The lattice of pre-complements of a classic interval valued fuzzy graph

Abstract

We prove that a complement Interval Valued Fuzzy Graph (IVFG), unlike the crisp and fuzzy cases, may have several non-isomorphic pre-complements. We introduce the notion of complement numbers, and show that, by assigning complement numbers to the edges of a complement IVFG, we can ensure uniqueness of pre-complement. We introduce the concepts superior pre-complement G * and inferior pre-complement G * , for any given classic IVFG G. A partial order ⊆ P is defined on P = C −1 (G), the collection of all pre-complements of G. It is proved that (P, ⊆ P) is a lattice with G * as the greatest element and G * as the least element. We derive a necessary and sufficient condition for this lattice to become a chain.