A New family of hypergroups and hypergroups of type U on the right of size five

We determine a new family of hypergroups of type U on the right and find necessary and sufficient conditions so that such hypergroups are of type C on the right or cogroups. Moreover, we prove some fundamental properties of the famili P_e={ex | x\in H}, determined by the right scalar identity e of a hypergroup H of type U on the right. If H has sizefive, then there exist six possibilities for the family P_e. In three of these cases, we find, up to isomorphisms, the tables of the corresponding hypergroups. The remaining cases will be studied in a forthcoming paper.