On the Classes of Boolean Functions Generated by Maximal Partial Ultraclones

The sets of multifunctions are considered. A multifunction on a finite set A is a function de fined 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 vertical bar A vertical bar-valued logic and on the other, is a subset of functions of 2(vertical bar A vertical bar)-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.