Coherent ultrafilters and nonhomogeneity
Jan Star´y
Abstract. We introduce the notion of acoherentP-ultrafilter on a complete ccc Boolean algebra, strengthening the notion of aP-point onω, and show that these ultrafilters exist generically underc=d. This improves the known existence re- sult of Ketonen [On the existence ofP-points in the Stone- ˇCech compactification of integers, Fund. Math.92(1976), 91–94]. Similarly, the existence theorem of Canjar [On the generic existence of special ultrafilters, Proc. Amer. Math. Soc.
110(1990), no. 1, 233–241] can be extended to show that coherently selective ultrafilters exist generically underc= covM.
We use these ultrafilters in a topological application: a coherentP-ultrafilter on an algebraBis anuntouchable point in the Stone space ofB, witnessing its nonhomogeneity.
Keywords: nonhomogeneity; ultrafilter; Boolean algebra; untouchable point Classification: 54G05, 06E10
1. Introduction
The article is organized as follows.
In Section 2, we describe the lattice P art(B) of partitions of a complete ccc Boolean algebra B and see how a given ultrafilter U on B interplays with this lattice.
In Section 3, we define coherentP-ultrafilters andcoherently selective ultrafil- ters on a complete ccc algebra and show that they exist generically, i.e., every filter with a small base can be extended into such ultrafilter, under conditions isolated in [K] and [C].
In Section 4, we recall the homogeneity problem for extremally disconnected compact Hausdorff spaces — the Stone spaces of complete Boolean algebras. We show the relevance of coherent ultrafilters to this question: a coherentP-ultrafilter on a complete ccc Boolean algebra is anuntouchable point in the corresponding Stone space.
2. The lattice of partitions
Recall that a partition of a Boolean algebra is a maximal antichain. We will denote the set of all infinite partitions of an algebra B by P art(B). We only consider infinite algebras.
DOI 10.14712/1213-7243.2015.123
Definition 2.1. Let Bbe a Boolean algebra. For two partitions P, Q ofB, say that P refines Qand write P Qif for eachp∈P there is exactly one q∈Q such that p≤q. For a family Q of partitions, say that a partition P of B is a common refinement ofQ ifP Qfor everyQ∈ Q.
The minimal possible size of a family of partitions without a common refine- ment is thedistributivity number h(B) of the algebraB.
ClearlyP∧Q={p∧q;p∈P, q∈Q}\{0}is a common refinement of partitions PandQ, and the relationP Qis easily seen to be a partial order onP art(B); in fact,P∧Qis the infimum of{P, Q}in (P art(B),), making (P art(B),∧,{1B},) a semilattice with unit.
Observation 2.2. For a complete Boolean algebraB, the order(P art(B),)is a h(B)-complete lattice. In particular,P art(B)is complete iff Bis a power algebra.
Proof: We show that, in fact, every system{Pα;α∈κ} ⊆P art(B) has a supre- mum. Fix any P from the system. For p ∈ P, put p0 = p and inductively define
pn+1=_ n q∈[
Pα;q∧pn6= 0o .
Obviously, p ≤ pn ≤ pn+1 for each n ∈ ω; put u(p) = W{pn;n∈ω}. It is easily verified that the setWPα={u(p);p∈P} does not depend on the choice of the starting partitionP. ClearlyWPα is a partition refined by every Pα; we show that it is the finest among such partitions, and therefore a supremum.
LetPα R for every α∈κ. It suffices to see that whenever p≤r for some p∈Pα and r∈ R, we also have u(p)≤r. This can be shown by induction for everypn as defined above. Let p∈P and letr be the only member of R such thatp≤r. Everyq∈S
Pαis below exactly oner′ ∈R, and ifr6=r′, thenq⊥p;
hencep1 =W {q∈S
Pα;q∧p6= 0} ≤ r. By the same argument, pn+1 ≤ r for everyn∈ω, henceu(p)≤randW
PαR.
We have shown that any system of partitions has a supremum. Hence to have an infimum for a systemQ of sizeκ <h(B), we only need to have a lower bound forQ. But this is precisely a common refinement ofQ, guaranteed byκ <h(B).
In the extreme case of an atomic algebra, the set of all atoms is clearly the finest partition, i.e. the smallest element ofP art(B).
Note that for atomless algebras, completeness is actually necessary in the pre- vious observation. The following example shows that in an atomless algebra B which is notσ-complete, two partitions can always be found that do not have a supremum inP art(B).
