On Function Classes in $P_3$ Precomplete with Respect to a Strengthened Closure Operator

A classical theorem of Post [1] describes five precomplete classes in the set of Boolean functions. In [2], it was shown that there exist $18$ precomplete classes of functions of three-valued logic. In [1], [2], the closure of sets of functions with respect to the substitution operator was studied. We consider two closure operators on functions of three-valued logic, which are obtained by supplementing the substitution operator by closures with respect to two identifications of function values, and prove the existence of three precomplete classes for one of these operators and five precomplete classes for the other.