What's the name of this function property?

Question

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: $A \subseteq B \implies F(A) \subseteq F(B)$).

Answer 1

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.)