Tensor products of complete lattices and their application in constructing quantales

Abstract

This paper aims to implement tensor products of complete lattices into fuzzy set theory. The most convenient approach for this purpose is to view the tensor product of two complete lattices as the family of all join reversing maps between those lattices. We show that some fundamental constructions in fuzzy set theory are tensor products. Examples of such circumstances include the following (to mention only three of them): the complete lattice of all lower semicontinuous maps from a topological space into a continuous lattice is the tensor product of the topology of the space and the range lattice; a binary operation coming from Zadeh's extension principle is the tensor product of the Minkowski multiplication with the multiplication of the underlying unital quantale; triangle functions on nonnegative left-continuous distribution functions are tensor products of the real unit interval and the extended nonnegative half-line equipped, respectively, with a left-continuous t-norm and the usual addition.