On closed classes in partial $k$-valued logic that contain the class of monotone functions

Let $A$ be a precomplete class (a maximal clone) in $k$-valued logic and $T(A)$ be the family of all closed classes (under superposition) in partial $k$-valued logic that contain $A$. A simple test is put forward capable of finding out from a partial order defining the precomplete class $A$ of monotone functions whether the family $T(A)$ is finite or infinite. This completes the solution of the problem of finiteness of $T(A)$ for all precomplete classes of $k$-valued logic. The proof depends on new families of closed classes founded by the author of the present paper.