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Convex normal functions revisited

By John Harding, Carol Walker and Elbert Walker

Abstract

The lattice Lu of upper semicontinuous convex normal functions with convolution ordering arises in studies of type-2 fuzzy sets. In 2002, M. Kawaguchi and M. Miyakoshi [5] showed that this lattice is a complete Heyting algebra. Later, J. Harding, C. Walker, and E. Walker [4] gave an improved description of this lattice and showed it was a continuous lattice in the sense of Gierz et al. [3]. 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 Hausdor 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 identi es two convex normal functions if they agree almost everywhere and their intervals of increase and decrease agree almost everywhere

Topics: lattice, metric
Year: 2010
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.5070
Provided by: CiteSeerX
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