  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.
