I will begin with a mostly motivational thought about the projective plane. In this plane, every two lines intersect at a singular point. Let's mark the lines set as $\mathcal{L}$ and the points set as $\mathcal{P}$.
So you can think of two binary operations as follows: $f_1: \mathcal{P}^2 \rightarrow \mathcal{L}$ and $f_2: \mathcal{L}^2 \rightarrow \mathcal{P}$.
Where $f_1$ takes some 2 points and returns the line that passes through both and $f_2$ takes 2 lines and returns the intersection. This works well, partially due to the duality between points and lines. (Edit: In this example, we must use $f_1$ and $f_2$ as partial functions without pairs of the same point. It may also be possible to give some arbitrary line that goes through the point/ point on the line.)
This made me think about a generalization for some collection of $m$ sets $\{A_i\}_{i=1}^m$. Where between some of them there is some binary operation(/maybe partial without duplicate elements). A way to think about that is by putting the sets in some directed graph where there is a binary operation between neighboring sets.
This made me think that almost everything I know of in algebra includes a set with a closed operation. My questions are the following:
Is it a known structure or some far-fetched idea?
a. Are there any known algebraic structures with more than one set? (like the given example of the projective plane)
Can this be reduced to a classic single-set structure?
What tools are there when encountering some unknown/exotic algebraic structure?
(I added the tag of category theory because I thought it may be related, but I neither see any way to use it nor am proficient enough in the subject)