Multi-valued homomorphisms between some groups and hypergroups

Abstract

A hypergroup is a system (H,o) where H is a nonempty set and o is a hyper-operation such that x o(y o z) = (x o y) oz and x o H = H o x = H for all x,y,z [is an element of a set] H By a multi-valued homomorphism from a hypergroup (H,o) into a hypergroup (H’, o’) we mean a multi-valued function f from H into H’ such that f(x o y) = f(x) o’ f(y) for all x,y [is an element of a set] H and we say that f is surjective if f(H){=[union[subscript x[is an element of a set] H]]f(x)} = H’ For a positive integer k let (Z, o[subscript k]) be the hypergroup with the hyperoperation o[subscript k] on Z defined by x o[subscript k]y = x+y+kZ for all x,y [is an element of a set] Z The hypergroup (Z[subscript n], o[subscript k]) is defined analogously, that is, [x][subscript n] o[subscript k] [y][subscript n] = [x][subscript n]+ [y][subscript n]+kZ[subscript n] for all x,y [is an element of a set] Z In this research, we characterize the multi-valued homomorphisms and the surjective multi-valued homomorphisms between the groups (Z,+),(Z[subscript n],+) and the hypergroups of the forms (Z, o[subscript k]),(Z[subscript n], o[subscript k]). In addition, the cardinalities of the sets of such multi-valued functions are determined.