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.

That might be a silly question but : what is the semi group structure on $\beta \mathbb{Z}$ ? as far as i know cartesian product of ultrafilter are no longer ultrafilter, so the sum of two ultrafilter has no reason to be an ultrafilter in general ?
– Simon HenrySep 25 '12 at 7:33

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If you are interested in algebraic structures on the compactification of a topological group (here the whole numbers as a discrete group) then I would suggest that perhaps you should look at the Bohr rather than the Stone-Cech compactification
– jbcSep 25 '12 at 8:35

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There is a standard definition. A set X is in U+V iff {i in Z : {j in Z : i+j in X} in V} in U. See Hindman and Strauss's book or either of the following books by Todorcevic: "Topics in Topology", "introduction to Ramsey Spaces."
– Justin MooreSep 25 '12 at 11:55

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I agree with Justin's suggestion to see the book of Hindman and Strauss, "Algebra in the Stone-Cech Compactification". I suspect the answer to your question isn't known, but if it is (or, rather, if it was a few years ago) then this book would very probably contain it.
– Andreas BlassSep 25 '12 at 13:13

I have to disagree with jbc. The perspective "let's only look at compactifications which are groups" would ignore a whole bunch of interesting and actively studied semigroup compactifications (WAP and LUC being the obvious ones).
– Yemon ChoiSep 29 '12 at 5:24