Substitution Principle and semidirect products

TL;DR

This paper extends the logical and algebraic framework of regular language recognition to more general languages using Stone duality, introducing semidirect products of Boolean spaces with internal monoids and generalizing key theorems.

Contribution

It formalizes the Substitution Principle for arbitrary languages via Stone duality and introduces semidirect products of BiMs, extending classical results to broader language classes.

Findings

Generalized Decomposition Theorem for BiMs

Constructed topo-algebraic recognizers for logical fragments

Extended semidirect product construction to BiMs

Abstract

In the classical theory of regular languages the concept of recognition by profinite monoids is an important tool. Beyond regularity, Boolean spaces with internal monoids (BiMs) were recently proposed as a generalization. On the other hand, fragments of logic defining regular languages can be studied inductively via the so-called "Substitution Principle". In this paper we make the logical underpinnings of this principle explicit and extend it to arbitrary languages using Stone duality. Subsequently we show how it can be used to obtain topo-algebraic recognizers for classes of languages defined by a wide class of first-order logic fragments. This naturally leads to a notion of semidirect product of BiMs extending the classical such construction for profinite monoids. Our main result is a generalization of Almeida and Weil's Decomposition Theorem for semidirect products from the profinite setting to that of BiMs. This is a crucial step in a program to extend the profinite methods of regular language theory to the setting of complexity theory.