Semihypergroups in Which the Radical of Every Hyperideal Is a Subsemihypergroup

Abstract

The concept of a semihypergroup is a notable generalization of the concept of a semigroup. For nonempty subset \(A\) of a semihypergroup \(H\), the radical of \(A\) denoted by \(\sqrt{A}\), is \(\sqrt{A} = \{a \in H \mid a^n \subseteq A \) for some positive integer \(n \}\). A characterization when the radical of every hyperideal of \(H\) is a subsemihypergroup of \(H\) is investigated and given in this paper. Indeed, we characterize when the radical of every hyperideal of \(H\) is a bi-hyperideal of \(H;\) when the radical of every bi-hyperideal of \(H\) is a left hyperideal of \(H\); and when the radical of every subsemihypergroup of \(H\) is a hyperideal of \(H\).