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 $k\ge2$ 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 precomplete classes in $P_k$ and prove a criterion of equational completeness. Some results are extended from equationally closed classes to positively closed classes.