A variant of the proof of a completeness criterion for functions of $k$-valued logic

We suggest a new version of the proof of completeness criterion in terms of precomplete classes of $k$-valued logic functions. As before, the basis of the proof is the idea of preserving relations (predicates) by these functions, which was suggested by Post and developed by Yablonskii, Kuznetsov, Rosenberg, Lo Chu Kai, Kudryavtsev, Zakharova, etc. The essence of our reasoning consists in a rather different approach to the process of finding relations such that the classes preserving them coincide with precomplete ones. This approach arose while studying the $r$-completeness problem in the class of determinate functions. It allows us to shorten the well-known proof due to Rosenberg.
The work was supported by the Russian Foundation for Basic Research, grant 93–01–00382.