Syntactic semigroups

This chapter gives an overview on what is often called the algebraic theory of finite automata. It deals with languages, automata and semigroups, and has connections with model theory in logic, boolean circuits, symbolic dynamics and topology. Kleene’s theorem [70] is usually considered as the foundation of this theory. It shows that the class of recognizable languages (i.e. recognized by finite automata), coincides with the class of rational languages, which are given by rational expressions. The definition of the syntactic monoid, a monoid canonically attached to each language, was first given in an early paper of Rabin and Scott [128], where the notion is credited to Myhill. It was shown in particular that a language is recognizable if and only if its syntactic monoid is finite. However, the first classification results were rather related to automata [89]. A break-through was realized by Schützenberger in the mid sixties [144]. Schützenberger made a non trivial use of the syntactic monoid to characterize an important subclass of the rational languages, the star-free languages. Schützenberger’s theorem states that a language is star-free if and only if its syntactic monoid is finite and aperiodic. This theorem had a considerable influence on the theory. Two other important “syntactic” characterizations were obtained in the early seventies: Simon [152] proved that a language is piecewise testable if and only if its syntactic monoid is finite and J -trivial and Brzozowski-Simon [41] and independently, McNaughton [86] characterized the locally testable languages. These successes settled the power of the semigroup approach, but it was Eilenberg who discovered the appropriate framework to formulate this type of results [53]. A variety of finite monoids is a class of monoids closed under taking submonoids, quotients and finite direct product. Eilenberg’s theorem states that varieties of finite monoids are in one to one correspondence with certain classes of recognizable languages, the varieties of languages. For instance,