Example 2.3. LetA={an;n∈ω} ⊆ B be a countable subset without a supre- mum inB; without loss of generality,Ais an antichain. LetC be the completion ofB, and considerc=WC
A∈ C \ B. The element−c∈ C can be partitioned into some{xα;α∈κ}=X ⊆ B, as Bis dense inC.
Split every an ∈ A into a0n∨a1n, put b0 = a00, bn+1 = a1n∨a0n+1 and B = {bn;n∈ω}. Then clearlyWC
B =WC
A=c. Put P =A∪X, Q=B∪X. Now P, Qare partitions ofB, and we show that{P, Q}has no supremum inP art(B).
LetR∈P art(B) satisfyP, QR. Then there must be some r∈R such that r≥an, bn for alln; butr∈ B cannot be a supremum ofan, hencermeets some x∈X. In fact, we havex≤r, asX ⊆P∩QandP, QR. Then the partition R0 which containsr−x, x∈R0 instead ofr∈RsatisfiesP, QR0≺R. Hence Ris not a supremum.
2.1 The structure induced by partitions. LetBbe a complete ccc Boolean algebra. For P ∈ P art(B), let BP be the subalgebra completely generated by P ⊆ B. Denote the inclusion aseP :BP ⊆ B. IfP Q, leteQP be the inclusion of BQ in BP. The family {BP;P ∈P art(B)} together with the mappings ePQ forms a directed system of complete Boolean algebras indexed by the directed set (P art(B),).
Observation 2.4. In the setting described above,
(a) for eachP ∈P art(B), the algebraBP is isomorphic toP(ω);
(b) BP∧Q is completely generated byBP∪ BQ, and BP∨Q=BP∩ BQ; (c) BP∩ BQ={0B,1B}iffP∨Q={1B};
(d) forP Q, the embedding eQP :BQ⊆ BP is regular;
(e) for eachP ∈P art(B), the embedding eP :BP ⊆ B is regular.
Lemma 2.5. The algebra B, with the regular embeddings eP : BP → B, is a direct limit of the directed system of algebrasBP and mappingseQP.
For P ∈P art(B), letJP be the ideal on B generated byP ⊆ B. Note that JP∧Q =JP∩ JQ andJP ⊆ JQ for P Q. WriteB/P forB/JP andBP/P for BP/JP. WheneverP Q∈P art(B), we haveJP ⊆ JQ, hence the algebraB/Q is a quotient ofB/P; denote the quotient mapping by fPQ : B/P → B/Q. The family of algebrasB/P and mappingsfPQ for P, Q ∈ P art(B) forms an inverse system indexed by (P art(B),).
Observation 2.6. In the setting described above,
(a) for eachP ∈P art(B), the quotientBP/P is isomorphic toP(ω)/f in;
(b) the inclusionBP/P ⊆ B/P is a regular embedding.
Lemma 2.7. The algebraB, with the quotient mappingsfP : B → B/P, is an inverse limit of the inverse system of algebrasB/P and mappingsfPQ.
Employing the Stone duality, we can summarize that
Corollary 2.8. (a) Every infinite complete ccc algebra is a limit of a directed system of copies of P(ω). Dually, every infinite ccc extremally discon- nected compact space is an inverse limit of an inverse system of copies of βω.
(b) Every infinite complete ccc Boolean algebra is an inverse limit of an in- verse system of copies of P(ω)/f in. Dually, every infinite ccc extremally disconnected compact space is a direct limit of a system of copies of ω∗. For an ultrafilterU onB and P a partition of B, letUP =U ∩ BP, which is clearly an ultrafilter onBP. AsBP is isomorphic to P(ω), the ultrafilterUP can be viewed as an ultrafilter onω.
Observation 2.9. LetB be a complete atomless ccc algebra, letP, Q be parti- tions of B, and letU be an ultrafilter onB. Then
(a) P∩ U 6=∅if and only if UP is trivial,
(b) {P ∈P art(B);U ∩P =∅}is an open dense subset of (P art(B),), (c) UQ=UP∩ BQ forP Q,
(d) B=S
{BP;P∩ U=∅}.
3. Coherent ultrafilters
Definition 3.1. LetBbe a complete, atomless, ccc algebra. For a propertyϕof families of subsets ofω, we say that a subsetX ⊆ Bis acoherentϕ-family onBif for every partitionP ={pn;n∈ω}ofB, the family{A⊆ω;W{pn;n∈A} ∈X} of subsets ofωsatisfies ϕ.
