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Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left topological semigroup.

  1. What are the invertible (left/right/both) elements of $\beta(G)$ as a semigroup?
  2. Is right invertiblility the same as left invertibility? (That is, if $xy=1$ does this mean that $yx=1$?)
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    $\begingroup$ No element outside of G is left or right invertible if G is discrete. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 21, 2022 at 21:54
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    $\begingroup$ The nonpricipal ultrafilters form a two sided ideal for any cancellative semigroup. See chapter 4 of Hindman and Strauss's book on Algebra in the Stone Cech compactification $\endgroup$
    – Benjamin Steinberg
    Commented Aug 21, 2022 at 21:56

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Corollary 4.33 of Hindman and Strauss's book on Algebra in the Stone Cech Compactification says that if $S$ is an infinite cancellative (discrete) semigroup, then the nonprincipal ultrafilters in $\beta S$ form a two-sided ideal. In particular every left invertible element is invertible and the units of $\beta S$ and $S$ coincide. If $S$ is a group this means there are no left or right invertible elements outside of $S$.

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