Syntactic complexity of suffix-free languages

We solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a suffix-free language with n left quotients (that is, with state complexity n) is at most (n - 1)(n-2) + n - 2 for n >= 6. Since this bound is known to be reachable, this settles the problem. We also reduce the alphabet of the witness languages reaching this bound to five letters instead of n + 2, and show that it cannot be any smaller. Finally, we prove that the transition semigroup of a minimal deterministic automaton accepting a witness language is unique for each n. (c) 2017 Elsevier Inc. All rights reserved.