Let $S$ be a discrete semigroup, define $\beta S$ as the set of ultrafilters on $S$ with the identification of S as the set of $p_s$ for $s\in S$ where $p_s$ is the principal ultrafilter that is determined by wehther $s\in A$ or not.
One can define a multiplication on $\beta S$ by extending the definition on $S$, charactarization theorem for multiplication states that:
$$\forall p,q\in \beta S :A\in pq \iff\exists B\in p \ \forall b\in B \ \exists C_b\in q : b\cdot C_b\subseteq A$$
I'm currently working on a small project which my interest is the structure of $C_{\beta S}(p) = {q\in \beta S : pq=qp}$, I tried to find some metirial on this subject online but i didn't.
Now let $S$ be a commutative semigroup, let $p,q\in \beta S$, it is fairly easy to show that:
$$(\forall A: A\in pq\iff\exists C\in q\ \exists B\in p \ \forall b\in B:b\cdot C\subseteq A )\implies pq=qp$$
I tried to prove the converse in few ways but it seems like that it is very strong assumption, do someone knows an example which countradicts this statement?