On the structure of equationally closed classes

We study the structure of equationally closed classes. We prove a theorem on representation of the graph of a function in an equationally closed class in the form of a union of the sets of values of special vector functions. For any k >= 2 we establish the equational generability of any equationally closed class in P-k by the set of all its k-place functions. We find all equationally pre-complete classes in Pk and prove a criterion of equational completeness. Some results are extended from equationally closed classes to positively closed classes.