On the classes of Boolean functions generated by maximal partial ultraclones

S. A. Badmaev

The Bulletin of Irkutsk State University. Series Mathematics, Tome 27 (2019), pp. 3-14

Résumé

The sets of multifunctions are considered. A multifunction on a finite set $A$ is a function defined on the set $A$ and taking its subsets as values. Obviously, superposition in the usual sense does not work when working with multifunctions. Therefore, we need a new definition of superposition. Two ways of defining superposition are usually considered: the first is based on the union of subsets of the set $A$, and in this case the closed sets containing all the projections are called multiclones, and the second is the intersection of the subsets of $A$, and the closed sets containing all projections are called partial ultraclones. The set of multifunctions on $A$ on the one hand contains all the functions of $|A|$-valued logic and on the other, is a subset of functions of $2^{|A|}$-valued logic with superposition that preserves these subsets. For functions of $k$-valued logic, the problem of their classification is interesting. One of the known variants of the classification of functions of $k$-valued logic is one in which functions in a closed subset $B$ of a closed set $M$ can be divided according to their belonging to the classes that are complete in $M$. In this paper, the subset of $B$ is the set of all Boolean functions, and the set of $M$ is the set of all multifunctions on the two-element set, and the partial maximal ultraclones are pre-complete classes.

Détail

Keywords: multifunction, clone, ultraclone
Mots-clés : superposition, maximal clone.