Let $T$ be a topological semigroup. For a net $\{t_i\}$ in $T$, $t_i\to \infty$ is to mean that the net $\{t_i\}$ is ultimately outside each element of compact set $K$ in $T$.
In my research, I need to find conditions if $t_n\to \infty$, then $st_n\to\infty$. Clearly every topological semigroup does not have such property,Consider topological semigroup $T$ with right-identitystructure i.e. for all $s, t\in T$, $st=s$. But if $T$ does have left-identitystructure i.e. for all $s, t\in T$, $st=t$, then for every $\{t_n\}\subseteq T$ with $t_n\to \infty$, we have $tt_n\to \infty$ for all $t\in T$. Also, one can check that if $s^{-1} K$ is relatively compact for all $s\in T$ and all compact sets $K$ in $T$ then for every $\{t_n\}\subseteq T$ with $t_n\to \infty$, we have $tt_n\to \infty$ for all $t\in T$.
Can you help me to know another conditions on $T$ to implies that if $t_n\to \infty$, then $st_n\to\infty$ for all $s\in T$.