Let $(N, \times)$ be a commutative semigroup and assume that a countably infinite subset $P$ of $N$ algebraically generates $N$, and let ${\mathcal F}(P)$ denote the set of all non-empty finite subsets of $P$. Assume that there exists a mapping
$$\quad \quad \mathtt P: N \to {\mathcal F}(P)$$
satisfying the following properties:
$$\tag 1 \forall \, p \in P, \mathtt P (p) = \{p\}$$
$$\tag 2 \forall \, a,b,c \in N, \; \text{If } c = ab \text{ then } \mathtt P(c) = \mathtt P(a) \cup \mathtt P(b)$$
Example: The function that maps every integer in $(\mathbb N^{\ge 2}, *)$ to its prime factors.
Question 1: Is every such structure the quotient of a universal one generated by the 'alphabet of letters' in $P$ creating 'words'?
Question 2: If the answer is yes how to we create the quotients? Are they all defined by factoring out relations, satisfying some rules, so that the conditions of $\mathtt P$ are still guaranteed to hold?
My Work
It looks like $(\mathbb N^{\ge 2}, *)$ is the universal structure with free generators the set of prime numbers.