Conditions for the $\alpha$-completeness of systems of many-valued logic functions

Conditions of $\alpha$-completeness are presented for systems of functions of $k$-valued logic consisting of functions whose one-place subfunctions obtained by arbitrarily fixing all variables except one, are substitutions. For $k\ne2,3,4$, $\alpha$-complete systems of two binary operations with right cancellation are constructed, and for $k=2$, the nonexistence of finite $\alpha$-complete systems is proved.