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
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