Eilenberg's variety theorem without Boolean operations

Eilenberg's variety theorem marked a milestone in the algebraic theory of regular languages by establishing a formal correspondence between properties of regular languages 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 utilize the duality between algebraic completely distributive lattices and posets. (c) 2022 Elsevier Inc. All rights reserved.