Let $\beta\omega$ be collection of all ultrafilters on $\omega$ (principal and non-principal). We endow $\beta\omega$ with an operation $+$ in the following way. For ${\bf a}, {\bf b}\in \beta\omega$, set $${\bf a}+{\bf b} = \big\{ N\subseteq \omega:\{x \in \omega:\{y\in\omega: x+y\in N\}\in {\bf b}\}\in {\bf a}\big\}.$$


Standard exercises show that ${\bf a}+{\bf b}\in \beta\omega$, and that the operation $+:\beta\omega \times \beta\omega \to \beta\omega$ is associative.


If we identify a member $n\in\omega$ with the principal ultrafilter containing the singleton $\{n\}$ as an element, the addition just introduced agrees with the addition given in $\omega$. In $\omega$, we have the following "completion property": given $a, b\in \omega$, there is $x\in \omega$ such that $a+x=b$ or $b+x=a$.


This motivates the following question: if ${\bf a},{\bf b} \in \beta\omega$, is there ${\bf x}\in\beta\omega$ with ${\bf a}+{\bf x} = {\bf b}$ or ${\bf b}+{\bf x} = {\bf a}$?



"Completion property" in $(\beta\omega,+)$
No. Kunen proved that there are many weak P-points in $\beta\omega-\omega$, which means they are nonprincipal ultrafilters that are not in the closure of any countable set of other nonprincipal ultrafilters. In particular, a weak P-point is never a sum of nonprincipal ultrafilters because $\mathbf a+\mathbf b$ is in the closure of the set $\{\mathbf b+n:n\in\omega\}$.


