For any difficult math that’s way beyond me, I try to relate _in some fashion_ to mathematical morphology, which is used in image processing, particularly in the medical field.
As it is “hands on” (using ImageJ) I was able to quickly (within a month or so) understand it and through it, any “complete lattice” as mathematical morphology is a complete lattice.
Boosted my math intuition by uncountable amounts.
Now it seems that while Loewner’s theorem on monotone matrix functions does NOT fit into a complete lattice model, Loewner order CAN get part way there via a notion of Sponges.
Alas, as I looked at it,
“5.6 A non-sponge:
The Loewner order
The Loewner order considers a (symmetric) matrix A less than or equal to another (symmetric) matrix B if the difference B – A is positive semidefinite. This is a partial order compatible with the vector space structure of (symmetric) matrices, but it does not give rise to a lattice, or even a sponge.”
So, I didn’t get a satisfying shortcut. BUT this tells me that: a) This is not only NOT in a set of complete lattices but also b) it can barely be approached in that fashion.
Loewner order CAN be computed WITH CARE such as to fit the join/meet requirements in a limited fashion… but I’m getting a strong sense that whatever magic is with Loewner’s Theorem on Monotone Matrix Functions it’s simply beyond me. Still, it’s satisfying to get a tiny bit there, and I can at least chew on sponges instead for a bit.