For some properties ϕ, the coherent ϕ is actually no stronger than ϕ it- self. As an easy example, any antichain in B is a coherent antichain; and any filter F on B is a coherent filter, as for every partition P of B, the family {A⊆ω;W{pn;n∈A} ∈ F} is a filter on ω. Similarly, every ultrafilter on B is a coherent ultrafilter, and an ultrafilter that is coherently trivial is a generic ultrafilter onB. We are interested in ultrafilters with special properties, for which the coherent version becomes nontrivial.
It can be seen from the very definition that the ZFC implications between various classes of ultrafilters onω continue to hold for the corresponding classes of coherent ultrafilters onB. For instance, every coherent selective ultrafilter onB is a coherentP-ultrafilter onB, as every selective ultrafilter onωis aP-ultrafilter onω.
3.1 Coherent P-ultrafilters.
Definition 3.2. An ultrafilter U on a complete ccc algebraB is a coherent P- ultrafilter if for every partitionP ofB, the family {A⊆ω;W
{pn;n∈A} ∈ U}is aP-ultrafilter onω.
Seeing that the subalgebraBP is a copy ofP(ω), we can equivalently charac- terize coherentP-ultrafilters as follows.
Observation 3.3. Let B be a complete ccc algebra. An ultrafilter U on B is a coherent P-ultrafilter iff for every pair of partitions P and Q of B such that P Q, either U ∩Q6= ∅, or there is a set X ⊆P such that W
X ∈ U and for everyq∈Q, the set{p∈X;p∧q6= 0}is finite.
We show now that coherentP-points consistently exist. The proof is an itera- tion of the Ketonen argument of [K] for the existence ofP-points onω.
Proposition 3.4. Let B be a complete ccc Boolean algebra of size at most c.
Every filter onB with a base smaller than c can be extended to a coherentP- ultrafilter onBif and only if c=d.
Proof: Assumec=d and letF ⊆ B be a filter with a base smaller thanc. We will construct an increasing chain of filtersFα extending F, eventually arriving at a filterSFα, where eachFα takes care of a pair of partitions, as per 3.3.
Start withF0=Fand enumerate all partition pairsP Qas (Pα, Qα), where α < d runs through all isolated ordinals. If an increasing chain (Fβ;β < α) of filters has already been found such that everyFβ has a base smaller than cand has theP-ultrafilter property 3.3 with respect to the partition pairsPγ Qγ for γ < β, proceed as follows.
Ifαis a limit, take forFαthe filter generated byS
{Fβ;β < α}; thenFαstill has a base smaller thanc=d. We didn’t miss a partition pair here.
Ifα=β+ 1 is a successor, consider the partition pairPβQβ. If someq∈Qβ
is compatible with Fβ, let Fα =Fβ+1 be the filter generated by Fβ∪ {q} and be done with (Pβ, Qβ). If there is no suchqin Qβ, enumerateQβ as {qn;n∈ω}
and consider the refinement Pβ of Qβ. Without loss of generality, every qn ∈ Qβ is partitioned into infinitely many p ∈ Pβ; enumerate {p∈P;p < qn} as {pmn;m∈ω}. Let{aξ;ξ < κ} be a base ofFβ, for someκ <c.
Now perform the Ketonen construction for this step: for each ξ < κ, put fξ(n) = min{m;aξ∧pmn 6= 0}if there is such anm. The value offξ(n) is defined for infinitely manyn, corresponding to those qn whichaξ meets. In the missing places, fill the value offξ(n) with the next defined value (there must be some).
This yields a family{fξ :ω→ω;ξ < κ}of functions which cannot be dominating, asκ <c=d. Therefore, there is a functionf :ω→ωwhich is not dominated by any fξ; that is, for each ξ, we have f(n)> fξ(n) for infinitely manyn. We can assume thatf is strictly increasing.
Put a = W{pmn;n∈ω, m≤f(n)}. The element a is compatible with Fβ, because it meets everyaξ, as witnessed byf 6≤fξ. LetFαbe the filter generated by Fβ∪ {a}. This filter obviously extends Fβ, is generated by fewer than c elements, and has theP-ultrafilter property with respect to (Pβ, Qβ).
Now every ultrafilter extendingS{Fα;α <c} is a coherentP-ultrafilter onB that extendsF, because we have taken care of all possible partition pairsPQ, as requested by 3.3.
