Existence of proper semihypergroups of type U on the right

We generalize the classical definition of hypergroups of type U on the right to semihypergroups, and we prove some properties of their subsemihypergroups and subhypergroups. In particular, we obtain that a finite proper semihypergroup of type U on the right can exist only if its order is at least 6. We prove that one such semihypergroup of order 6 actually exists. Moreover, we show that there exists a hypergroup of type U on the right of cardinality 9 containing a proper non-trivial subsemihypergroup. In this way, we solve a problem left open in [D. Freni, Sur les hypergroupes de type U et sous-hypergroupes engendrés par un sous-ensemble, Riv. Mat. Univ. Parma 13 (1987) 29-41].