Geodesic
A completeness theorem in the class of quasimonotonic functions
Diskretnyj analiz i issledovanie operacij, Tome 13 (2006) no. 3, pp. 62-82

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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 $k\to\infty$. 
@article{DA_2006_13_3_a4,
     author = {N. G. Parvatov},
     title = {A completeness theorem in the class of quasimonotonic functions},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {62--82},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2006_13_3_a4/}
}
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N. G. Parvatov. A completeness theorem in the class of quasimonotonic functions. Diskretnyj analiz i issledovanie operacij, Tome 13 (2006) no. 3, pp. 62-82. http://geodesic.mathdoc.fr/item/DA_2006_13_3_a4/