The other direction follows from [K] immediately. Being able to extend every small filterF ⊆ Binto a coherentP-ultrafilter is apparently stronger than being able to extend every small filterFonωto aP-point, which itself impliesc=d.
For completeness, we translate the Ketonen argument for the opposite direction into the algebraB, showing howd<ccan break the coherenceanywhere.
Assumed<cand let{fα;α <d}be a dominating family of functions. Choose any two countable partitions P Q ofB such that every qn ∈Qis partitioned
into countably many pmn ∈ P. For each α < d, put aα = S{pmn;m > fα(n)}.
The family {aα;α <d} ∪ {−qn;n∈ω} ⊆ B is centered, and the filter F that it generates has d < c generators. No ultrafilter on B that extends F can be a coherentP-ultrafilter, as witnessed byP Q.
We have shown that coherentP-ultrafilters consistently exist on complete ccc algebras of size not exceeding the continuum. On the other hand, there is consis- tently no coherentP-ultrafilter on any complete ccc algebra, as even the classical P-points need not exist [W]. Hence the existence of coherent P-ultrafilters is undecidable in ZFC.
Question 3.5. The consistency we have shown is what [C] calls “generic exis- tence”. Under our assumptions, coherentP-ultrafilters not only exist, but every small filter can be enlarged into one.
(a) Is it consistent that P-points exist on ω, but there are no coherentP- ultrafilters on complete atomless ccc algebras?
(b) Is it consistent that a coherentP-ultrafilter exists on a complete atomless ccc algebraB, but it does not exist on another?
(c) Is there a single “testing” algebraBwith the property that if there is a co- herentP-ultrafilter onB, then necessarilyc=d, and henceP-ultrafilters exist generically?
3.2 Coherent selective ultrafilters. Similarly to coherentP-ultrafilters, the arguments from [K] and [C] on existence of selective ultrafilters on ω can be strengthened to coherent selective ultrafilters on complete ccc algebras.
Definition 3.6. Let B be a complete ccc algebra. An ultrafilter U on B is a coherent selective ultrafilter iff for every pair of partitionsP andQofBsuch that P Q, either U ∩Q6= ∅, or there is a set X ⊆P such that W
X ∈ U and for everyq∈Q, the set{p∈X;p∧q6= 0}is at most a singleton.
Proposition 3.7. Let B be a complete ccc Boolean algebra of size at most c.
Then every filterFonBwith a base smaller thanccan be extended to a coherent selective ultrafilter onBif and only if c= cov(M).
Proof: Assume c = cov(M) and let F be a filter with a base smaller than c.
We will construct an increasing chain of filters extending F. Put F0 = F and enumerate all partition pairsP Qas{(Pα, Qα);α <cov(M) isolated}.
If an increasing chain (Fβ;β < α) of filters has been found such that every Fβ has a base smaller than cand has the selective property with respect to all {(Pγ, Qγ);γ < β}, proceed as follows.
Ifαis a limit, take forFαthe filter generated byS
{Fβ;β < α}. ThenFαstill has a base smaller thanc.
If α = β + 1 is a successor, consider (P, Q) = (Pβ, Qβ). Without loss of generality, both partitions are infinite, and everyqn ∈Q is infinitely partitioned intopmn ∈P.
If there is someq∈ Qcompatible with Fβ, letFα be the filter generated by Fβ∪ {q}. If there is no suchq∈Q, consider some base{aξ;ξ < κ} ofFβ, where
κ <c. Everyaξ intersects infinitely manyq∈Q: if aξ only meetsq1, . . . , qn∈Q, chooseaiξ disjoint with qi, respectively; then aξ ≤W
qi is disjoint with V aiξ — a contradiction.
Consider the setT = Πn∈ω{pmn;m∈ω}; the functionsϕ∈T are the selectors forQ. ViewT as a copy of the Baire spaceωω. If no selector forQis compatible withFβ, putTξ={ϕ∈T;W
rng(ϕ)⊥aξ}; then we have T =S
ξ<κTξ. But the sets Tξ cannot coverT, as κ < cov(M) and everyTξ is a nowhere dense subset ofT, which is seen as follows.
For a basic clopen subset [s] of T, there is some n > |s| such that aξ meets qn ∈Q, becauseaξ meets infinitely many qn. Hence somepmn meets aξ. Extend sintot so thatt(n) =m. Then [t]⊆[s] is disjoint withTξ.
