Functional completeness and weak completeness in set logic

The functional completeness problems in r-valued set logic, which is the logic of functions mapping n-tuples of subsets into subset over r values, is discussed. It is shown that r-valued set logic is isomorphic to 2^r-valued logic, meaning that the well-known completeness criteria in multiple-valued Post algebras apply to set-valued logic. Since Boolean functions are convenient choice as building blocks in the design of set logic functions, the notion of weak completeness of a set is introduced; i.e., a set is weak complete if it becomes complete once all Boolean functions are added to the set. A full description of weak complete sets, weak maximal sets, weak bases, and weak Sheffer functions is given for the case of two-valued set logic