Varieties

TL;DR

This paper explores the theory of varieties in universal algebra, emphasizing their role in classifying regular languages, and discusses key aspects like decidability, operations, and logical characterizations.

Contribution

It provides an overview of the fundamental concepts of varieties and their application to regular languages, building on recent work and highlighting important theoretical aspects.

Findings

Varieties serve as a crucial tool for classifying regular languages.

Decidability of certain properties within varieties is analyzed.

Operations on languages within the variety framework are characterized.

Abstract

This text is devoted to the theory of varieties, which provides an important tool, based in universal algebra, for the classification of regular languages. In the introductory section, we present a number of examples that illustrate and motivate the fundamental concepts. We do this for the most part without proofs, and often without precise definitions, leaving these to the formal development of the theory that begins in Section 2. Our presentation of the theory draws heavily on the work of Gehrke, Grigorieff and Pin (2008) on the equational theory of lattices of regular languages. In the subsequent sections we consider in more detail aspects of varieties that were only briefly evoked in the introduction: Decidability, operations on languages, and characterizations in formal logic.