A completeness theorem in the class of quasimonotonic functions

The problem of functional completeness is solved in the class $Q_L$ of quasimonotonic functions on a finite semilattice $L$ under superposition with all so-called weakly essential functions. An effective description of the precomplete classes in $Q_L$ containing all weakly essential functions is given. The asymptotics of the number of such classes on the semilattice of all nonempty subsets of a $k$-element set is found as $k\to\infty$.