On strongly conjugable extensions of hypergroups of type U with scalar identity

Let S_n denote the class of hypergroups of type U on the right of size n with bilateral scalar identity. In this paper we consider the hypergroups (H,◦) ∈ S_7 which own a proper and non-trivial subhypergroup h. For these hypergroups we prove that h is closed if and only if (H − h) ◦ (H − h) = h. Moreover we consider the set E_7 of hypergroups in S_7 that own the above property. On this set, we introduce a partial ordering induced by the inclusion of hyperproducts. This partial ordering allows us to give a complete characterization of hypergroups in E_7 on the basis of a small set of minimal hypergroups, up to isomorphisms. This analysis gives a partial answer to a problem raised in [5] concerning the existence in S_n of proper hypergroups having singletons as special hyperproducts.