Syntactic complexity of bifix-free regular 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)2n−3 for n⩾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⩽5 are known, this completely settles the problem. We also prove that (n−2)n−3+(n−3)2n−3−1 is the minimal size of the alphabet required to meet the bound for n⩾6. Finally, we show that the largest transition semigroups of minimal DFAs which recognize bifix-free languages are unique up to renaming the states.