Overview of Lattices of Convex Normal Functions

AbstractThe algebra of truth values of type-2 fuzzy sets is the set of maps from the unit interval to itself with convolution ordering. In applications of type-2 fuzzy sets, the full algebra is seldom used, but rather certain subalgebras that satisfy useful algebraic properties. The algebra of truth values of type-2 fuzzy sets is not itself a lattice, but the subalgebras considered here are lattices and, in fact, are complete distributive lattices. The subalgebras of special interest are the lattice of convex normal maps, the lattice of convex strongly normal maps, and the lattice of upper semicontinuous convex normal maps. We review and summarize some interesting properties of these subalgebras. A special feature of our treatment is a representation of these algebras as sets of monotone functions with pointwise order, making the operations more intuitive.