Sets with type-2 operations

The algebra of truth values of type-2 fuzzy sets consists of all mappings of the unit interval to itself, with type-2 operations that are convolutions of ordinary max and min operations. This paper is concerned with a special subalgebra of this truth value algebra, namely the set of nonzero functions with values in the two-element set {0, 1}. This algebra can be identified with the set of all non-empty subsets of the unit interval, but the operations are not the usual union and intersection. We give simplified descriptions of the operations and derive the basic algebraic properties of this algebra, including the identification of its automorphism group. We also discuss some subalgebras and homomorphisms between them and look briefly at t-norms on this algebra of sets.