Talk:Knaster–Tarski theorem

Reference for this form of the theorem?
I have found this theorem in various textbooks (including the references mentioned in the article) only in the form the the set of fixed points is non-empty or that there is the largest/smallest fixed point. Are you aware of any textbook using the same formulation as in the article (the set of all fixed points is a complete lattice)? I would be grateful for any pointer.

BTW it is probably useful to mention that this theorem (or some form of it) is sometimes mentioned also in connectin with the name Kleene (as far as I remember, it was in some books from computer science and logic, but it was a long time ago when I stumbled upon this, so I am not able to give any exact reference).

--158.195.15.9 09:37, 23 November 2006 (UTC)


 * The theorem is presented in this article essentially as it was proved in Tarski's paper listed in the references section. The fact that the collection of fixed point forms a complete lattice implies that there is a fixed point — although Tarski proved the collection was nonempty before proving it to be complete.  As for Kleene, I suspect you are thinking of the Kleene fixpoint theorem.  That theorem assumes that the map for which we want fixed points is continuous and not merely monotone. Michael Slone (talk) 15:52, 23 November 2006 (UTC)


 * But the content here as written also assumes continuity: "If f(lim xn)=lim f(xn) for all ascending sequences xn, then the least fixpoint of f is lim fn(0).." If continuity is not required, it needs to be explicitly stated

2605:6000:E9CD:F700:C0F0:D454:AA17:F453 (talk) 07:16, 30 March 2015 (UTC)


 * Over at Kleene fixpoint theorem we had a references problem, and I found this paper on who found out what, regarding fixed point theorems. You could read it and find out who to reference, unfortunately I do not have the time now. The link is http://www.sciencedirect.com/science/article/pii/0020019082900655 131.155.234.39 (talk) 13:14, 23 April 2012 (UTC)

Any way to write $$\sqsubseteq$$ as a single character?
I tried to change the ≤ to a character $$\sqsubseteq$$, but my trials failed and the math code is biger than the rest of the text. If someone knows how to write this in HTML, as &xxxx; or as a unicode character, (s)he could embellish this article. — Preceding unsigned comment added by 189.178.44.14 (talk) 08:16, 30 September 2014 (UTC)


 * That would be ⊑, U+2291(SQUARE IMAGE OF OR EQUAL TO), or  as a numeric character reference. See Mathematical Operators for related characters. — Tobias Bergemann (talk) 09:55, 30 September 2014 (UTC)