Fully simple semihypergroups

In this paper we consider the class of semihypergroups H such that all subsemihypergroups K⊆H are simple and, when |K|⩾3 the fundamental relation βK is not transitive. For these semihypergroups we prove that hyperproducts of elements in H have size ⩽2 and the quotient semigroup H/β⁎ is trivial. This last result allows us to completely characterize these semihypergroups in terms of a small set of simple semihypergroups of size 3. Finally, we solve a problem on strongly simple semihypergroups introduced in [11].