Some remarks on the lattice of fuzzy intervals

  • Authors:

  • Ath. Kehagias

  • Affiliations:

  • Division of Mathematics, Department of Mathematics, Physics and Computer Sciences, Faculty of Engineering, Box 464, Aristotle University of Thessaloniki, Thessaloniki, GR 54124, Greece

  • Venue:
  • Information Sciences: an International Journal
  • Year:
  • 2011

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Abstract

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.