EXTENSIONS OF FINITE COMMUTATIVE HYPERGROUPS

Abstract

The purpose of this paper is to investigate extension problems for the cat- egory of finite commutative hypergroups. In fact, sufficiently many extensions will be provided by applying the notion of a field of finite commutative hypergroups. More- over, the duality of such extensions will be studied via fields of finite commutative hypergroups. 1 Introduction Let H and L be finite commutative hypergroups. A finite commuta- tive hypergroup K is called an extension of L by H if the sequence 1 → H → K → L → 1 is exact, i.e. if the quotient hypergroup K/H is isomorphic to L. Here, the notions of subhypergroup, quotient hypergroup and isomorphism between hypergroups are taken from (B-H), a source from which all the elementary knowledge needed in the sequel will be taken. In the previous papers (H-J-K-K) and (K-I) we constructed extensions K(H, G, α) and K( ˆ H, ˆ G, ˆ α) for a regular action α of a finite abelian group G on a finite commutative hypergroup H which satisfies by definition the exact sequence : 1 → H α → K(H, G, α) → K(G) → 1 and 1 → K( ˆ G) → K( ˆ H, ˆ G, ˆ α) → ˆ H ˆ α → 1.