The subpower membership problem for bands

Fix a finite semigroup S and let a1,…,ak,b be tuples in a direct power Sn. The subpower membership problem (SMP) for S asks whether b can be generated by a1,…,ak. For bands (idempotent semigroups), we provide a dichotomy result: if a band S belongs to a certain quasivariety, then is in P; otherwise it is NP-complete.Furthermore we determine the greatest variety of bands all of whose finite members induce a tractable . Finally we present the first example of two finite algebras that generate the same variety and have tractable and NP-complete SMPs, respectively.