Let $S$ be a semigroup with infinite cardinality, $A\subset S$ with $|A|<|S|$. Under what condition we may find a infinite net $\{s_\alpha; \alpha\in \Gamma\}$, such that $s_\alpha A \cap s_\beta A =\emptyset$ whenever $\alpha\neq \beta\in \Gamma$?
I know that this is insured when $S$ is a right cancellative semigroup. Then for each pair $B, C\subset S$ with $|B|, |C|<|S|$, there always exist $s\in S$, such that $B\cap sC=\emptyset$. Otherwise, for each pair of $s\neq t\in S$ would correspond with 2 different points in $B\times C$. Hence $|S|\le|B||C|$, which is impossible.