Connections between measures and multiplicative functions

Let us define an interesting object as a triple $(P, M, f)$ s.t.

1. $(P, \preceq , ×)$ is a poset equipped with commutative monoid structure s.t. its neutral element $1\in P$ is minimal (i.e. for any $x\in P$ it's true that $1\preceq x$),
2. $(M, +)$ is a commutative monoid,
3. $f\colon P\to M$ is a map such that $f(1)=0$ and the implication $$(c\preceq x \ \text{and} \ c\preceq y \Leftrightarrow c =1) \Rightarrow f(x\times y)= f(x)+f(y)$$ holds.

It seems pretty nice observation that a measure on a set and arithmetic multiplicative functions (this object usually named as just "multiplicative functions" in number theory) may be seen as examples of interesting objects:

1. If $P:= (\mathfrak{S} , \subseteq, \cup)$, where $\mathfrak{S}$ is a ring of subsets, $M:=([0,+\infty], +)$, then $f$ is a measure defined on the ring $\mathfrak{S}$,
2. If $P:=(\mathbb{Z}_{>0}, \mid, \times)$, $M:=(\mathbb{Z}_{>0}, \times)$, then $f$ is an arithmetic multiplicative function.

Are the previous observations just a fancy coincidence or there really exist some deep and non-trivial connections between measures and arithmetic multiplicative functions?