Generalized fuzzy hyperideals of hypernear-rings and many valued implications

Abstract

The hypernear-rings generalize the concept of near-rings, in the sense that instead of the operation + the hyperoperation + is defined on the set R, that is a R× R→ p* (R), where p*(R) is the set of all the non-empty subsets of R. The study of hypernear-rings is extremely challenging, effering curiously beautiful results to one who is willing to look for structure where symmetry is not so abundant. In this paper, using the notion of "belongingness (∈)" and "quasi-coincidence (q)" of fuzzy points with fuzzy sets, the concept of (∈, ∈∨ q)-fuzzy sub-hypernear-ring (hyperideal) is introduced. Characterization and some of the fundamental properties of such fuzzy sub-hypernear-rings (hyperideals) are obtained. (∈, ∈ ∨ q)-fuzzy cosets determined by (∈, ∈ ∨ q)-fuzzy sub-hypernear-rings are discussed. Finally, we give the definition of a fuzzy sub-hypernear-ring (hyperideal) with thresholds which is a generalization of an ordinary fuzzy sub-hypernear-ring (hyperideal) and an (∈, ∈, ∨ q)-fuzzy sub-hypernear-ring (hyperideal).