Editing Semigroup (section)

== Structure theorem for commutative semigroups ==

There is a structure theorem for commutative semigroups in terms of [[semilattice]]s.{{sfn|ps=|Grillet|2001}} A semilattice (or more precisely a meet-semilattice) {{math|(''L'', ≤)}} is a [[partially ordered set]] where every pair of elements {{math|''a'', ''b'' ∈ ''L''}} has a [[greatest lower bound]], denoted {{math|''a'' ∧ ''b''}}.  The operation ∧ makes ''L'' into a semigroup that satisfies the additional [[idempotence]] law {{math|1=''a'' ∧ ''a'' = ''a''}}.

Given a homomorphism {{math|''f'' : ''S'' → ''L''}} from an arbitrary semigroup to a semilattice, each inverse image {{math|1=''S''<sub>''a''</sub> = {{itco|''f''}}<sup>−1</sup>{{mset|''a''}}}} is a (possibly empty) semigroup.  Moreover, ''S'' becomes '''graded''' by ''L'', in the sense that {{math|''S''<sub>''a''</sub>''S''<sub>''b''</sub> ⊆ ''S''<sub>''a''∧''b''</sub>}}.

If ''f'' is onto, the semilattice ''L'' is isomorphic to the [[quotient]] of ''S'' by the equivalence relation ~ such that {{math|''x'' ~ ''y''}} if and only if {{math|1=''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''<sub>''a''</sub> are all [[Special classes of semigroups|Archimedean semigroups]].  An Archimedean semigroup is one where given any pair of elements ''x'', ''y '', there exists an element ''z'' and {{math|''n'' > 0}} such that {{math|1=''x''<sup>''n''</sup> = ''yz''}}.

The Archimedean property follows immediately from the ordering in the semilattice ''L'', since with this ordering we have {{math|''f''(''x'') ≤ ''f''(''y'')}} if and only if {{math|1=''x''<sup>''n''</sup> = ''yz''}} for some ''z'' and {{math|''n'' > 0}}.