Homomorphisms onto effectively separable algebras

We study variations of the concept of separable enumeration and, basing on that, describe a series of algorithmic and algebraic concepts. In this framework we characterize negative equivalences, describe enumerated algebras with the most general separability conditions, give a separability criterion for the enumerated algebras satisfying the descending chain condition for the lattices of congruences, and consider some questions related to the algorithmic complexity of enumerations of the algebras satisfying various separability axioms.