Title: On Varieties of Ordered Automata,
Abstract: 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
$\mathcal C$-varieties of ordered semiautomata (i.e. no initial and final
states are specified) is introduced and their correspondence with positive
$\mathcal C$-varieties of languages is proved.