Is a submonoid of a commutative, finitely generated monoid, always finitely generated?

Question

Let $M$ be a commutative, finitely generated monoid and $N$ its submonoid. Is $N$ finitely generated as well?

Answer 1

Let M be $Z\oplus Z$, and let N be the subset of M of those pairs $(x,y)$ such that $x \geq 0$ and either $y=0$ or $x>0$.