Some remarks on the lattice of fuzzy intervals

In this paper we study the connections between three related concepts which have appeared in the fuzzy literature: fuzzy intervals, fuzzy numbers and fuzzy interval numbers (FIN’s). We show that these three concepts are very closely related. We propose a new definition which encompasses the three previous ones and proceeds to study the properties ensuing from this definition. Given a reference lattice (X, ⊑), we define fuzzy intervals to be the fuzzy sets such that their p-cuts are closed intervals of (X, ⊑). We show that, given a complete lattice (X, ⊑), the collection of its fuzzy intervals is a complete lattice. Furthermore we show that, if (X, ⊑) is completely distributive, then the lattice of its fuzzy intervals is distributive. Finally we introduce a new inclusion measure, which can be used to quantify the degree in which a fuzzy interval is contained in another, an approach which is particularly valuable in engineering applications.