Semihyperrings Characterized by Their Hyperideals

TL;DR

This paper introduces the concept of semihyperrings and characterizes them using hyperideals, exploring their properties and the topology of their irreducible hyperideals.

Contribution

It defines semihyperrings and various types of hyperideals, and establishes a topology on the lattice of irreducible hyperideals, advancing the algebraic theory of hyperstructures.

Findings

Defined semihyperrings and hyperideals.

Characterized semihyperrings using hyperideals.

Established topology on the irreducible spectrum of hyperideals.

Abstract

The concept of hypergroup is generalization of group, first was introduced by Marty [9]. This theory had applications to several domains. Marty had applied them to groups, algebraic functions and rational functions. M. Krasner has studied the notion of hyperring in [11]. G.G Massouros and C.G Massouros defined hyperringoids and apply them in generalization of rings in [10]. They also defined fortified hypergroups as a generalization of divisibility in algebraic structures and use them in Automata and Language theory. T. Vougiouklis has defined the representations and fundamental relations in hyperrings in [14, 15]. R. Ameri, H. Hedayati defined k-hyperideals in semihyperrings in [2]. B. Davvaz has defined some relations in hyperrings and prove Isomorphism theorems in [7]. The aim of this article is to initiate the study of semihyperrings and characterize it with hyperideals. In this article we defined semihyperrings, hyperideals, prime, semiprime, irreducible, strongly irreducible hyperideals, m-systems, p-systems, i-systems and regular semihyperrings. We also shown that the lattice of irreducible hyperideals of semihyperring R admits the structure of topology, which is called irreducible spectrum topology.