The Reducibility Concept in General Hyperrings

Abstract

By using three equivalence relations, we characterize the behaviour of the elements in a hypercompositional structure. With respect to a hyperoperation, some elements play specific roles: their hypercomposition with all the elements of the carrier set gives the same result; they belong to the same hypercomposition of elements; or they have both properties, being essentially indistinguishable. These equivalences were first defined for hypergroups, and here we extend and study them for general hyperrings—that is, structures endowed with two hyperoperations. We first present their general properties, we define the concept of reducibility, and then we focus on particular classes of hyperrings: the hyperrings of formal series, the hyperrings with P-hyperoperations, complete hyperrings, and (H,R)-hyperrings. Our main aim is to find conditions under which these hyperrings are reduced or not.

Hypergroup Theory and Algebrization of Incidence Structures

“…A large number of papers has been published on the hyperfields and hyperrings, starting from the pioneer work of J. Mittas [37][38][39][40][41][42][43][44] and continuing with a plenitude of researchers, such as Ch. Massouros [29,35,[45][46][47][48][49][50][51], A. Nakassis [36], G. Massouros [50][51][52][53][54], R. Rota [55,56], S. Jančic-Rašović [57][58][59], I. Cristea [58][59][60][61][62][63][64], H. Bordbar [59][60][61], M. Kankaraš [62], V. Vahedi et al [63][64][65], M. Jafarpour et al [63][64][65][66], A. Connes and C. Consani [67,68], O. Viro [69,70], R. Ameri, M. Eyvazi and S. Hoskova-Mayero...…”

A Note on the Connection between Ordered Semihyperrings

“…In [21], Cristea and Kankaraš studied the reducibility of hyperrings. In [22,23], the (m-)k-hyperideals of an ordered semihyperring are studied.…”