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 ≥ 7. 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 such a witness language is unique for each n ≥ 7.
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机译：我们解决了一个关于句法复杂性的开放问题：我们证明了使用N个左引用的后缀语法的语法半群的基数（即，具有状态复杂性n）最多（n-1）〜（n-2 ）+ n - 2对于n≥7.由于已知该绑定可到达，这奠定了问题。我们还减少了证人语言的字母表，达到了五个字母而不是n + 2，并表明它不能更小。最后，我们证明，接受这种见证语言的最小确定性自动机的过渡半群对于每个N≥7是独一无二的。
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