On the Borderline of Fields and Hyperfields

Abstract

The hyperfield came into being due to a mathematical necessity that appeared during the study of the valuation theory of the fields by M. Krasner, who also defined the hyperring, which is related to the hyperfield in the same way as the ring is related to the field. The fields and the hyperfields, as well as the rings and the hyperrings, border on each other, and it is natural that problems and open questions arise in their boundary areas. This paper presents such occasions, and more specifically, it introduces a new class of non-finite hyperfields and hyperrings that is not isomorphic to the existing ones; it also classifies finite hyperfields as quotient hyperfields or non-quotient hyperfields, and it gives answers to the question that was raised from the isomorphic problems of the hyperfields: when can the subtraction of a field F’s multiplicative subgroup G from itself generate F? Furthermore, it presents a construction of a new class of hyperfields, and with regard to the problem of the isomorphism of its members to the quotient hyperfields, it raises a new question in field theory: when can the subtraction of a field F’s multiplicative subgroup G from itself give all the elements of the field F, except the ones of its multiplicative subgroup G?

Oriented matroids from triangulations of products of simplices

“…The base of the theory is the notion of hyperfields, which can be thought as ordinary fields with multi-valued addition. Besides unifying parallel notions and propositions among these theories, the new language allows one to explain features in a particular matroid theory using the property of its base hyperfield, which we do so here by classifying all hyperfields in which Theorem A is valid: they satisfy the inflation property [5,35]. Unlike the case of oriented matroids, such an extension to hyperfields is new even for coherent matching fields.…”

“…The following definition was probably first considered by Massouros [35] under the name of monogene hyperfields, but we follow the terminology of Anderson [5]. Definition 5.…”

“…We refer the reader to [35] and [11,Example 2.14] for examples and constructions of hyperfields that have the IP. Proposition 5.7 ([5, Proposition 6.10]) H having the IP is equivalent to any of the following properties:…”

“…We elaborate on a construction of hyperfields with the IP from an arbitrary hyperfield H = (H, , ⊗, 0, 1) [35]. It allows one to turn a χ constructed in Theorem 5.8 over H into a (weak) matroid by pushforward.…”

Spectrum of Zariski Topology in Multiplication Krasner Hypermodules

“…Hypergroup theory, which was defined in [1] as a more comprehensive algebraic structure of group theory, has been investigated by different authors in modern algebra. It has been developed using hyperring and hypermodule theory studies by many authors in a series of papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Following these papers, let us start by giving the basic information necessary for the algebraic structure that we will study as Krasner S-hypermodule in studying the S-hypermodule class on a fixed Krasner hyperring class S. Let N be a nonempty set; (N, •) is called a hypergroupoid if for the map defined as • : N × N −→ P * (N) is a function.…”