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Say $X$ is a set of subsets of some arbitrary set. Is there a name for a function $f:X\to X$ satisfying $f(A)\subseteq A$ for all $A\in X$?

More specifically, is there a name for an $f:X\to X$ with $f(A)\subseteq A$ and $f(f(A)) = f(A)$? In decision theory, this is how choice functions behave on sets of acts, but I wonder if there is a mathematical name for these properties.

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How can $f(X) \subseteq X$ possibly fail to hold? – Qiaochu Yuan Feb 15 '12 at 17:14
Do you mean to say $f(A)\subseteq A$ for $A\in X$? – David Mitra Feb 15 '12 at 17:16
@DavidMitra: Do you mean $A\subset X$ instead of $A\in X$? If $f(A)\subseteq A$ for $A\subset X$, then we can say that $f$ is $A$-invariant. – Damian Sobota Feb 15 '12 at 17:26
Yes. I was being foolish. See edit. Note especially the for all $A$ part. – Seamus Feb 15 '12 at 17:28
@Kevin C.: Equality of the two outer sets is not assumed! Taking the interior on a topological space is an interesting example of a function as in the question. – Stefan Geschke Feb 15 '12 at 17:48
up vote 4 down vote accepted

(I'm stating the following definitions for function defined on the entire power set, but they make sense for any subset of the power set...)

A function $f\colon \mathcal{P}(X)\to\mathcal{P}(X)$ such that $f(A)\subseteq A$ for all $A$ is said to be decreasing. (If $A\subseteq f(A)$ for all $A$, we say the function is increasing).

If the function satisfies $A\subseteq B\Rightarrow f(A)\subseteq f(B)$ for all $A$ and $B$, then we say $f$ is isotone.

If the function satisfies $f(f(A)) = f(A)$, then we say the function is idempotent.

A function that is increasing, isotone, and idempotent is called a closure operator. If, in addition, $$f(A) = \bigcup_{B\subseteq A,\ B{\rm\ finite}} f(B)$$ for all $A$, then we say the closure operator $f$ is algebraic. If $f(A\cup B) = f(A)\cup f(B)$ for all $A$ and $B$, then we say the closure operator is topological.

A function that is decreasing, isotone, and idempotent is called an interior operator. If in addition $f(A\cap B)=f(A)\cap f(B)$ for all $A$ and $B$, then we say the interior operator $f$ is topological. If $$f(A) = \bigcap\limits_{B\subseteq A, B\text{ finite}} f(B)$$ for all $A$, then we say the interior operator $f$ is algebraic.

So it looks like you might have an interior operator; it is certainly decreasing and idempotent, but you don't say enough to tell whether it is also isotone.

You can find some of this in George Bergman's Invitation to General Algebra and Universal Constructions, Section 5.3, pages 134-139.

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Thanks! This is perfect. And thanks for pointing out isotonicity (?). I think this will be an interesting extra property to explore. I have a feeling most of the interesting examples will be isotone. – Seamus Feb 15 '12 at 18:15

If you have a function with the properties in your question, then the function $g:A\mapsto X\setminus f(X\setminus A)$ satisfies $A\subseteq g(A)$ and $g(g(A))=g(A)$ for all $A$. This is called a hull operator. What you are looking for is the dual, and I am not aware of a name for this. Maybe we should call it core-operator. Also see my comment: The interior operator on a topological space satisfies your conditions.

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