I am wondering if there is an easy answer to the following question:

Let us consider a finite partially ordered set $P$. It is clear that there exists a $k$ such that there is an order embedding $P\rightarrow\mathbb{N}^k$. Furthermore, one can even deduce a strategy for finding a minimal $k$. Actually one has to determine the order dimension of $P$ (for instance, defined by realizers).

In the literature, there are many more approaches. However, I am wondering if there is an easy or at least existing answer to the following question:

Consider partial orders on antisymmetric (i.e. no inverse elements) cancellative monoids given by multiplication, say, from the left. So, $ab\leq b$ (or, dually, $b\leq ab$) for all $a,b\in H$. Now, what is the minimal number $l$ such that there exists an antisymmetric cancellative monoid $H$ with $l$ generators and an order embedding $P\rightarrow H$?

Thank you for any hints!