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The proof is currently available in the section “Some inequalities” of this PDF file.

The proof isn’t yet thoroughly checked for errors.

Note that I have not yet proved , but the proof is expected to be similar to the above.

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It considerably simplifies the formulas.

If you want to be on this topic, learn what is called “dependent lambda calculus”. (Sadly, I do not use it in my book explicitly, in order to make my book easier to understand. But I weight the possibility to rewrite my book in a dependent lambda calculus proof-assistant language, that is in the language of an automatic proof verification software, to make it even greater.)

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The calculations are currently presented in the chapter 3 “Some (example) values” of addons.pdf.

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After proving this I always felt that there is some “mystery” about meet of funcoids: It behaves in a weird way and what it is in general (not this one special counterexample case) is not known.

Today I noted a simple formula which decomposes : for every funcoids and and more generally for a set of funcoids. (It follows from that is an upper adjoint and that for every funcoid .) This way it looks much more clear and less counterintuitive.

So now it looks more clear, but I have not yet found particular implications of these formulas leading to interesting results.

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- ;
- ;
- .

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**Example** For a set of binary relations

does not imply that there exists funcoid such that .

The proof is currently available at this PDF file and this wiki page.

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**Conjecture** The following are equivalent (for every lattice of funcoids between some sets and a set of principal funcoids (=binary relations)):

- .
- (for every natural ).
- There exists a funcoid such that .

and are obvious.

I welcome you to actively participate in the research!

Please write your comments and idea both in the wiki and as comments and trackbacks to this blog post.

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