Follow

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?

@amiloradovsky hmmmm I have no idea

@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 *not posets, lattices.

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

@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.

¹ https://en.wikipedia.org/wiki/Monoidal_category#Monoidal_preorders

@popefucker Generally, this seems to be a relation between the product and co-product in a #monoidal #category, or at least #preorder. 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:

https://forum.azimuthproject.org/discussion/2082/lecture-21-chapter-2-monoidal-preorders

Andrew Miloradovsky@amiloradovsky@mathstodon.xyz• Positive (incl. 0) numbers, ordered by subtractability, ≤.

• Integers, ordered by divisibility, |.

• Measurable sets / probabilities, ordered by inclusion / entailment, ⊆.