Syntactic complexity of bifix-free languages

We study the properties of syntactic monoids of bifix-free regular languages. In particular, we solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a bifix-free language with state complexity $n$ is at most $(n-1{)}^{n-3}+(n-2{)}^{n-3}+(n-3)2^{n-3}$ for $n\ge 6$. The main proof uses a large construction with the method of injective function. Since this bound is known to be reachable, and the values for $n \le 5$areknown,thiscompletelysettlestheproblem\mathrm{.}Wealsoprovethat$(n-2)^{n-3} + (n-3)2^{n-3} - 1$ is the minimal size of the alphabet required to meet the bound for $n\ge 6$. Finally, we show that the largest transition semigroups of minimal DFAs which recognize bifix-free languages are unique up to renaming the states.