# 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? –  Andrew Stacey 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 ... –  Andrew Stacey 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

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