A *po-groupoid* is a groupoid $\langle A,\cdot\rangle $ such that the relation
defined by
$$
x \leq y \text{ if and only if } x \cdot y = x
$$
is a partial order on $A$, the order *related to $\langle A,\cdot\rangle $*.

For every poset $\langle A,\leq\rangle $ one can define a po-groupoid operation $*$ on $A$ setting $$ x * y := \begin{cases} x & \text{if } x\leq y \\ y & \text{if } x\not\leq y. \end{cases} $$ such that $\leq$ is the order related to $\langle A,*\rangle $.

Po-groupoids are obviously idempotent, but need not be associative
nor commutative. In spite of this, I think that every down-directed poset is related to a commutative po-groupoid. An important example is given by *semilattices:* Commutative idempotent semigroup;
these are exactly the po-groupoids related to posets $\langle A,\leq\rangle $ such that for every $x,y\in A$, $\inf\{x,y\}$ exists (where the product is given by the infimum).

I think I have an example of a poset that has no associative po-groupoid (a *po-semigroup*) related to it. My question is,

Is there any characterization in the literature of posets related to po-semigroups?

An important piece of information is that po-semigroups form a variety axiomatized by the following identities:

\begin{align*} (x \cdot y) \cdot z &\approx x \cdot (y \cdot z) \\ x \cdot x &\approx x \\ x \cdot y \cdot x &\approx y \cdot x. \end{align*}