ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXI (1979)
Alex a n d er Abian (Ames, Iowa)
/с-inductive Boolean algebras and the existence of special ultrafilters
Abstract. For obvious reasons an atomless Boolean algebra has no complete ultrafilter U (i.e., S £ U implies gib S e U). In this paper we prove that any (atomless or not) /c-inductive Boolean algebra В has a complete ultrafilter U with respect to a family {Hyi<k of subsets H, of B, in the sense that if H t £ U and if gib Ht exists, then gib Я; e U for every i < k.
In what follows (В, is a Boolean algebra. Thus, B, with respect to is a complemented distributive lattice with minimum 0 and maximum 1. As usual, for every element z of B, we let z' denote the complement of z.
Let H be a subset of В whose greatest lower bound exists and is denoted by h, i.e.,
(1) fc = g lb H .
Corresponding to every subset such as H of B, we let H* denote the set of all the non-zero elements of В which are less than or equal to gib H or which are less than or equal to the complement of an element of H, i.e., (2) H* = {y| 0 < y ^ h or 0 < у ^ z' for some z e H } ,
Based on (1) and (2) we prove the following two principal lemmas.
Lem m a 1. Let U be an ultrafilter of В such that p e U for some p eH *.
I f H Я U , then h e U .
P ro o f. Assume on the contrary that H <= U and From (2) it follows that two cases may occur: (i) p ^ h or (ii) p ^ z', where z' is the complement of an element z of H. However, (i) cannot occur since U being an ultrafilter, (i) would imply that h s U . Similary, (ii) cannot occur since (ii) together with H ç U would imply that both z and z' are elements of the ultrafilter U. Thus, our assumption is false and the lemma is proved.
Lem m a 2. For every element x > 0 of В there exists an element у of H*
such that у ^ x (Le., H* is a dense subset of В —{0)).
P roof. From (1) and (2) it follows that lub H* = 1. Hence 0 < x
= (хл1иЬЯ*) = lub {х л t \ t e H* } . Thus (x a i) > 0 for some element t of H*. But then the conclusion of the lemma follows by choosing у = х л t.
2 A. Abian
De f in it io n. Let к be a cardinal number. The Boolean algebra (В, is called k-inductive if and only if every inversely well ordered (w.r.t.
subset S of non-zero elements of В with 5 < к has a non-zero lower bound.
Based on the above definition, we prove:
Th e o r e m. Let p > 0 be an element of a k-inductive Boolean algebra В and (Hi)i < k a family of subsets Ht of В such that
(3) ^ = gib Hi for every i < k.
Then, В has an ultrafilter U such that
(4) , p e U and H, e U implies ht e U for every i < k.
P roof. From Lemma 2 it follows that there exists an element p0 of Я*
such that p0 ^ p. Clearly p0 > 0 by (2). Similarly, from Lemma 2 and (2) it follows that there exists a chain such as:
(5) 0 < Pi < Po ^ P with р0 еНо and p t e Hf .
Now, let us assume that for a non-zero ordinal и < к there exists a chain such as (5) with i < u, i.e.,
(6) 0 < ... ^ р( ^ ... ^ Pi ^ Po ^ p with pi gHf and i < и < k.
If и is not a limit ordinal (and since и > 0), then pu^ 1 exists and by Lemma 2 and (2) there exists pue H * such that 0 < pu ^ pu- i - If и is a limit ordinal, then since и < к and since В is Zc-inductive there exists a non-zero element q of В such that q ^ to a lower bound of {pt \i < u}. But then again, by Lemma 2 and (2) there exists such that 0 < pu ^ q and consequently, 0 < pu ^ ... ^ Pi ^ ... ^ pi ^ Po ^ P- Thus, the chain in (6) can be extended to u. Therefore, by Zorn’s Lemma there exists a chain (7)
0 < ... ^ Pi ^ ... ^ pi ^ Po ^ p with Pi ^Hf and i < k.
In the above Zorn’s Lemma (or the Axiom of Choice) is needed since the choice of p* form Hf is not unique.
From (7) it follows that M — {p, p0, p x, ..., p{, ...} with i < к is a multi
plicative system of В such that 0 фМ. This is because for every element pt and pj of M, if i ^ j, then from (7) it follows that рг • pj = pt Ф 0. Hence, indeed M is closed under multiplication and 0 фМ. Thus there exists [1]
an ultrafilter U of В such that p e U and р,-бС/ with i < k. But then, in view of (3) and (4), the conclusion of the theorem follows from Lemma 1.
Re ma r k 1. Every Boolean algebra is obviously N0-inductive. Therefore, without additional hypothesis on a Boolean algebra, the result [2], which motivated our work, is the basic special case (i.e., for к = N0) of our theorem stated in the dual language.
k-inductive Boolean algebras 3
Co r o lla ry (Rasiowa-Sikorski). Let q Ф 1 be an element of a Boolean algebra A and (Ei)i<(0 a denumerable sequence of subsets Et of A such that e{ = . lub£ f for every i < со. Then A has a prime ideal P such that q e P and Et ç= P implies e i e P for every i < со.
Re ma rk 2. As mentioned below the definition of a /с-inductive Boolean algebra В and an ultrafilter of В can be extended to partially ordered sets and their generic subsets. Thus, a partially ordered set (P, without a mini
mum element is called k-inductive if and only if every inversely well ordered (w.r.t. subset S of P with 5 < к has a lower bound. Moreover, with respect to a family D = (Д ),< к of dense subsets of P a subset G of P is called D-generic [3] if and only if (i) x e G and y e G imply z e G with 2 ^ x and z ^ y, (ii) x e G and y ^ x imply y e G , (iii) G has a non-empty inter
section with every member of D.
Based on the above notions, a counterpart of the theorem runs as follows:
Co r o l l a r y. Let p be an element of a k-inductive partially ordered set P without a minimum element and D = (Di)i<k a family of dense subsets Dt of P.
Then P has a D-generic subset G such that p e G.
Proof. Just as in the case of the proof of the theorem, there exists a chain
• •• ^ Pi < ... < Pi ^ Po < p with PieDi and i < k.
But then obviously, [3] the subset G of P given by G = {x\ x ^ pt with i < k] is a D-generic subset of P and p e G , as desired.
References
[1] N. H. M cC oy, Rings and ideals, Carus Monograph (1948), p. 106.
[2] H. R a sio w a and R. S ik o rs k i, A proof o f the completeness theorem of Goedel, Fund.
Math. 37 (1950), p. 193-200.
[3] J. R. S h o e n fie ld , Unramified forcing, Proc. Symposia in Pure Math., Amer. Math.
Soc. (1) 13 (1971), p. 360.
DEPARTMENT OF MATHEMATICS IOWA STATE UNIVERSITY AMES, IOWA
