Characterizing Classes of Regular Languages Using Prefix Codes of Bounded Synchronization Delay

In this paper we continue a classical work of Schützenberger on codes with bounded synchronization delay. He was interested to characterize those regular languages where the groups in the syntactic monoid belong to a variety H. He allowed operations on the language side which are union, intersection, concatenation and modified Kleene-star involving a mapping of a prefix code of bounded synchronization delay to a group G ∈ H, but no complementation. In our notation this leads to the language classes SDG(A ∞) and SDH(A ∞). Our main result shows that SDH(A ∞) always corresponds to the languages having syntactic monoids where all subgroups are in H. Schützenberger showed this for a variety H if H contains Abelian groups, only. Our method shows the general result for all H directly on finite and infinite words. Furthermore, we introduce the notion of local Rees products which refers to a simple type of classical Rees extensions. We give a decomposition of a monoid in terms of its groups and local Rees products. This gives a somewhat similar, but simpler decomposition than in Rhodes’ synthesis theorem. Moreover, we need a singly exponential number of operations, only. Finally, our decomposition yields an answer to a question in a recent paper of Almeida and Kĺıma about varieties that are closed under Rees products.