Let X be a countably infinite set. We call a property P of subsets of X special if it satisfies the rule:

(A U B) has property P if and only if at least one of A and B has property P.

Now obviously, for an element x of X, the property "the set contains x" is special.

Also, if S is a (possibly infinite) set of special properties, we can define the properties:

ANY(S) = "the set has at least one property of S", and

INF(S) = "the set has infinitely many properties of S"

It is easy to prove that the properties defined by those rules are special.

Are "special properties" known by a different name? What other methods for constructing special properties are there?

## "Union-compatible" set properties

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- jestingrabbit
- Factoids are just Datas that haven't grown up yet
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### Re: "Union-compatible" set properties

A filter satisfies an extra property.

http://en.wikipedia.org/wiki/Filter_%28mathematics%29

Namely, if P is special, P(A) and P(B), then [imath]P(A\cap B).[/imath]

I don't know if it has a particular name, but there it is.

Edit: fixed some clumsiness (of course meet and join are well defined for sets doofus!)

http://en.wikipedia.org/wiki/Filter_%28mathematics%29

Namely, if P is special, P(A) and P(B), then [imath]P(A\cap B).[/imath]

I don't know if it has a particular name, but there it is.

Edit: fixed some clumsiness (of course meet and join are well defined for sets doofus!)

Last edited by jestingrabbit on Thu Sep 30, 2010 8:17 am UTC, edited 1 time in total.

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- imatrendytotebag
**Posts:**152**Joined:**Thu Nov 29, 2007 1:16 am UTC

### Re: "Union-compatible" set properties

Not sure about the "special name", but here is an observation:

**Spoiler:**

Hey baby, I'm proving love at nth sight by induction and you're my base case.

### Re: "Union-compatible" set properties

Your notion of special is essentially "partition regularity". The definition of partition regular given on wikipedia is that whenever A U B has the property, either A or B has the property, but I've only ever seen it used in the context of properties that are preserved by supersets (indeed, every example given on the wikipedia page is), which gives the other direction.

### Re: "Union-compatible" set properties

This was dumb.

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- MartianInvader
**Posts:**809**Joined:**Sat Oct 27, 2007 5:51 pm UTC

### Re: "Union-compatible" set properties

No, no! Ultrafilters are really really cool!

One way to construct an ultrafilter is by picking an element t and using the property "t is in A". Such ultrafilters are called "principal" ultrafilters. Nonprincipal ultrafilters exist, but they are unconstructable. Nonetheless, they play a very important role in some parts of math, such as geometric group theory.

Sadly, I don't think ultrafilters need to be "special", though I could be wrong. I certainly doubt anyone could construct a counterexample

One way to construct an ultrafilter is by picking an element t and using the property "t is in A". Such ultrafilters are called "principal" ultrafilters. Nonprincipal ultrafilters exist, but they are unconstructable. Nonetheless, they play a very important role in some parts of math, such as geometric group theory.

Sadly, I don't think ultrafilters need to be "special", though I could be wrong. I certainly doubt anyone could construct a counterexample

Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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