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While playing around with least-fixed-point constructions on a powerset lattice, I've found this property to be useful. Let's say that $F : \mathcal{P(U)} \to \mathcal{P(U)}$, and that $A \subseteq F(A)$ for all $A \in \mathcal{P(U)}$. What's the name of this property $A \subseteq F(A)$? I've read that $F$ is sometimes called "monotone" or "isotone," but I can't Google for those terms without running into the wrong definitions (i.e. what most call monotone: $A \subseteq B \implies F(A) \subseteq F(B)$).

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A map $F\colon\mathcal{P}(X)\to\mathcal{P}(X)$ is said to be increasing if $A\subseteq F(A)$ for all $A\in\mathcal{P}(X)$.

The map is said to be isotone if $A\subseteq B$ implies $F(A)\subseteq F(B)$.

E.g., see Lemma 5.3.1 and Definition 5.3.2 in George Bergman's An Invitation to General Algebra and Universal Constructions. (Link is to the PDF of Chapter 5; the definitions are on page 16 of that document, which corresponds to page 140 of the book; other parts of the book can be accessed through the links in this page.)

(Other common properties are decreasing, if $F(A)\subseteq A$ for all $A$; and idempotent, if $F(F(A)) = F(A)$ for all $A$. A map $F$ is a closure operator if it is increasing, isotone, and idempotent; and it is an interior operator if it is decreasing, isotone, and idempotent.)

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I think there should be a word added on the Knaster-Tarski theorem, don't you? :-) –  Asaf Karagila May 8 '11 at 22:31
Thanks for the quick answer! I guess I'm expecting too much of English, because most hits for "increasing" take it to mean what you called "isotone". Also, I've just discovered that what you called "increasing" is also called "extensive," but most hits for "extensive" are talking about extensionality. Geez. –  Neil Toronto May 8 '11 at 22:35
@Neil: Google tends to be useless if you try to google a mathematical term that is a single common word without any context. Google "interior", and good luck finding a link that gives you the interior of a set in topology! On the other hand, googling "interior" and "topology" gives it to you in the top link. "Increasing", "isotone", and "extensive" are overloaded words, but there you are. –  Arturo Magidin May 8 '11 at 22:41
@Asaf: I'm not sure I understand... –  Arturo Magidin May 8 '11 at 22:42
@Asaf: I'm totally down with Mr. Knaster and Mr. Tarski. But I'm after a more constructive fixed point - one that's built "from below" by transfinite recursion instead of "from above" by an intersection. Most of what I find on fixed-point theory uses continuity (supremum/join/union-preserving) to do it, but I don't think I have continuity. –  Neil Toronto May 8 '11 at 22:44

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