On Language Varieties Without Boolean Operations

TL;DR

This paper introduces basic varieties of regular languages that lack boolean closure, extending algebraic recognition theory through lattice bimodules and duality, with implications for quantum automata.

Contribution

It defines and proves a variety theorem for basic varieties of regular languages without boolean closure, generalizing classical algebraic language theory.

Findings

Established a variety theorem for basic varieties of regular languages

Developed algebraic recognition via lattice bimodules

Utilized duality between distributive lattices and posets

Abstract

Eilenberg's variety theorem marked a milestone in the algebraic theory of regular languages by establishing a formal correspondence between properties of regular languages and properties of finite monoids recognizing them. Motivated by classes of languages accepted by quantum finite automata, we introduce basic varieties of regular languages, a weakening of Eilenberg's original concept that does not require closure under any boolean operations, and prove a variety theorem for them. To do so, we investigate the algebraic recognition of languages by lattice bimodules, generalizing Klima and Polak's lattice algebras, and we utilize the duality between algebraic completely distributive lattices and posets.