Convex normal functions revisited

The lattice Lu of upper semicontinuous convex normal functions with convolution ordering arises in studies of type-2 fuzzy sets. In 2002, Kawaguchi and Miyakoshi [Extended t-norms as logical connectives of fuzzy truth values, Multiple-Valued Logic 8(1) (2002) 53–69] showed that this lattice is a complete Heyting algebra. Later, Harding et al. [Lattices of convex, normal functions, Fuzzy Sets and Systems 159 (2008) 1061–1071] gave an improved description of this lattice and showed it was a continuous lattice in the sense of Gierz et al. [A Compendium of Continuous Lattices, Springer, Berlin, 1980]. In this note we show the lattice Lu is isomorphic to the lattice of decreasing functions from the real unit interval [0,1] to the interval [0,2] under pointwise ordering, modulo equivalence almost everywhere. This allows development of further properties of Lu. It is shown that Lu is completely distributive, is a compact Hausdorff topological lattice whose topology is induced by a metric, and is self-dual via a period two antiautomorphism. We also show the lattice Lu has another realization of natural interest in studies of type-2 fuzzy sets. It is isomorphic to a quotient of the lattice L of all convex normal functions under the convolution ordering. This quotient identifies two convex normal functions if they agree almost everywhere and their intervals of increase and decrease agree almost everywhere.