 3 or more letters is actually PSPACE-complete, while on 2 letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup S embeds into a direct product of a Clifford semigroup and a nilpotent semigroup, then SMP(S) is in P; otherwise it is NP-complete.

Abstract. 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 bands (idempotent semigroups), we provide a dichotomy result: if a band S belongs to a certain quasivariety, then SMP(S) is in P; otherwise it is NP-complete.Furthermore we determine the greatest variety of bands all of whose finite members induce a tractable SMP. Finally we present the first example of two finite algebras that generate the same variety and have tractable and NPcomplete SMPs, respectively.