As in Unique minimal generator set to any numerical semigroup?
Let $M$ be a multiplicatively closed set of natural numbers greater than one. $L\subseteq M$ is a generator set of $M$ if for any $m\in M$ there exist numbers $l_1\leq l_2\leq\cdots\leq l_k$ in $L$ such that $m=\prod l_i$. A minimal generator set of $M$ is a generator set $L$ such that for all $l\in L$ it holds that $l=l_1 \cdots l_n\implies n=1$ for all $l_1,\dots,l_n\in L$.
Let $\mathcal G(l_1,\dots,l_n)$ be the semigroup generated by $\{l_1,\dots,l_n\}$. In the comments to the question above a unique minimal generator set $\{l_1,l_2,\dots \}$ is constructed for a numerical semigroup $M$ with a generator set $L$:
$l_1=\min L$
$l_{n+1}=\min(L\setminus\mathcal G(l_1,\dots,l_n))$
If $M$ and $M'$ are numerical semigroups with generator sets $L$ and $L'$, then $M\cdot M'=\{m\cdot m'|m\in M\wedge m'\in M'\}$ is a semigroup with generator set $L\cup L'$.
Also $M\cap M'$ is a semigroup and my question is: is it possible to generally construct a generator set for $M\cap M'$, from (eventually minimal) generator sets $L$ and $L'$?