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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.