Finitely generated commutative Archimedean semigroups. (Q1418265)

Let \(S\) be an additive, commutative semigroup. Then \(S\) is said to be an Archimedean (resp., \(MJ\)-) semigroup if for all \(x,y\in S\), there is a \(z\in S\) and an integer \(k\geq 1\) such that \(kx=y+z\) (resp., there are integers \(m,n\geq 1\) such that \(mx=ny\)), and \(S\) is an \(\mathcal N\)-semigroup if \(S\) is a cancellative, Archimedean semigroup with no idempotents. The authors show that \(S\) is a finitely generated Archimedean (resp., \(MJ\)-) semigroup if and only if either (1) \(S\) is an ideal extension of a finitely generated (resp., finite) group by a finitely generated nilsemigroup or (2) \(S\) is an ideal extension of a finitely generated \(\mathcal N\)-semigroup by a finite nilsemigroup.