Intersection operation on union-preserving mappings in completely distributive lattices

Abstract

In his classic paper [ 11, Zadeh introduced the notion of a fuzzy set. Subsequently, Goguen (21 extended this to the more general notion of an L-fuzzy set. Thereafter the completely distributive lattice [3] became a suitable framework to expound the theory of the L-fuzzy set. Meanwhile, much research has been carried out in the area of fuzzy topology(cf. [6-91). Some recent articles [4, 51 have considered the uniformities and metrizations on fuzzy sets and obtained rather profound results. In these articles, the importance of the investigation of the intersection operation on union-preserving mappings in completely distributive lattices became apparent. Especially, Hutton’s formula on the intersection operation [4; Lemma 31 is useful. In this paper, by counterexample, we shall show that this formula does not hold for all completely distributive lattices, except the lattice (0, I}. Moreover, under an additional assumption, a proof of this formula is given. We shall also show some properties about the intersection operation which will be needed in our latter work on fuzzy uniformity and fuzzy metrization.