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.
A well known result in fuzzy group theory states that “level subgroups of any fuzzy subgroup of a finite group form a chain”. We check the validity of this statement in the intuitionistic fuzzy perspective. We do this using Dihedral Group $D_3$, which is a non-cyclic group. We prove that $D_3$ has 100 distinct intuitionistic fuzzy subgroups (IFSGs) upto isomorphism. The intuitionistic level subgroups (ILSGs) of exactly 76 among them make chains, and hence it can be concluded that the result is not true in the intuitionistic fuzzy perspective. We also enlist all the 100 distinct intuitionistic fuzzy subgroups of $D_3$ upto isomorphism.
It is well known that the set of all level subgroups of any fuzzy subgroup of a finite group forms a chain. In this paper we prove that this result does not extend to Intuitionistic Fuzzy Subgroups (IFSGs) by providing a counterexample. For any two distinct prime numbers p and q, we prove that the cyclic group Z pq has 36 non-isomorphic IFSGs. The Intuitionistic Level Subgroups (ILSGs) of only 28 of them form chains, while those of remaining 8 do not form chains. The list of all the 36 distinct IFSGs is also provided; and those whose ILSGs form a chain, and not, are identified. The case is illustrated using a specific example. We have also obtained a characterisation of IFSGs of Z pq , whose ILSGs form a chain.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.