Completions of semilattices of cancellative semigroups

Abstract

K. Shoji has pointed out to me that construction [1] does not always yield a completion. In the notation of [1], the homomorphism from the strong semilattice of cancellative semigroups S to its purported completionTin Abian's order is not always a monomorphism. The difficulty arises when there iseɛE, e=sup{e'ɛEe'<e>e} but {‐e,e'}e' is not faithful, i.e. there arex, ywithx¬yinSesuch that φe,e'(x)=φe,e'(y) for alle'<e. A modification of the construction saves all parts of Theorem 1 except the fact that the new embeddingS⊆Tneed not preserve suprema existing inS; it does ifSis a semilattice of groups. The sequel [2] also needs amodification in the form of an additional hypothesis.