Classification of functions and enumeration of bases of set logic under Boolean compositions

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. B-maximal sets are maximal sets containing all Boolean functions, where Boolean functions are those obtained from ∪, ∩ and ^- by composition (constants are not involved in the compositions). We give the number of n-place functions in each B-maximal set and find some properties of intersection of B-maximal sets in r-valued set logic. These properties are used to classify all 2-valued and 3-valued set logic functions according to the B-maximal sets to which they belong to. We prove that there are 8 and 200 such classes of functions respectively in 2-valued and 3-valued set logic. For each class of functions we give a one-place example function and its total number of one-place set logic functions. Finally, we study the B-Sheffer functions, i.e. functions which are complete under compositions with Boolean functions. We find the number of n-place B-Sheffer functions of 2-valued set logic and give a lower bound and an upper bound on the number of n-place B-Sheffer functions of 3-valued set logic. Also we enumerate all the classes of B-bases of 2-valued and 3-valued set logic