Eilenberg correspondence for Stone recognition

TL;DR

This paper extends the Eilenberg correspondence to topological algebras, linking language varieties with Stone topological algebra varieties, and demonstrates how this framework surpasses classical regular language theory.

Contribution

It introduces a novel Eilenberg correspondence for Stone recognition, connecting language varieties with ordered Stone topological algebras, and extends the theory beyond regular languages.

Findings

Established an Eilenberg correspondence for Stone recognition.

Showed how to characterize languages outside regular languages using Stone completions.

Provided a method to prove languages do not belong to certain varieties.

Abstract

We develop and explore the idea of recognition of languages (in the general sense of subsets of topological algebras) as preimages of clopen sets under continuous homomorphisms into Stone topological algebras. We obtain an Eilenberg correspondence between varieties of languages and varieties of ordered Stone topological algebras and a Birkhoff/Reiterman-type theorem showing that the latter may me defined by certain pseudo-inequalities. In the case of classical formal languages, of words over a finite alphabet, we also show how this extended framework goes beyond the class of regular languages by working with Stone completions of minimal automata, viewed as unary algebras. This leads to a general method for showing that a language does not belong to a variety of languages, expressed in terms of sequences of pairs of words, which is illustrated when the class consists of all finite intersections of context-free languages.