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 $yx=1$?)
  • 2
    $\begingroup$ No element outside of G is left or right invertible if G is discrete. $\endgroup$ Aug 21, 2022 at 21:54
  • 3
    $\begingroup$ 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 $\endgroup$ Aug 21, 2022 at 21:56

1 Answer 1


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 $S$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.