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 $l_1,\dots,l_n\in L$.
I want to know if a minimal generator set of $M$ has to be unique.