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](https://en.wikipedia.org/wiki/Nielsen%E2%80%93Schreier_theorem) hold in this case, that is, will $S$ still be free? Moreover, will $S$ be finitely generated as in the groups case?



Nielsen-Schreier theorem for monoids
For the question of finite generation, here's a a counterexample that appears in D.B. McAlister and L. O'Carroll,
*Finitely generated commutative semigroups.* Glasgow Math. J. 11 1970 134–151. MR0269765 (42 #4660:


If $S$ is the free commutative semigroup on $a$ and $b$, and $K=\{a^nb^m\mid n,m>0\}$, then $K$ is not finitely generated: note that elements of the form $ab^k$ and of the form $a^kb$ cannot be the product of two or more elements of $K$ (since a product with $m$ factors will have both exponents greater than or equal to $m$), so you need all of them in any generating set for $K$.


Nielsen-Schreier ("subsemigroup of free is free") fails for even the cyclic free semigroup, as noted; and in general does not hold for non-commutative semigroups either. L.N. Shevrin proved, inter alia, that a subsemigroup $T$ of a free semigroup $S$ is free if and only if for every $a\in T$ and $s\in S$, if $as$ and $sa$ are both in $T$, then $s\in T$. (L.N. Shevrin, *On subsemigroups of free semigroups*, Dokl. Akad. Nauk SSSR 133 (1960), 537-539 (Russian)
Soviet Math. Dokl.1 (1960), 892-894.)


Alternatively, P.M. Cohn proved that a subsemigroup $T$ of a free semigroup $S$ is free if and only if for any $a,a',b,b'\in T$, if $ab'=ba'$ then $a=bx$ or $b=ax$ for some $x\in T$. ( P.M. Cohn, [*On Subsemigroups of Free Semigroups*](http://www.jstor.org/stable/2034935) Proceedings of the American Mathematical Society Vol. 13, No. 3 (Jun., 1962), pp. 347-351
DOI: 10.2307/2034935). Though these are for non-commutative semigroups, I expect similar conditions would be needed to guarantee that a subsemigroup si free. In any case, the easy example given by Todd Trimble shows you cannot hope to have a submonoid/subsemigroup of a free commutative monoid/semigroup always be free.


