On hypergroups with a β-class of finite height

In every hypergroup, the equivalence classes modulo the fundamental relation β are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a β-class and we analyze properties of hypergroups where the height of a β-class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one β-class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is n ≤ 4, apart from isomorphisms.