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# Introduction: Jay Kangel

edited June 2018 in Chat

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.

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1.

Your link is broken as it has a trailing ]

Comment Source:Your link is broken as it has a trailing ]
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2.

Thanks. The link works now.

Comment Source:Thanks. The link works now.
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3.

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.

Comment Source: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.
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4.

Matthew, corrected. Thanks!

Comment Source:Matthew, corrected. Thanks!
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5.

Great! I'm looking forward to your proof!

Comment Source:Great! I'm looking forward to your proof!
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6.

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

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