On recognizable languages in divisibility monoids

Abstract

Kleene's theorem on recognizable languages in free monoids is considered to be of eminent importance in theoretical computer science. It has been generalized into various directions, including trace and rational monoids. Here, we investigate divisibility monoids which are deFIned by and capture algebraic properties suffcient to obtain a characterization of the recognizable languages by certain rational expressions as known from trace theory. The proofs rely on Ramsey's theorem, distributive lattice theory and on Hashigushi's rank function generalized to our divisibility monoids. We obtain Ochmański's theorem on recognizable languages in free partially commutative monoids as a consequence.