A new representation of fundamental relations on fuzzy hyperrings

The \(\gamma ^{*}\) ${\gamma }^{\ast }$ -relation is the smallest fuzzy strongly regular relation on a fuzzy hyperring such that its quotient space is a ring. In this paper, a subclass of fuzzy hyperrings is considered, where a new equivalence relation, \(\mathcal {T}^{*}_{m}\) $\mathcal{T}_m^{\ast }$ , smaller than \(\gamma ^{*}\) ${\gamma }^{\ast }$ , is defined and proved that the quotient \(R/\mathcal {T}^{*}_{m}\) $R/\mathcal{T}_m^{\ast }$ is a ring. Moreover, the notion of m-idempotent fuzzy hyperrings is introduced and it is shown that \(\mathcal {T}^{*}_{m}\) $\mathcal{T}_m^{\ast }$ is a new representation for \(\gamma ^{*}\) ${\gamma }^{\ast }$ on the above mentioned subclass of m-idempotent fuzzy hyperrings. Also, m-fuzzy complete parts on fuzzy hyperrings are defined and compared with fuzzy complete parts, and also its connections with m-complete parts, as generalizations of complete parts in the associated hyperrings, is investigated. Finally, it is shown how m-fuzzy complete parts help us to study the transitivity of \(\mathcal {T}_{m}\) ${\mathcal{T}}_m$ -relation.