MULTIHOMOMORPHISMS FROM (Z,+) INTO CERTAIN HYPERGROUPS

Abstract

By a multihomomorphism from a hypergroup (H,◦) into a hypergroup( H´,◦´) we mean a multi-valued function f from H into H´ such that f(x◦ y)=f(x)◦´f(y) for all x,y ∈ H and f is called surjective if f(H)= H´. Denote by MHom((H,◦),(H´,◦´)) and SMHom((H,◦),(H´,◦´)) the set of all multihomomorphisms and the set of all surjective multihomomorphisms from (H,◦) into (H´,◦´), respectively. Characterizations of the elementsof MHom((Z,+),(Z,+)), SMHom((Z,+),(Z,+)), MHom ((Z,◦n),(Z,+)) and SMHom((Z,◦n),(Z,+)) have been given where n is a positive integer and ◦n is the hyperoperation on Z defined by x ◦n y = x + y + nZ. It has also been shown that |MHom((Z,+),(Z,+))| = ℵ0 = |SMHom((Z,+),(Z,+))| and |MHom((Z,◦n),(Z,+))|=2 ℵ0 = |SMHom((Z,◦n),(Z,+))|. In this paper,characterizations of the elements of MHom((Z,+),(Z,◦n)) and SMHom((Z,+),(Z,◦n)) are provided. We also show that |MHom((Z,+),(Z,◦n))|= k∈Z+ k|n k and |SMHom((Z,+),(