Completions of semilattices of cancellative semigroups

Extract

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 completion T in Abian's order is not always a monomorphism. The difficulty arises when there is eɛ E, e=sup{e'ɛEe'<e>e} but {‐_e,e'}e' is not faithful, i.e. there are x, y with x¬y in S_e such that φ_e,e'(x)=φ_e,e'(y) for all e'<e. A modification of the construction saves all parts of Theorem 1 except the fact that the new embedding S⊆T need not preserve suprema existing in S; it does if S is a semilattice of groups. The sequel [2] also needs amodification in the form of an additional hypothesis.