Invariance of syndetic set in topological semigroup

Question

Let $T$ be a topological semigroup. $A\subseteq T$ is syndetic, if there is compact set $K\subseteq T$ with $T=KA$.

Let syndetic set $A\subseteq T$ and $g\in T$ be given. I think that $Ag^{-1}=\{y: yg\in A\}$ is syndetic. Indeed since $A$ is syndetic, there is compact set $K$ with $T=KA$. For every $t\in T$, since $tg\in KA$, hence $t\in KAg^{-1}$. This implies that $T= KAg^{-1}$ i.e.

if $A$ is syndetic, then $B=Ag^{-1}$ is syndetic.

What can say about situation of $Ag$, is it syndetic?

Please help me to know it.

Answer 1

The answer depends on $g$ in the following way:

Proposition. Let $A$ be syndetic. Then $Ag$ is syndetic iff $Tg = T$.

Indeed, if $Ag$ is syndetic, then $KAg = T$ for some compact $K$.

Then $KAg \subseteq Tg$, so that $T = Tg$.

Conversely, if $Tg = T$, and $KA = T$, then $KAg = Tg = T$.

Answer 2

The answer is no. Take the semigroup $T=\{0,1\}$, viewed as a subgroup of the multiplicative integers, and equipped with the discrete topology. Then $A=\{1\}$ is syndetic (take $K=T$), but if $g=0$, then $Ag$ is not.