It there any known way of obtaining the Banach fixedpoint theorem from the Tarski fixedpoint theorem or viceversa?

$\begingroup$ I'm intrigued  have you any indication that they might be? $\endgroup$– Andrew StaceyAug 4 '10 at 15:33

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 '10 at 15:54

$\begingroup$ Okay, you've sold me. I'll follow this question ... $\endgroup$– Andrew StaceyAug 4 '10 at 16:24

$\begingroup$ @Michal: you should make that an answer. It isn't exactly right, but close enough. $\endgroup$– Jacques CaretteMay 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. 8387.
I attended this talk last summer, and it addresses exactly your question.

2$\begingroup$ Here is a link: tcs.uj.edu.pl/~pqw/waszkificsfinal.pdf $\endgroup$ Jan 24 '11 at 11:16

$\begingroup$ Perfect! And my intuition was not too far off either, which is nice! $\endgroup$ 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 yettobewritten book on ordered sets gives a proof of the Banach fixed point theorem using the KantorovitchTarski fixed point theorem: files.nyu.edu/eo1/public/BookPDF/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. 175226