On Varieties of Ordered Automata

The classical Eilenberg correspondence, based on the concept of the syntactic monoid, relates varieties of regular languages with pseudovarieties of finite monoids. Various modifications of this correspondence appeared, with more general classes of regular languages on one hand and classes of more complex algebraic structures on the other hand. For example, classes of languages need not be closed under complementation or all preimages under homomorphisms, while monoids can be equipped with a compatible order or they can have a distinguished set of generators. Such generalized varieties and pseudovarieties also have natural counterparts formed by classes of finite (ordered) automata. In this paper the previous approaches are combined. The notion of positive C-varieties of ordered semiautomata (i.e. no initial and final states are specified) is introduced and their correspondence with positive C-varieties of languages is proved.