Let $\langle S, \cdot, \leq \rangle$ be an ordered semigroup (or monoid).
Suppose, we have some element $a \in S$, such that for each $b \in S$, $a \cdot b \leq a \cdot b \cdot a$ and $b \cdot a \leq a \cdot b \cdot a$, or $(a \cdot b) \vee (b \cdot a) \leq a \cdot b \cdot a$, if we have supremum.
I've never seen such property before, so I have a question: is there some examples of such elements like $a$ in literature? (If this property is already investigated, I'm not even sure).
If $a$ is a zero of $S$, then your condition is trivially satisfied. More generally, your condition is also satisfied if $a$ is an idempotent (i.e. $a^2 = a$) in the center of $S$ (i.e. $ab = ba$ for every $b \in S$).