On logical descriptions of regular languages

Abstract

There are many examples in the research literature of families of regular languages defined by purely model-theoretic means (that is, in terms of the kinds of formulas of predicate logic used to define them) that can be characterized algebraically (that is, in terms of the syntactic monoids or syntactic morphisms of the languages). In fact the existence of such algebraic characterizations appears to be the rule. The present paper gives an explanation of the phenomenon: A generalization of Eilenberg’s variety theorem is proved, and then applied to logic. We find that a very wide assortment of families of regular languages defined in model-theoretic terms form varieties in this new sense, and that consequently membership in the family depends only on the syntactic morphism of the language.