Characterizations of hemirings by their $h$-ideals

TL;DR

This paper characterizes hemirings where all $h$-ideals or fuzzy $h$-ideals are idempotent, linking these properties to lattice distributivity and prime fuzzy $h$-ideals, advancing the algebraic understanding of hemirings.

Contribution

It introduces new characterizations of hemirings based on the idempotency of $h$-ideals and fuzzy $h$-ideals, including lattice and prime ideal properties.

Findings

All $h$-ideals are idempotent iff the lattice of fuzzy $h$-ideals is distributive.

Fuzzy $h$-ideals are intersections of prime fuzzy $h$-ideals containing them.

A non-constant $h$-ideal is prime iff each proper level set is a prime $h$-ideal.

Abstract

In this paper we characterize hemirings in which all $h$-ideals or all fuzzy $h$-ideals are idempotent. It is proved, among other results, that every $h$-ideal of a hemiring $R$ is idempotent if and only if the lattice of fuzzy $h$-ideals of $R$ is distributive under the sum and $h$-intrinsic product of fuzzy $h$-ideals or, equivalently, if and only if each fuzzy $h$-ideal of $R$ is intersection of those prime fuzzy $h$-ideals of $R$ which contain it. We also define two types of prime fuzzy $h$-ideals of $R$ and prove that, a non-constant $h$-ideal of $R$ is prime in the second sense if and only if each of its proper level set is a prime $h$-ideal of $R$.