(H,G)-hypergroups

The author studies a class of special hypergroups defined in the paper. Let (H,∘) be a hypergroup and (G,⋅) a group with the identity 1, and let {A_i:i∈G} be a family of non-empty sets such that A_1 =H and A_i ∩A_j =∅ if i≠j . Define the binary hyperoperation ∗ on K=⋃ A_i by taking x∗y=x∘y if (x,y)∈H×H and x∗y=A_k if (x,y)∈A_i ×A_j ≠H×H and i⋅j=k . Then (K,∗) is a hypergroup which is called an (H,G)-hypergroup. We quote some theorems giving the main results in the paper. Theorem 1: If (K_1 ,∗) and (K_2 ,∗) are (H_1 ,G_1 ) and (H_2 ,G_2 ) -hypergroups, respectively, and K_1 ≃K_2 , then H_1 ≃H_2 and G_1 ≃G_2 . Theorem 5: If (K,∗) is an (H,G) -hypergroup, then K is cyclic if and only if G is a cyclic group. Some combinatorial properties of these hypergroups are also considered.