On the closed classes of $k$-valued logic functions taking no more than three values

The paper considers $k$-valued logic functions where $k=2^m$, $m\geq2$. A $\beta$-closure operator is defined based on their encoding in the binary numeral system. A special mapping of all $\beta$-closed classes to a set of closed classes of Boolean functions is denoted. The cardinality of a set of $\beta$-closed classes which are mapped to a class $\mathcal B$ and contain only functions taking no more than three values is studied in this paper for each class $\mathcal B$ of Boolean functions.