On the $\alpha$-superposition of functions of a $k$-valued logic

We reveal a relation between the operations of $\alpha$-completion and closure for the systems of functions of a $k$-valued logic. For $k=3,4$ we construct the $\alpha$-bases consisting of two binary operations. We prove that the complete system $T$ of functions of a 4-valued logic containing all permutations of the set $E_4=\{0,1,2,3\}$ and the operation of addition modulo 4 is not $\alpha$-complete, whereas its $\alpha$-completion $[T]_\alpha$ will be an $\alpha$-complete system.