"Union-compatible" set properties

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"Union-compatible" set properties

Postby shrdlu » Thu Sep 30, 2010 5:45 am UTC

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?

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Re: "Union-compatible" set properties

Postby jestingrabbit » Thu Sep 30, 2010 6:40 am UTC

A filter satisfies an extra property.


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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Re: "Union-compatible" set properties

Postby imatrendytotebag » Thu Sep 30, 2010 6:44 am UTC

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

Consider the property "not P", let's call it P'. Now if A has P', then every subset of A also has P'. For if B is a subset of A and B has property P then A = B U (A - B) must have property P. Furthermore, if A has P' and B has P' then A U B must have P' (by the "if and only if" clause).

Furthermore, suppose P' is any property of sets in X such that if A has P' then all subsets of A have P' and if B also has P' then A U B has P'. Let P be the property not P'. Then if (A U B) has P, we must have A or B has P (otherwise both A and B would have P' and then their union would, too). Also, if either A or B has P then A U B must have P, since if A U B were P' then both A and B (being subsets of A U B) would be P'.

So basically, call a property Q of subsets of X "not special" if it satisfies both: A has Q implies subsets of A have Q and A, B have Q imply A U B have Q. Then the complement of a "special" property is a "not special" property and vise-versa.
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Re: "Union-compatible" set properties

Postby elseif » Fri Oct 01, 2010 12:09 am UTC

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.

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Re: "Union-compatible" set properties

Postby antonfire » Fri Oct 01, 2010 3:23 am UTC

A filter that satisfies this extra property is just an ultrafilter. Yes, they're useful. But nontrivial examples are I believe impossible to construct explicitly (without the axiom of choice).

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Re: "Union-compatible" set properties

Postby MartianInvader » Fri Oct 01, 2010 5:10 pm UTC

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 :)
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