Structure theorem for commutative semigroups

Statement

There is a structure theorem for commutative semigroups in terms of semilattices. A semilattice (or more precisely a meet-semilattice) $(L, \le)$ is a partially ordered set where every pair of elements $a, b \in L$ has a greatest lower bound, denoted $a \land b$. The operation ∧ makes L into a semigroup that satisfies the additional idempotence law $a \land a = a$.

Given a homomorphism $f : S \to L$ from an arbitrary semigroup to a semilattice, each inverse image $S_{a} = f^{-1}\{a\}$ is a (possibly empty) semigroup. Moreover, S becomes graded by L, in the sense that $S_{a}S_{b} \subseteq S_{a\land b}$.

If f is onto, the semilattice L is isomorphic to the quotient of S by the equivalence relation ~ such that $x \sim y$ if and only if $f(x) = f(y)$. 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 S, there is a finest congruence ~ such that the quotient of S by this equivalence relation is a semilattice. Denoting this semilattice by L, we get a homomorphism f from S onto L. As mentioned, S becomes graded by this semilattice.

Furthermore, the components S_a are all Archimedean semigroups. An Archimedean semigroup is one where given any pair of elements x, y, there exists an element z and $n > 0$ such that $x^{n} = yz$.

The Archimedean property follows immediately from the ordering in the semilattice L, since with this ordering we have $f(x) \le f(y)$ if and only if $x^{n} = yz$ for some z and $n > 0$.