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It there any known way of obtaining the Banach fixed-point theorem from the Tarski fixed-point theorem or vice-versa?

flag	asked Aug 4 2010 at 15:04 Jacques Carette 6,717●1●14●35

I'm intrigued - have you any indication that they might be? – Andrew Stacey Aug 4 2010 at 15:33

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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. – Jacques Carette Aug 4 2010 at 15:54

Okay, you've sold me. I'll follow this question ... – Andrew Stacey Aug 4 2010 at 16:24

@Michal: you should make that an answer. It isn't exactly right, but close enough. – Jacques Carette May 10 2012 at 3:07

Matteo Mio · Accepted Answer · 2011-01-24 10:16:19Z UTC

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.

Michał Kukieła · Answer 2 · 2012-05-15 12:11:57Z UTC

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

Driss Misane · Answer 3 · 2012-02-20 08:31:13Z UTC

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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

answered Feb 20 2012 at 8:31

Driss Misane
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Banach and Knaster-Tarski fixed point theorems — are they related?

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