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On the classes of Boolean functions generated by maximal partial ultraclones
S. A. Badmaev Buryat State University,
Ulan-Ude, Russian Federation
Abstract:
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.
Keywords:
multifunction, superposition, clone, ultraclone, maximal clone.
UDC:
519.716
MSC: 8A99,
03B50 Received: 01.02.2019
DOI:
10.26516/1997-7670.2019.27.3
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