in terms of semilattices. A semilattice (or more precisely a meet-semilattice) is a partially ordered set where every pair of elements has a greatest lower bound, denoted . The operation makes into a semigroup satisfying the additional idempotence law . Given a homomorphism from an arbitrary semigroup to a semilattice, each inverse image is a (possibly empty) semigroup. Moreover, becomes graded by , in the sense that If is onto, the semilattice is isomorphic to the quotient of by the equivalence relation such that if and only if . This equivalence relation is a semigroup congruence, as defined above. Whenever we take the quotient of a commutative semigroup by a congruence, we get another commutative semigroup. The structure theorem says that for any commutative semigroup , there is a finest congruence such that the quotient of by this equivalence relation is a semilattice. Denoting this semilattice by , we get a homomorphism from onto . As mentioned, becomes graded by this semilattice. Furthermore, the components are all Archimedean semigroups. An Archimedean semigroup is one where given any pair of elements , there exists an element and such that . The Archimedean property follows immediately from the ordering in the semilattice , since with this ordering we have if and only if for some and . Group of fractions The group of fractions or group completion of a semigroup S is the group generated by the elements of S as generators and all equations which hold true in S as relations. There is an obvious semigroup homomorphism which sends each element of S to the corresponding generator. This has a universal property for morphisms from S to a group: given any group H and any semigroup homomorphism , there exists a unique group homomorphism with k=fj. We may think of G as the "most general" group that contains a homomorphic image of S. 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 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