Homomorphisms of commutative cancellative semigroups into nonnegative real numbers

Abstract

Let S be a commutative cancellative semigroup and T 0 {T_0} be a cofinal subsemigroup of S. Let h 0 {h_0} be a homomorphism of T 0 {T_0} into the semigroup of nonnegative real numbers under addition. We prove that Kobayashi’s condition [2] is necessary and sufficient for h 0 {h_0} to be extended to S. Further, we find a necessary and sufficient condition in order that the extension be unique. Related to this, the “boundedness condition” is introduced. For further study, several examples are given.