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  "body": "\n## Statement\n\nThere 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|0QV2JS9RJ3]] 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 $a \\land a = a$.\n\nGiven 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 $S_{a}S_{b} \\subseteq S_{a\\land b}$.\n\nIf *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.\n\nWhenever 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.\n\nFurthermore, the components *S*\u003csub\u003e*a*\u003c/sub\u003e are all Archimedean semigroups. An Archimedean semigroup is one where given any pair of elements *x*, *y*, there exists an element *z* and $n \u003e 0$ such that $x^{n} = yz$.\n\nThe 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 $n \u003e 0$.\n",
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