On $\pi$-extensions of the semigroup $\mathbb{Z}_+$

In the paper inverse $\pi$-extensions of the semigroup $\mathbb{Z}_+$ are studied. It is shown that $pi$-extension of the semigroup $\mathbb{Z}_+$ is inverse, if and only if its $\pi$-extension coincides with $\pi(\mathbb{Z}_+)$. The existence of a non-inverse $\pi$-extension for any abelian semigroup is proved.