Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|cite|improve this question
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 3

up vote 12 down vote accepted


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.

share|cite|improve this answer
Here is a link: –  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

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

share|cite|improve this answer

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:

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.