Recent Results in Hyperring and Hyperfield Theory 209

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. K]/WORDS ANt)IIRASS. Hyperoperation, hyperring, hyperfield, quotient hyperring, partition hyperring. AHS 1980S CLASSIFICATIO CODE: 16A78 I. The purpose of this survey paper is to present some recent results in hyperring theory. At the same time, it is hoped that it will draw the attention of English speaking mathematicians to the work of Marc Krasner and of his students, who have published, mainly, in French. The notion of a hyperoperation is a straightforward generalization of the notion of an operation. Given a non empty set S, a hyperoperation maps SxS into the set of all non empty subsets of S. If the hyperoperation is commutative and associative, then it is called a hyperaddition. And if we generalize the usual axioms for addition, we obtain structures that are generalizations of the usual algebraic structures and we call them Abelian hypergroups, hyperrings, hypermodules, and hyperfields (these terms are defined in section 2). The notion of a hypergroup appears at least as early as 1935 in the work of F. Marty [1,2], while the notion of a hyperring was introduced by Krasner [3], who used it as a technical tool in a study of his on the approximation of valued fields. Later, two students of Krasner, Mittas and Stratigopoulos, earned their theses by studying the structure of the hyperrings. The thesis of Stratigopoulos and the articles based on his thesis that were subsequently published, fueled a discussion on the merits of studying these structures rather than channeling the efforts of mathematicians into more traditional subjects. During this discussion it was noted that all known hyperrings were obtained via a construction that Krasner had introduced. Krasner observed that if R is a ring and G is a subset of R such that (G,-) is a group, then G can define an equivalence relation