Characterizations of hemirings by their hh-ideals

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 $R$

Fuzzy hh-ideals of hemirings

A characterization of an $h$-hemiregular hemiring in terms of a fuzzy $h$-ideal is provided. Some properties of prime fuzzy $h$-ideals of $h$-hemiregular hemirings are investigated. It is proved that a fuzzy subset $\zeta$ of a hemiring $S$ is a prime fuzzy left (right) $h$-ideal of $S$ if and only if $\zeta$ is two-valued, $\zeta(0) = 1$, and the set of all $x$ in $S$ such that $\zeta(x) = 1$ is a prime (left) right $h$-ideal of $S$. Finally, the similar properties for maximal fuzzy left (right) $h$-ideals of hemirings are considered