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

We study inverse $\pi$-extensions of the semigroup $\mathbb{Z}_+$. It is shown that $\pi$-extension of the semigroup $\mathbb{Z}_+$ is inverse, iff its $\pi$-extension coincides with $\pi(\mathbb{Z}_+)$. The existence of a non-inverse $\pi$-extension for semigroup $\mathbb{Z}_+$ is proved.