On Fuzzy semihyperrings

TL;DR

This paper introduces fuzzy semihyperrings and fuzzy R-semihypermodules, exploring properties of fully idempotent semihyperrings and their fuzzy hyperideals, including their lattice structure and topological spectrum.

Contribution

It establishes the equivalence between full idempotency and lattice distributivity of fuzzy hyperideals, and constructs the fuzzy prime spectrum as a topological space.

Findings

Fully idempotent semihyperrings have distributive fuzzy hyperideal lattices.

The set of fuzzy prime hyperideals forms a topological space.

Characterization of fuzzy hyperideals in semihyperrings.

Abstract

In this article we introduce the study of fuzzy semihyperrings and fuzzy R-semihypermodules, where R is a semihyperrings and R-semihypermodules are represntations of R. In particular, semihyperrings all of whose hyperideals are idempotent, called fully idempotent semihyperrings, are investigated in a fuzzy context. It is proved, among other results, that a semihyperring R is fully idempotent if and only if the lattics of fuzzy hyperideals of R is distributive under the sum and product of fuzzy hyperideals. It is also shown that the set of proper fuzzy prime hyperideals of a fully idempotent semihyperring R admits the structure of a topological space, called the fuzzy prime spectrum of R.