  The Eilenberg correspondence relates varieties of regular languages to
pseudovarieties of finite monoids. Various modifications of this correspondence
have been found with more general classes of regular languages on one hand and
classes of more complex algebraic structures on the other hand. It is also
possible to consider classes of automata instead of algebraic structures as a
natural counterpart of classes of languages. Here we deal with the
correspondence relating positive $\mathcal C$-varieties of languages to
positive $\mathcal C$-varieties of ordered automata and we present various
specific instances of this correspondence. These bring certain well-known
results from a new perspective and also some new observations. Moreover,
complexity aspects of the membership problem are discussed both in the
particular examples and in a general setting.
