Problem of checking the functional completeness in the class of piecewise parallel functions

The author showed previously that it is possible to implement an arbitrary piecewise parallel function by circuits of linear elements and one nonlinear element implementing the θ-function. Thus, the functional completeness of a particular system of functions in the class of piecewise parallel functions was established. In this paper we consider the same class and study its completeness property for a system of functions whose closures contain all linear functions. We formulate and prove a criterion of functional completeness for such systems in terms of precomplete classes.