The group of fractions of a semigroup S is the group G = G(S) generated by the elements of S as generators and all equations xy=z which hold true in S as relations. This has a universal property for morphisms from S to a group. There is an obvious map from S to G(S) by sending each element of S to the corresponding generator.
An important question is to characterize those semigroups for which this map is an embedding. This need not always be the case: for example, take S to be the semigroup of subsets of some set X with set-theoretic intersection as the binary operation (this is an example of a semilattice). Since A.A = A holds for all elements of S, this must be true for all generators of G(S) as well: which is therefore the trivial group. It is clearly necessary for embeddability that S have the cancellation property. When S is commutative this condition is also sufficient and the Grothendieck group of the semigroup provides a construction of the group of fractions. The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups, (Suschkewitsch 1928). Anatoly Maltsev gave necessary and conditions for embeddability in 1937.
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