Finite J-Trivial Monoids and Partially Ordered Monoids

Abstract

An important result in the theory of automata, due to Imre Simon, characterizes the recognizable languages whose syntactic monoids are J-trivial. This theorem has an easy, although not entirely obvious, restatement as a global structure theorem for finite J-trivial monoids, asserting that every finite J-trivial monoid is a quotient of a finite monoid admitting a partial order compatible with the multiplication and having 1 as the maximum element. We have proved the theorem in this form using semigroup expansion techniques. In the present note we discuss the background and significance of this work and give a brief sketch of our proof. The full details of the proof will appear elsewhere [ST].