12:32 AM
Yeah, I just excitedly barfed jargon there, after the connection hit. Here's an attempted unpacking:
Sticking to the set of integers `Z`, we can consider sets like `Z×n` for all `n∊Z`, i.e. just sets of all multiples of `n`. These are the so-called "ideals" of `Z`. In this setting, my word salad amounts to noticing that `(Z×n∧m)≡(Z×n)∩Z×n`.
If you throw away the semantics and just look at how `∩` and `∪` interact as abstract binary operations, then you get something called a (distributive) lattice. The abstract names of the lattice operations are "meet" and "join", written `∧` and `∨`, and they correspond to `∩` and `∪`, respectively.
More broadly, though, we can consider broader situations where we `+` and `×` are just binary operations on the objects of interest. When the distributive rule holds (and modulo a few minor technicalities), then you get something called a ring, deoted `R`. This is the most general setting where the concept of "ideal" holds. When all your ideals look like `R×a` for some `a∊R`, we call it the mouthful "principal ideal domain", or PID for short.
PIDs look and feel a whole lot integers in many respects. Examples abound and include booleans, the integers as mentioned, real numbers, polynomials with real coefficients, the Gaussian integers, etc.
So, arguably, the natural domain of `∧` and `∨` is the category of PIDs, which nicely motivates their extended domain over the complex numbers. A hardline stance might even declare that they are not scalar functions, instead treating numeric vector inputs as polynomial coefficients.

3 hours later…
4:13 AM
@B.Wilson OK, that was easier to follow than ChatGPT's longwinded babble. How do you meet and join polynomials?