Timeline for Which complete lattices arise as images of the Galois connections induced by binary relations?
Current License: CC BY-SA 3.0
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| Apr 20, 2014 at 5:56 | comment | added | მამუკა ჯიბლაძე | @JosephVanName Right. And yes, you've been right, I checked Davey&Priestley (second edition), it is Theorem 3.9 there. | |
| Apr 19, 2014 at 17:52 | comment | added | Joseph Van Name | For finite lattices, the collection of all join (meet)-irreducible sets is the smallest join (meet)-dense subset, so that will be a canonical choice. More generally, if a complete lattice has no infinite chains, then the join(meet)-irreducible sets forms the smallest join(meet)-dense subset. However, I can only think of canonical choices of join-dense subsets for certain specialized classes of complete lattices. I therefore cannot think of a canonical join-dense subset in general. | |
| Apr 19, 2014 at 17:40 | vote | accept | მამუკა ჯიბლაძე | ||
| Apr 19, 2014 at 17:40 | comment | added | მამუკა ჯიბლაძე | I like this very much, thank you! One question - in the last statement (about duality between relations and triples), could not one further restrict $f$ and $g$ in such a way as to obtain some sort of "canonical representation" of a given $L$ by a relation? E. g. by taking $f$ and $g$ inclusions? Of course one almost never has smallest join- or meet-dense subsets, but maybe there are some canonical choices... | |
| Apr 19, 2014 at 17:02 | history | edited | Joseph Van Name | CC BY-SA 3.0 |
added 1568 characters in body
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| Apr 19, 2014 at 16:46 | history | answered | Joseph Van Name | CC BY-SA 3.0 |