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. https://www.degruyter.com/view/j/mathm.2016.1.issue-1/mathm-2016-0002/mathm-2016-0002.xml 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.

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.
 
https://www.degruyter.com/view/j/mathm.2016.1.issue-1/mathm-2016-0002/mathm-2016-0002.xml
 
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.
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