It is possible to restate the characteristic property of a cancellative element in terms of a property held by the corresponding left multiplication La : S → S and right multiplication Ra : S → S maps defined by La(b) = ab and Ra(b) = ba: an element a in S is left cancellative if and only if La is injective, an element a is right cancellative if and only if Ra is injective.
The procedure for doing this is similar to that of embedding an integral domain in a field (Clifford & Preston 1961, p. 34) – it is called the Grothendieck group construction, and is the universal mapping from a commutative semigroup to abelian groups that is an embedding if the semigroup is cancellative.
[4] To obtain a sufficient (but not necessary) condition, it may be observed that the proof of the result that a finite cancellative semigroup S is a group critically depended on the fact that Sa = S for all a in S. The paper (Dubreil 1941) generalized this idea and introduced the concept of a right reversible semigroup.
[5] Though theoretically important, the conditions are countably infinite in number and no finite subset will suffice, as shown in (Malcev 1940).
The two embedding theorems by Malcev and Lambek were compared in (Bush 1963) and later revisited and generalized by (Johnstone 2008), who also explained the close relationship between the semigroup embeddability problem and the more general problem of embedding a category into a groupoid.