Some properties of $P$-sets of finite-automaton functions

Classes of deterministic finite functions are considered in the paper and each state of those functions realizes a function from some closed class $D$ of $k$-valued logic ($P$-sets). It is proved that there exists continuum of precomplete classes $C$ containing an arbitrary $P$-set. The problem of existence of a completeness criterion for systems containing $P$-sets is also considered.