Geodesic
$A$-closed classes of many-valued logic that contain constants
Diskretnaya Matematika, Tome 10 (1998) no. 3, pp. 10-26 
S. S. Marchenkov. $A$-closed classes of many-valued logic that contain constants. Diskretnaya Matematika, Tome 10 (1998) no. 3, pp. 10-26. http://geodesic.mathdoc.fr/item/DM_1998_10_3_a1/
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     author = {S. S. Marchenkov},
     title = {$A$-closed classes of many-valued logic that contain constants},
     journal = {Diskretnaya Matematika},
     pages = {10--26},
     year = {1998},
     volume = {10},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1998_10_3_a1/}
}
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The $A$-closure in the set $P_k$ of functions of $k$-valued logic is defined as the closure with respect to the operations of superposition and passing to the dual functions for even permutations of the set $E_k=\{0,1,\ldots, k-1\}$. For any $k$, $k\ge4$, all $A$-closed classes of $P_k$ containing constants are described. As a corollary, we obtain the description of all $A$-closed classes contained in the Slupecki class as well as an $A$-classification of the symmetric semigroup of mappings of the set $E_k$ into itself.This research was supported by the Russian Foundation for Basic Research, grant 97–01–00089.