Connections between similar looking theorems

Question

Consider the following three theorems:

• If $m,n$ are relatively prime, then $\varphi(mn) = \varphi(m)\varphi(n)$ (Where $\varphi$ is the totient function, the Euler-phi function)
• If $f: A \rightarrow B$ is injective and $C,D\subseteq A$ then $f(C \cap D) = f(C) \cap f(D)$
• If $X,Y$ are independent random variables then $Pr[X \cap Y] = Pr[X]Pr[Y]$

These all look a bit similar. The notions of injectivity, relatively prime, and independence seem related at the surface level, in that things are disjoint or distinguishable. Given that notion in the premise, the conclusion seems like a conditional homomorphism. Is there some more general theory that these can all be taken as corollaries of, or perhaps is this a coincidence? The proofs of these three are quite different. They are not just homomorphisms, but conditional on some notion of separation. I am hoping that there is some more general concept which can be instantiated as coprime/injective/independent objects.

This question is a bit vague and ambiguous, but I am hoping someone can provide a direction to read in. Perhaps in algebra or category theory. I am not too sure what to search for.

Thanks.

Answer 1

Well what are you asking is intriguing. In Universal Algebra there is a concept of "Homomorphism" basically the same that can be applied to groups, rings etc... But there is more, there are some partial functions that can act like "homomorphisms" and they are treated like "partial homomorphisms". If you take for example $\varphi\colon (\mathbb{N},\cdot)\rightarrow (\mathbb{N},\cdot)$ is not a total homomorphy but is partial if you restrict it to coprime numbers.

Read https://en.wikipedia.org/wiki/Partial_algebra for "Partial Algebras" where "Partial Homomorphisms" are described.