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 β_K 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].