Brzozowski, JanuszSzykuła, Marek2017-09-282017-09-282017-09-05http://dx.doi.org/10.1016/j.ic.2017.08.014http://hdl.handle.net/10012/12500The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.ic.2017.08.014 © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/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.enAttribution-NonCommercial-NoDerivatives 4.0 InternationalRegular languageSuffix-freeSyntactic complexityTransition semigroupUpper boundArticle