Half-homomorphisms of Groups

Let G and G' be multiplicative systems. A half-homomorphism of G into G' will mean a mapping a—>a' of G into C such that for all a, bEG, (ab)'=a'V or b'a'. An anti-homomorphism is a mapping such that always (ab)' = b'a'. The terms half-isomorphism, etc., are defined similarly. It will be shown that any half-homomorphism of a group G into a group G' is either a homomorphism or an anti-homomorphism (Theorem 2). The corresponding theorem for nonassociative rings (with the added requirement that (a + b)' =a'-\-b') was proved by Hua [l] (see also Jacobson and Rickart [2, Lemma l]). For halfisomorphisms, it is sufficient to assume that G and G' are cancellation semigroups in order to obtain the analogous result (Theorem 1). Examples are given to show that Theorem 1 is false for semi-groups and for loops.