An example of a monoid whose set of invertibles does not commute setwise with every element

Question

This is a follow-up to my previous question, here: Does there exist a non-commutative, non-pure, non-group monoid such that its set of invertible elements commutes with every element?. In my current question, I am asking for an example of a monoid $(M,*)$, such that its set of invertible elements $U$ does not commute setwise under multiplication with every element $a$ of $M$. That is, for at least one element $a$ of $M$, the set $a * U \neq U * a$. I prefer a finite example, if possible.

Answer 1

Take $M$ to be the finite monoid of functions from $[n] = \{ 1, 2, \dots n \}$ to itself, for $n \ge 2$. The group of invertible elements is $U = S_n$. Now consider the function $a \in M$ with constant value $1$. We have that $a U = a$ but $U a$ is the set of constant functions.