Minimal Partial Ultraclones on a Two-Element Set

Abstract

Set of functions from a finite set A to set of all subsets of A is a natural generalization of the set of many-valued functions on A (k-valued logic functions). These generalized functions, which are called multifunctions, often are regarded as incompletely defined functions. Partial functions, hyperfunctions, ultrafunctions, partial hyperfunctions, partial ultrafunctions on A are arised depending on the type of multifunctions and superposition. In the theory of discrete functions the classical problem is description of lattice of clones - sets of functions that are closed with respect to superposition and contain all projections. Full description of a lattice is obtained only for Boolean functions by Emil Post in 1921. Thus this problem remains open more than 90 years for other discrete functions. Because of difficulty of this problem lattice fragments are studied, for example, the minimum and maximum elements, different intervals. In particular, we note that the descriptions of all minimal clones are known for Boolean functions, 3-valued logic functions, partial functions on two-element and three-element sets, hyperfunctions and partial hyperfunctions on a two-element set. In this paper we consider ultrafunctions and partial ultrafunctions on a two-element set. A description of all minimal clones for these classes of multifunctions is got.

Keywords

• minimal clone
• partial ultraclone
• multifunction
• partial ultrafunction
• superposition