On maximal clones of partial ultrafunctions on a two-element set

Class of discrete functions from a finite set $A$ to set of all subsets of $A$ is a natural generalization of the class of many-valued functions on $A$ ($k$-valued logic functions). Functions of this type are called multifunctions or multioperations on $A$, and are used, for example, in the solution of the functional equations, in logical and technical systems. It is obvious that the superposition in the usual sense not appropriate for multifunctions, therefore, we need to expand the standard concept of superposition. We note there are various ways to determine the operation of superposition of multifunctions, one of such methods is considered in this paper. Multifunctions on $A$ with this superposition are called partial ultrafunctions on $A$. In this article starting set $A$ is two-element set and we consider classical problem of theory of discrete functions — description of clones — sets of functions closed with respect to the operation of superposition and containing all the projections. We got a description of the two maximal clones of partial ultrafunctions of a two-element set by the predicate approach.