Abstract:The γ * -relation de ned on a general hyperring R is the smallest strongly regular relation such that the quotient R γ * is a ring. In this note we consider a particular class of hyperrings, where we de ne a new equivalence, called ε * m , smaller than γ * and we prove it is the smallest strongly regular relation on such hyperrings such that the quotient R ε * m is a ring. Moreover, we introduce the concept of m-idempotent hyperrings, show that they are a characterization for Krasner hyper elds, and that ε * m is a new exhibition for γ * on the above mentioned subclass of m-idempotent hyperrings.
The α* -relation is a fundamental relation on hyperrings, being the smallest strongly regular relation on hyperrings such that the quotient structure R/α* is a commutative ring. In this paper we introduce on hyperrings the relation ζm, which is smaller than α*, and show that, on a particular class of m-idempotent hyperrings R, it is the smallest strongly regular relation such that the quotient ring R/ζ*m is commutative. Some properties of this new relation and its differences from the α* -relation are illustrated and discussed. Finally, we show that ζm is a new representation for α* on this particular class of m-idempotent hyperrings.
Using a new definition, with respect to [21], for normal hyperideals in Krasner (m, n)-hyperrings, we show that the corresponding quotient structures are (m, n)-rings. We prove equivalency of (strongly) regular and (strongly) compatible relations on n-ary hypergroups. Also, the connection between (m, n)-hyperfields and maximal normal hyperideals is investigated.
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