Fundamental relation on m-idempotent hyperrings

What is it about?

The γ*-relation defined 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 define 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 hyperfields, and that ε*m is a new exhibition for γ* on the above mentioned subclass of m-idempotent hyperrings.

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Why is it important?

Our findings show that there exist a special class of hyperrings, called m-idempotent hyperrings, where the relation ε*m is equal with the relation γ*, but on general hyperrings we have that ε*m is smaller than γ*.

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