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 \(\mathscr{G}^*\) and inferior pre-complement \(\mathscr{G}_*\), for any given classic IVFG \(\mathscr{G}\). A partial \(\operatorname{order} \frac{\subset}{P}\) is defined on \(\mathscr{P}=C^{-1}(\mathscr{G})\), the collection of all pre-complements of \(\mathscr{G}\). It is proved that \(\left(\mathscr{P}, \frac{C}{P}\right)\) is a lattice with \(\mathscr{G}^*\) as the greatest element and \(\mathscr{G}_*\) as the least element. We derive a necessary and sufficient condition for this lattice to become a chain.