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.

中文翻译：

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关于没有布尔运算的语言种类

Eilenberg 的变体定理通过在正则语言的性质和识别它们的有限幺半群的性质之间建立正式的对应关系，标志着正则语言代数理论的一个里程碑。受量子有限自动机接受的语言类的启发，我们引入了正则语言的基本变体，弱化了 Eilenberg 的原始概念，即不需要在任何布尔运算下闭包，并为它们证明了变体定理。为此，我们通过格双模研究语言的代数识别，推广 Klima 和 Polak 的格代数，并利用代数完全分配格和偏序集之间的对偶性。