Talk:Galois connection
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There was some confusion about the definition (whether it is "F(a) <= b if and only if a ≤ G(b)" or "F(a) <= b if and only if G(b) ≤ a"). I changed it back to the former version (sorry Axel), since it is more common in the order-theory literature I know of. This definition is also the only one that is currently cited in other articles (see adjoint functors, heyting algebra and order theory glossary). However, to make it still possible to use the historical definition, I include this one too. It should be called antitone Galois connection in Wikipedia.
At least the following authors use the current definition of Galois connection in a book published after 1990: Abramski, Borceux, Davey, Gierz, Hofmann, Jung, Keimel, Lawson, Mislove, Priestley, Scott. Especially [Davey & Priestley] is probably the introductory text-book in this field.
--Markus Krötzsch 20:53, 3 Feb 2004 (UTC)
- That's fine, we should definitely use the modern definition. When I found the discrepancy, I went back to the original papers and copied their antitone definition, not realizing that conventions had changed in the meantime. AxelBoldt 15:56, 14 Feb 2004 (UTC)
- I don't think that conventions changed completely. Where it is convenient (in some applications, such as formal concept analysis) the antitone definition is used as well. So it might be good to mention "monotone" or "antitone" in articles that use Galois connections. --Markus Krötzsch 18:55, 29 Mar 2004 (UTC)
It would be nice if somebody could amplify about applications, including the application cited to programming. My vague understanding is that it has something to do with programming semantics, perhaps denotational semantics where data objects form a complete partial order and operations on them are partial functions. I seem to recall that Galois connections are used in relation to formal methods, perhaps to ensure that program refinements preserve some correctness properties (liveness and safety) from the unrefined to the refined domain, but I don't know the details. I heard this from somebody who audited a course in program semantics at Oxford U, which used Galois connections as a unifying concept.
Fred Kintanar
Cebu City
- I'm afraid we have to wait until someone shows up who has audited that course at Oxford U. Can't you give a call to your friend? :-) AxelBoldt 15:53, 14 Feb 2004 (UTC)
As part of the whole Tony Hoare tradition, there are 'weakest preconditions' for something to happen with a program, and so on. This leads very quickly (if somewhat superficially) to the use of Galois connections. I don't see anything substantive about this on WP, currently.
Charles Matthews 09:16, 5 Apr 2004 (UTC)
About the definitions (again):
We currently have F(a) <= b if and only if G(b) ≤ a for antitone Galois connections. In contrast, a book on my desk (Ganter & Wille) says it should be b <= F(a) if and only if a ≤ G(b). This is not the same, since orders may be partial ("not ≤" is not the same as "≥"). Is the definition in the article also used somewhere (it certainly affects the results, since one gets no closure operators in this case, but obtains their duals -- kernel operators -- instead)? If not, it's an error and needs to be corrected. --Markus Krötzsch 23:39, 19 Feb 2005 (UTC)
Is it possible to change ≤ and <= to ≤a and ≤b to make it clear that there are 2 different orders that are related to the sets A and B.
Like this: (A,≤a) and (B,≤b) F(a) ≤b b if and only if a ≤a G(b).