Similitude and isomorphism in K_H hypergroups

Let (H,⊙) be a hypergroupoid and {A(x):x∈H} be a family of nonempty and pairwise disjoint sets indexed by H . Then the set K=⋃{A(x):x∈H} is a hypergroup under the hyperoperation ∗ defined for all a,b∈K and a∈A(x) , b∈A(y) by a∗b=⋃{A(z):z∈x⊙y} . This construction, introduced by the author in an earlier paper, is called a K_H-hypergroup generated by H . In the paper under review, first a similarity for K_H-hypergroups is defined and some properties are studied. The author uses in his study mainly the so-called Σ -hypergroups and isomorphisms between the underlying hyperstructures. In the second part of the paper, it is proved that for a K_H-hypergroup K generated by H there exists a regular equivalence R such that K/R≅H . Finally, it is proved that given a K_H-hypergroup, there exists a chain of K_H hypergroups which terminates when a trivial K_H-hypergroup is obtained, i.e. all the sets A(x) are singletons.