It there any known way of obtaining the Banach fixed-point theorem from the Tarski fixed-point theorem or vice-versa?
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$\begingroup$ I'm intrigued - have you any indication that they might be? $\endgroup$– Andrew StaceyAug 4, 2010 at 15:33
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4$\begingroup$ I was thinking that the metric (in the Banach version) induces a foliation of the space, which could be seen as a poset. If things 'line up' just right, contraction could preserve this foliation just right, so that the Tarski LFP exists and is the same as the Banach one. $\endgroup$– Jacques CaretteAug 4, 2010 at 15:54
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$\begingroup$ Okay, you've sold me. I'll follow this question ... $\endgroup$– Andrew StaceyAug 4, 2010 at 16:24
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$\begingroup$ @Michal: you should make that an answer. It isn't exactly right, but close enough. $\endgroup$– Jacques CaretteMay 10, 2012 at 3:07
3 Answers
Hello,
I just found the question, so the answer might come a bit too lat, but.. Have a look at:
Paweł Waszkiewicz, "Common patterns for metric and ordered fixed point theorems.", In Proceedings of the 7th Workshop on Fixed Points in Computer Science (Luigi Santocanale ed.), 2010, pp. 83-87.
I attended this talk last summer, and it addresses exactly your question.
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2$\begingroup$ Here is a link: tcs.uj.edu.pl/~pqw/waszkificsfinal.pdf $\endgroup$ Jan 24, 2011 at 11:16
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1$\begingroup$ Perfect! And my intuition was not too far off either, which is nice! $\endgroup$ Jan 29, 2011 at 14:20
As suggested by Jacques, I turn my comment into an answer.
This is not exactly what you ask for, but it is related. Efe Ok in Section 3.4 in Chapter 6 of his yet-to-be-written book on ordered sets gives a proof of the Banach fixed point theorem using the Kantorovitch-Tarski fixed point theorem: files.nyu.edu/eo1/public/Book-PDF/CHAPTER%205.pdf https://sites.google.com/a/nyu.edu/efeok/books/CHAPTER%205.pdf
Look the article : M.Jawahiri, D. Misane, M. Pouzet. Retracts: graphs and ordrerd sets from the metric point of view. Contemporary Mathematics, 1986, vol. 57 pp. 175-226