CHAPTER 13

Filters and Ideals in Partial Orders

This chapter contains assorted results on filters and partial orders. The first section

is devoted to the general notion of a filter in a p.o. This material will be used in

several later chapters and probably should be read up front. Section 13.2 is devoted

to applications of ultrafilters to an important construction in model theory. This

material will not be needed until Chapter 24. In Section 13.3 we develop the theory

of Boolean algebras (which can be looked at as the theory of a very special kind of

partial order) to the extent required for the proofs of Lemmas 19.7 and 19.8.

13.1. The general concept of a filter

Recall that if X is a nonempty set, then a filter on X is a subfamily J7 of V(X)

such that

(i)

J7

is closed under supersets, i.e., VY e FVZ C X (Y C Z -* Z e

J7)',

(ii)

J7

is closed under finite intersections, i.e., f\ H e T for all H G

[F]u;.

Recall also that a filter T is proper if 0 £ T. A maximal proper filter on X is called

an ultrafilter on X.

EXAMPLE

13.1. If a e X then Ta = {Y C X : a e Y} is a proper filter on X.

It is called the principal ultrafilter determined by a.

EXAMPLE

13.2. The subsets of the unit interval [0,1] of Lebesgue measure one

form a proper filter T\ on [0,1].

EXAMPLE

13.3. If (X, r) is a topological space and x G l , then the family

Afx = {Y C X : 3U e r (x e U CY)}

forms a proper filter on X, called the neighborhood filter of x or neighborhood system

at x.

A filter T on X is uniform if \Y\ = \X\ for all Y e

J7.

The filter in Example 13.1

is uniform only if \X\ = 1. Thus, every uniform ultrafilter on an infinite set X is

nonprincipal. Moreover, if X is denumerable, then the notions of a nonprincipal

and a uniform ultrafilter on X coincide. An ultrafilter on a set X of cardinality Ki

is uniform if and only if it contains all cocountable subsets of Xy i.e., all subsets of X

with countable complement. The filter of Example 13.2 is uniform. Neighborhood

filters as in Example 13.3 are uniform in some but not all topological spaces.

Given a proper filter T on a set X, the dual ideal to T is the family

r =

{ r c i :

X\YG

J7}.

The sets in

V(X)\J7*

are called the stationary sets with respect to

J7,

or simply the

J7-stationary sets.

i

http://dx.doi.org/10.1090/gsm/018/01