Recent results in hyperring and hyperfield theory

This survey article presents some recent results in the theory of hyperfields and hyperrings, algebraic structures for which the “sum” of two elements is a subset of the structure. The results in this paper show that these structures cannot always be embedded in the decomposition of an ordinary structure (ring or field) in equivalence classes and that the structural results for hyperfields and hyperrings cannot be derived from the corresponding results in field and ring theory.