A REMARK ON SOME SEMIGROUPS OF HYPERGROUP HOMOMORPHISMS

Abstract

Let Z be the set of integers, n a positive integer and (Z,◦n) the hypergroup where x ◦n y = x + y + nZ for all x,y ∈ Z. Denote by Hom(Z,◦n) the semigroup, under composition, of all homomorphisms of (Z,◦n). It has been shown that for f : Z → Z,f ∈ Hom(Z,◦n) if and only if f(x + nZ)=xf(1) + nZ for all x ∈ Z and |Hom(Z,◦n)| =2 ℵ0. Using this characterization, we show in this paper that the relation δ on Hom(Z,◦n) defined by fδg ⇔ f(1) ≡ g(1) mod n is a congruence on Hom(Z,◦n) and Hom(Z,◦n)/δ ∼