The 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.

The elements of the truth value algebra of type-2 fuzzy sets are the 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 the elements of that algebra. These partial orders and their intersection are the principal objects of 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.

The main concern of this paper is with the equations satisfied by the algebra of truth values of type-2 fuzzy sets. That algebra has elements all mappings from the unit interval into itself with operations given by certain convolutions of operations on the unit interval. There are a number of positive results. Among them is a decision procedure, similar to the method of truth tables, to determine when an equation holds in this algebra. One particular equation that holds in this algebra implies that every subalgebra of it that is a lattice is a distributive lattice. It is also shown that this algebra is locally finite. Many questions are left unanswered. For example, we do not know whether or not this algebra has a finite equational basis, that is, whether or not there is a finite set of equations from which all equations satisfied by this algebra follow. This and various other topics about the equations satisfied by this algebra will be discussed.

We introduce modal compact Hausdorff spaces as generalizations of modal spaces, and show these are coalgebras for the Vietoris functor on compact Hausdorff spaces. Modal compact regular frames and modal de Vries algebras are introduced as algebraic counterparts of modal compact Hausdorff spaces, and dualities are given for the categories involved. These extend the familiar Isbell and de Vries dualities for compact Hausdorff spaces, as well as the duality between modal spaces and modal algebras. As the first step in the logical treatment of modal compact Hausdorff spaces, a version of Sahlqvist correspondence is given for the positive modal language.

The algebra of truth values of type-2 fuzzy sets contains isomorphic copies of the algebra of truth values of type-1 fuzzy sets and the algebra of truth values of interval-valued fuzzy sets. The algebra of truth values of type-2 fuzzy sets is not a lattice, but these subalgebras are lattices, and in fact, are complete lattices. There are many other subalgebras that are lattices, for example, the subalgebra of convex normal elements. This paper examines several subalgebras which are lattices, with the goal of determining whether or not they are complete.

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