ABSTRACT
There is a well developed structure theory for finite commutative semigroups and, to a lesser extent, for finitely generated commutative semigroups. The classic approach to commutative semigroups in general is through semilattice decompositions. For finitely generated commutative semigroups one uses presentations, subdirect decompositions, or embedding into finite-like semigroups. The Ponizovsky decomposition of a finite commutative semigroup 5 is a special subdirect decomposition of S, due to Ponizovsky. The chapter discusses the construct finite commutative nilsemigroups from suitable finite sets on free commutative semigroups. Redei's Theorem also implies that every finitely generated commutative semigroup S is finitely presented. At most one nil factors are necessary since a subdirect product of finitely many nilsemigroups is a nilsemigroup.
