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For some time I have been interested in closure operators in partially ordered sets whose elements are viewed as regions in some abstract space, something like point-free topology.

One interesting question is: Suppose that \( a \) is an element of a partially ordered set. Are there conditions that ensure there exists a collection \(E \) of elements, each of which is smaller than \( a \) but \( a \) is the least upper bound of \(E \)? If so, is there a sense in which the elements of \(E \) are as small as possible. Informally, maximize the amount of \( a \) that is generated by the set of elements. In a special case there is a theorem, the Krein-Milman theorem https://en.wikipedia.org/wiki/Krein–Milman_theorem, that applies to locally convex, Hausdorff, topological vector spaces that does just this.

A careful analysis of the proof of the Krein-Milman theorem shows that the statement of this theorem and its proof can be adapted so that a generalized version applies to partially ordered sets. The result of this analysis provides necessary and sufficient conditions for this result.

In a day or two I will post a proof of this result.

Again, thinking of elements of a partially ordered set as abstract regions, Suppose that \(A \) and \( B \) are to collections of elements, how does one say the regions in \(A \) "contain" the same part of the space as \( B \).

This is the kind of thing I like to do.

I would like to learn enough about category theory to translatematerial like this into the language of category theory.

## Comments

Your link is broken as it has a trailing ]

`Your link is broken as it has a trailing ]`

Thanks. The link works now.

`Thanks. The link works now.`

Hey Jay!

Welcome to the forums.

\(\LaTeX\) is a little non-standard on this website. You need to write

`\\( A \\)`

instead of`$A$`

for inline equations.`Hey Jay! Welcome to the forums. \\(\LaTeX\\) is a little non-standard on this website. You need to write `\\( A \\)` instead of `$A$` for inline equations.`

Matthew, corrected. Thanks!

`Matthew, corrected. Thanks!`

Great! I'm looking forward to your proof!

`Great! I'm looking forward to your proof!`

Welcome to the forum and course, Jay! It will be fun to see your result!

`Welcome to the forum and course, Jay! It will be fun to see your result!`