min a b + max a b = a + b
GCD a b * LCD a b = a * b
μ(A ∩ B) + μ(A ∪ B) = μ(A) + μ(B)

What is the general name for these correspondences? And what are other examples?

• Positive (incl. 0) numbers, ordered by subtractability, ≤.
• Integers, ordered by divisibility, |.
• Measurable sets / probabilities, ordered by inclusion / entailment, ⊆.

@amiloradovsky maybe you could call it like, each of those operations on the left takes out one part of the things on the right, and it commutes? I don't think there's a term for it. Very interesting pattern though.

@amiloradovsky it's probably some kind of abstract category-theoretic adjunction or somesuch.

@popefucker I also suspect this. This could even justify CT for me, if true…

@amiloradovsky for the first two, it is meet(a,b)*join(a,b) = a*b, then.

@amiloradovsky third one is pretty much the same but you have to use mu on the sets too

@amiloradovsky this is possibly a property of the group action on those posets?

@amiloradovsky maybe just call it a "compatible group structure on the lattice" or some such :P

@popefucker Monoid action, perhaps: incredibility doesn't participate. I need to think more about it as an action.

@popefucker Action is a functor.
• Here we have a ("flat") category (of positive numbers as objects) w.r.t. ≤. There we have (co)product (min, max).
• We also have the addition monoid: viewed as a category, the numbers are the morphisms, on a single object.

What now…

@popefucker This seems to fall into the definition of a monoidal¹ preorder (generally, monoidal category), but I wasn't able to find this construction as an example.


@popefucker Generally, this seems to be a relation between the product and co-product in a , or at least . If so, then I'm wondering why such an interesting examples aren't better known…

Also found this, maybe will dig this stuff up a little, if the time will permit:

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