Partial Orders on Fuzzy Truth Value Algebras

The elements of the truth value algebra of type-2 fuzzy sets consist of 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. First, we consider the more general case of all mappings of a bounded chain with an involution into a complete chain, and delimit some of the properties of the resulting algebra. These include two binary operations each of which give a partial order on its elements. These partial orders and their intersection are the principal interest. We specialize this situation in two cases: (1) All mappings of the unit interval into itself, the original version of the truth value algebra of type-2 fuzzy sets introduced by Zadeh, and (2) all mappings of a finite chain into another finite chain. Again, each of these two cases yields two partial orders on the elements of the resulting algebras, and in each case, our principal interest is in these partial orders and their intersection.