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)forallx,y∈Handfiscalledsurjectiveiff(H)=

H ́. Denote by MHom((H, ◦), (H ́, ◦ ́)) and SMHom((H, ◦), (H ́, ◦ ́)) the

set of all multihomomorphisms and the set of all surjective multihomo-

morphisms from (H, ◦) into (H ́, ◦ ́), respectively. Characterizations of

the elementsofMHom((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, +))| = 2א0 =

|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 and |SMHom((Z,+), (Z, ◦n ))| = n. k∈Z+

k|n