This paper deals with cancellative semigroups \(S\) which admit a semigroup \(Q\) of left quotients in the sense of \textit{J. Fountain} and \textit{V. Gould} [in: Contributions to General Algebra 3, Proc. Conf., Vienna 1984, 115-124 (1985; Zbl 0574.20047)]\ (as is well known, \(Q\) is not uniquely determined by \(S\)). It is shown that, in the above case, \(Q\) is a semilattice of left groups and \(S\) is embedded in \(Q\) in a nice way -- e.g., for each \(\mathcal H\)-class \(H\) of \(Q\), \(H\) is the group of left quotients of \(S\cap H\). The semigroup \(S\) itself has a group \(G\) of left quotients, and there is a homomorphism of \(Q\) onto \(G\) whose restriction to \(S\) is the identical map. Then the authors describe how the semigroups of left quotients of \(S\) correspond to certain semilattice decompositions of \(S\).