Let $G$ be a discrete countable group and let $\beta G$ be the Stone-Cech compactification of $G$, which has the structure of a semigroup.
Is $\beta G$ left cancellable? What about right cancellable? (With respect to the semigroup operation).
By that I mean if $xy=xz$ does that mean that $y=z$? What about the if $yx=zx$?
This question is related to the question proposed in https://mathoverflow.net/questions/428924/stone-%c4%8cech-compactification-as-a-semigroup.
The answer is negative if the group is infinite. In fact, Theorems 5 and 6 of [1] give a complete characterization of the semigroups $S$ such that $\beta S$ is left [right] cancellative. Condition (f) of these theorems is the most interesting part:
Theorem 6. Let $S$ be a discrete semigroup. Then $\beta S$ is left cancellative if and only if there exist a finite group $G$ and a right zero semigroup $R$ such that $S$ is isomorphic to $G \times R$.
Theorem 7. Let $S$ be a discrete semigroup. Then $\beta S$ is right cancellative if and only if there exist a finite group $G$ and a left zero semigroup $L$ such that $S$ is isomorphic to $L \times G$.
Thus if $S$ is a discrete countable infinite group, it is not of the above form and hence is neither left nor right cancellative.
[1] N. Hindman and D. Strauss, Characterization of simplicity and cancellativity in $\beta S$. Semigroup Forum 75 (2007), no. 1, 70--76.