TITLE: Nielsen-Schreier theorem for monoids
QUESTION [7 upvotes]: Let $S$ be a finitely generated free abelian semigroup (or monoid), and let $T \subset S$ be a sub-semigroup (sub-monoid). Does the Nielsen-Schreier theorem hold in this case, that is, will $S$ still be free? Moreover, will $S$ be finitely generated as in the groups case?

REPLY [3 votes]: Here is (partially) what is known about commutative monoids and their submonoids. 

The elementary theory of every finitely generated commutative semigroup (monoid) is decidable (Taiclin). 
The isomorphism problem for finitely generated commutative semigroups (monoids) is decidable (Taiclin). 
Every finitely generated commutative monoid is representable by matrices over $\mathbb Z$, hence is residually finite and Hopfian (Malcev). 
Every finitely generated commutative monoid is finitely presented (Redei) moreover it admits a finite Church-Rosser presentation and for every finite presentation, the Knuth-Bendix algorithm terminates (all these essentially follow  from the Hilbert Finite Basis theorem). 
The lattice of subsemigroups of the infinite cyclic semigroup does not satisfy any non-trivial identity (Repnitskii, Katsman). In contrast, the lattice of subgroups of any commutative group is modular.