Categories of Fuzzy Sets with Values in a Quantale or Projectale


Properties of the lattice L are reflected in the properties of the categories Set(L), Set(L)/(A,α), and the lattice U(A, α). The lattices U(A,α) best reflect the structures on the lattice if the structure is inherited by closed down segments and direct products. Operations at the level of the slice categories require distributivity too. The first object of this paper is to see how the additional structures given by an associative operation & which distributes over sups and hence has a right adjoint (a quantale in the sense of Rosenthal [7]) shows up at the three different levels for fuzzy sets. This structure generalizes the Heyting operation and the t-norms which give Lukaciewcz logic by allowing non-commutative connectives. Since a quantale structure does not always restrict to a quantale structure on down segments, we do not always get a quantale structure on U(A, α). The second object is to see what a projectale structure on L as in Khatcherian ([3]), given as a consistent family of projection functors with right adjoints, gives in each of the levels. Here we no longer have a binary operation, though we do have notions of projection onto an element of L .