LATTICE OF KEYCHAINS

In this paper we consider the set of all n + 1-tuples of real numbers, not necessarily all distinct, in the decreasing order from the unit interval under the usual ordering of real numbers, always including 1. Such n + 1-tuples inherently arise as the membership values of fuzzy subsets and are called keychains. An natural equivalence relation is introduced on this set and the equivalence classes of keychains are studied here. The number of such keychains is finite and the set of all keychains is a lattice under the coordinate-wise ordering. Thus keychains are subchains of a finite chain of real numbers in the unit interval. We study some of their properties and give some applications to counting fuzzy subsets of finite sets.