Geodesic
Completions of semilattices of cancellative semigroups
Glasgow mathematical journal, Tome 26 (1985) no. 2, pp. 157-160

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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 Se such that φe,e'(x)=φe,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. 
Burgess, W. D. Completions of semilattices of cancellative semigroups. Glasgow mathematical journal, Tome 26 (1985) no. 2, pp. 157-160. doi: 10.1017/S0017089500005930
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Burgess, W. D., Completions of semilattices of cancellative semigroups, Glasgow Math J. 21 (1980), 29–37. Google Scholar

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