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4 13. FILTERS AND IDEALS IN PARTIAL ORDERS EXAMPLE 13.7. Suppose (P, ) is a l.o. Then F is a filter in (P, ) if and only if it is a final segment of (P, ), i.e., iff Vp E F \/q e¥(pq^qe¥). Now that we are equipped with the general notion of a filter, let us see how the other concepts from the beginning of this section translate into the broader context. An ultrafilter in (P, ) is a maximal filter in (P, ). Thus an ultrafilter in (V(X)\®, C) is the same thing as an ultrafilter on X, but the only ultrafilter in (P(X),C) is V{X) itself. Similarly, if (P, ) is a l.o., then the only ultrafilter in (P, ) is P itself. EXERCISE 13.5 (G). Show that if F is an ultrafilter in a p.o. (P, ), thenF ^ P if and only if P contains two incompatible elements. Are there ultrafilters in every p.o.? Yes. THEOREM 13.4. If (P, ) is a p.o., then there exists an ultrafilter F C P. Moreover, every filter F in (P, ) can be extended to an ultrafilter in (P, ). EXERCISE 13.6 (PG). Prove Theorem 13.4. EXAMPLE 13.8. Let I, J be nonempty sets. Define: Fn(I, J) = {p : p is a function from a finite subset of I into J} . EXERCISE 13.7 (G). (a) Convince yourself that p and q are compatible in (Fn(7, J), 3) iff p U q is a function. (b) Prove that a filter F in (Fn(I, J), 3) is an ultrafilter in this p.o. iff (JF is a function from / into J. EXAMPLE 13.9. If a is a minimal element in a p.o. (P, ), then the set F a = {p e¥ : a p} is an ultrafilter in (P, ). Ultrafilters of the form F a will be called principal ultrafilters or fixed ultrafilters. Ultrafilters which do not contain a smallest element will be called free ultrafilters. Let us now try to generalize the other notions introduced for filters on X. As you might have guessed, an ideal in a p.o. (P, ) is a subset I of P such that (I)* I is closed downwards, i.e., Vp G I\/q G P (q p — q G I) (II)* Every finite subset of I has an upper bound in I. Subsets of p.o.'s that are closed downwards are also called open. The following exercise explains why. EXERCISE 13.8 (G). Let (P, ) be a p.o. Then { I C P : I satisfies (I)*} is a topology on P. The notion of the dual ideal of a filter has no generalization to the context of all p.o.'s, because not every partial order admits a meanigful notion of the complement of an element p of P. One important class of p.o.'s for which such a notion is defined, the class of Boolean algebras, will be introduced in Section 13.3 and studied in detail in Chapter 25. The concept of a filter base has a straightforward generalization: A subset B of P is a filter base in (P, ) if it satisfies condition (II) of the definition of a filter, i.e., if every finite subset of B has a lower bound in B. It is easy to see that the family ¥(B) = {q G P : 3p G B (p q)} is a filter in (P, if and only if B is a filter base in (P,).