Partial orders on the truth value algebra of finite type-2 fuzzy sets

The elements of the truth value algebra of type-2 fuzzy sets are all mappings of the unit interval into itself, with operations given by various convolutions of the pointwise operations. This algebra can be specialized and generalized in various interesting ways. Here we replace each copy of the unit interval by a finite chain, and define operations analogously. Among these are two binary operations which are idempotent, commutative, and associative, and thus each yields a partial order. Here we investigate these partial orders. It is easy to show that each is a lattice. One principal concern is with the partial order given by the intersection of these two partial orders, which we call the double order. Some results are that two functions are incomparable under the double order unless they have the same least upper bound, and that the set of functions with a given least upper bound is a lattice under the double order. Thus the algebra itself is an antichain of lattices in a natural way.