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We study the syntactic complexity of finite/cofinite, definite and reversedefinite languages. The syntactic complexity of a class of languages is definedas the maximal size of syntactic semigroups of languages from the class, takenas a function of the state complexity n of the languages. We prove that (n-1)!is a tight upper bound for finite/cofinite languages and that it can be reachedonly if the alphabet size is greater than or equal to (n-1)!-(n-2)!. We provethat the bound is also (n-1)! for reverse definite languages, but the minimalalphabet size is (n-1)!-2(n-2)!. We show that \lfloor e\cdot (n-1)!\rfloor is alower bound on the syntactic complexity of definite languages, and conjecturethat this is also an upper bound, and that the alphabet size required to meetthis bound is \floor{e \cdot (n-1)!} - \floor{e \cdot (n-2)!}. We prove theconjecture for n\le 4.

Syntactic Complexity of Finite/Cofinite, Definite, and Reverse Definite Languages