Syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that n is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix-, bifix-, and factor-free regular languages, conjecture tight upper bounds on their size to be (n− 1) + (n− 2), (n− 1) +(n− 2) +(n− 3)2, and (n− 1) +(n− 3)2 +1, respectively, and exhibit languages with these syntactic complexities. keyword bifix-free, factor-free, finite automaton, monoid, prefix-free, regular language, reversal, semigroup, suffix-free, syntactic complexity