Characterizing classes of regular languages using prefix codes of bounded synchronization delay

TL;DR

This paper characterizes classes of regular languages using prefix codes with bounded synchronization delay, linking their syntactic monoids to subgroup varieties, and introduces local Rees products for monoid decomposition.

Contribution

It extends Schützenberger's work to all subgroup varieties H, providing a new monoid decomposition method using local Rees products and addressing open questions about Rees product closure.

Findings

Languages in SD_H(A^∞) have syntactic monoids with all subgroups in H.

The method applies directly to finite and infinite words.

Decomposition uses a singly exponential number of operations.

Abstract

In this paper we continue a classical work of Sch\"utzenberger 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 $G\in H$, but no complementation. In our notation this leads to the language classes $SD_G(A^\infty)$ and $SD_H(A^\infty$). Our main result shows that $SD_H(A^\infty)$ always corresponds to the languages having syntactic monoids where all subgroups are in $H$. Sch\"utzenberger 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 Kl\'ima about varieties that are closed under Rees products.