Convex normal functions revisited
The lattice L 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 L(u) 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 L(u). It is shown that L(u) 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 L(u) 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. (C) 2008 Elsevier B.V. All rights reserved.
type-2 fuzzy set continuous lattice uniquely complemented lattice complete lattice metric topology