A criterion of properness for a family of functions

Proper families of functions are a convenient apparatus for specification of large parametric classes of quasigroups and $n$-quasigroups. K. D. Tsaregorodtsev noticed that in the Boolean case a family is proper if and only if every mapping specified by the family or any of its subfamilies has a unique fixed point. We extend this result to the case of $k$-valued logics for $k > 2$. We also show that reencoding transformations used in the extended criterion enriched (in terms of composition) with consistent renumbering of variables and functions form the stabilizer of the set of all proper families of the given size.