The commutative quotient structure 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 ζ_m, 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 ζ_m is a new representation for α^* on this particular class of m-idempotent hyperrings.