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

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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I'm intrigued - have you any indication that they might be? – Loop Space Aug 4 '10 at 15:33
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 '10 at 15:54
Okay, you've sold me. I'll follow this question ... – Loop Space Aug 4 '10 at 16:24
@Michal: you should make that an answer. It isn't exactly right, but close enough. – Jacques Carette May 10 '12 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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Here is a link: tcs.uj.edu.pl/~pqw/waszkificsfinal.pdf – Michael Greinecker Jan 24 '11 at 11:16
Perfect! And my intuition was not too far off either, which is nice! – Jacques Carette Jan 29 '11 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

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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

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