Enumeration of function and bases of three-valued set logic under compositions with Boolean functions

Abstract

This paper discusses some classification and enumeration problems in r-valued set logic, which is the logic of functions mapping n-tuples of subsets into subsets over r values. Boolean functions are convenient choice as building blocks in the design of set logic functions. Weak maximal sets are these containing all Boolean functions. The authors give the number of n-ary functions in each weak maximal set and and some properties of intersections of weak maximal sets in r-valued set logic. These properties are used to classify all three-valued set logic functions according to the weak maximal sets they belong to. They prove that there are 29 such classes of functions and give a unary function representative for each of them. Finally, they find the number of n-ary weak Sheffer functions of three-valued set logic, i.e. functions which are complete under compositions with Boolean functions.<>