Completeness criteria in set-valued logic under compositions with union and intersection

This paper discusses the Boolean completeness 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 circuits. Given a set S of Boolean functions, a set of functions F is S-complete if any set logic function can be composed from F once all Boolean functions from S are added to F. For the special case U = {union of all the sets, intersection of all the sets}, we characterize all U-maximal sets in r-valued set logic. A set F is then U-complete if it is not a subset of any of these U-maximal sets, which is a completeness criterion in r-valued set logic under compositions with U functions.