On properties of maximal homomorphic prototypes of $k$-valued logic in $l$-valued logic

The properties of the set $\mathcal L_{k}^{l}$ of all closed subsets of $l$-valued logic $P_l$, which may be reflected homomorphically onto $P_k$ are investigated. We determined all maximal elements of $\mathcal L_{k}^{l}$ and proved that any maximal element is generated by a single function. The asymptotic formula for the number of in pairs nonisomorphic maximal elements of $\bigcup_{k=2}^l\mathcal L_{k}^{l}$ was obtained.