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Is there any discontinuous semigroup automorphism on $\beta\mathbb{Z}$ that preserves $\mathbb{Z}$ and $\beta\mathbb{Z}\setminus\mathbb{Z}$?

In other words, can

Can the identity isomorphism on the additive group $\mathbb{Z}$ be extended to a non-identity semigroup isomorphism on $\beta\mathbb{Z}$, and still preserves $\beta\mathbb{Z}\setminus\mathbb{Z}$? Obviously this extension is discontinuous.

Is there any discontinuous semigroup automorphism on $\beta\mathbb{Z}$ that preserves $\mathbb{Z}$ and $\beta\mathbb{Z}\setminus\mathbb{Z}$?

In other words, can the identity isomorphism on the additive group $\mathbb{Z}$ be extended to a non-identity semigroup isomorphism on $\beta\mathbb{Z}$, and still preserves $\beta\mathbb{Z}\setminus\mathbb{Z}$? Obviously this extension is discontinuous.

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# A basic question on Stone-Cech compactification of $\mathbb{Z}$

Is there any discontinuous semigroup automorphism on $\beta\mathbb{Z}$ that preserves $\mathbb{Z}$ and $\beta\mathbb{Z}\setminus\mathbb{Z}$?