Surjective Multihomomorphisms between Cyclic Groups

S. Nenthein, P. Lertwichitsilp

Abstract

A multifunction f from a group G into a group G' is called a multi- homomorphism if

$f(xy)=f(x)f(y) ( = {st : s \in f(x) and t \in f(y)} )$

for all $x, y \in G$. Denote by MHom(G;G') the set of all multihomomorphisms from G into G'. We call $ f \in $MHom(G;G0) a surjective multihomomorphism if f(G) = G' where $f(G) = \cup _{x \in G}f(x)$. The elements of MHom((Z, +), (Z, +)), MHom((Zn, +), (Z, +)), MHom((Z, +), (Zn, +)) and MHom((Zm, +), (Zn, +))

have been already characterized and counted. Our purpose is to characterize when these multihomomorphisms are surjective. The cardinalities of the subsets of surjective multihomomorphisms are also determined.