Thus there must be a selectorϕ∈T withb=Wrng(ϕ) compatible with every aξ. Let Fβ+1 be the filter generated by Fβ ∪ {b}. Iterating this process, we obtain an increasing sequence of filters (Fα;α∈c) extendingF=F0. Now every ultrafilter extendingSFα is a coherent selective ultrafilter onBby 3.6.
4. Nonhomogeneity
Definition 4.1. A topological spaceX ishomogeneous if for every pair of points x, y∈X there is an autohomeomorphismhofX such thath(x) =y.
Extremally disconnected compact Hausdorff spaces, which are precisely the Stone spaces of complete Boolean algebras, are long known not to be homoge- neous. However, the original elegant proof due to Frol´ık [F] suggests no simple topological property of points to be a reason for this.
If a spaceXis not homogeneous, then pointsx, y∈Xfailing the automorphism property are often calledwitnesses of nonhomogeneity. In large subclasses of the extremally disconnected compacts, such witnesses have been found by isolating a simple topological property that is shared by some, but not all, points in the space.
Definition 4.2. A point xof a topological space X is an untouchable point if x /∈Dfor every countable nowhere dense subset D⊆X not containingx.
The subclass of extremally disconnected compact spaces where a witness of nonhomogeneity has not been explicitly described yet is currently reduced to the class of ccc spaces of weight at most continuum. In other extremally disconnected compacts, points with even stronger properties have been found. See [S] and [BS]
for history and pointers to the development of these questions.
4.1 An application to nonhomogeneity. Via Stone duality, the topic has a Boolean translation: we are looking for ultrafilters on complete ccc Boolean algebras of size (or, equivalently, algebraic density) at most continuum, which are discretely untouchable points in the corresponding Stone spaces. It is in this algebraic form that we actually deal with the question.
Proposition 4.3. Let B be a complete ccc algebra. Let U be a coherent P- ultrafilter onB. ThenU is an untouchable point in the Stone space of B.
Proof: We assume thatUis not generated by an atom, otherwise there is nothing to prove. Let R={Fn;n∈ω} be a countable nowhere dense set in St(B) such that Fn 6= U for all n. Choose some a0 ∈ F0 with −a0 ∈ U and put R0 = {F ∈R;a0∈ F} ⊆ R. Generally, if ai ∈ B+ for i < k are disjoint elements such that W
i<kai ∈ U/ and Ri = {F ∈R;ai∈ F}, consider S
i<kRi ⊆ R. If S
i<kRi =R, we are done, asW
i<kai ∈ U/ guaranteesU 6∈cl(R). Otherwise, let nk be the first index such that Fnk 6∈S
i<kRi and choose someak disjoint with W
i<kai such thatak∈ Fnk andak ∈ U/ .
This construction either stops at some k and we are done, or we arrive at an infinite disjoint system Q ={ai;i∈ω} ⊆ B+. Again, if W
Q 6∈ U, we have U 6∈cl(R). Otherwise, we can assume thatW
Q= 1, soQis a partition ofB. For eachai ∈Q, choose an infinite partition Pi ofai such that Pi∩S
Ri =∅ – this is possible, becauseRi⊆R is nowhere dense. NowP =S
PiQis a partition pair inB.
As U is a coherent P-ultrafilter and misses Q, there is some X ⊆ P with u=W
X ∈ U such that for everyi, the set{p∈X;p≤ai} is finite. This means thatu /∈ Fnfor alln: everyFnis in one particularai, sou∈ Fnwould mean that Fn contains one of the finitely many{p≤u;p≤ai}. But this is in contradiction withPi∩SRi=∅. Sou∈ U isolatesU from cl(R).
In fact, we have proven a slightly stronger statement: U escapes the closure of any nowhere dense set that can be covered by countably many disjoint open sets.
Acknowledgment. The author wishes to thank the anonymous referee for help- ful comments which cleaned and improved the presentation.
References
[BS] Balcar B., Simon P.,On minimalπ-character of points in extremally disconnected com- pact spaces, Topology Appl.41(1991), 133–145.
[C] Canjar R.M.,On the generic existence of special ultrafilters, Proc. Amer. Math. Soc.
110(1990), no. 1, 233–241.
[F] Frol´ık Z.,Maps of extremally disconnected spaces, theory of types, and applications, in Franklin, Frol´ık, Koutn´ık (eds.), General Topology and Its Relations to Modern Analysis and Algebra, Proceedings of the Kanpur topological conference (1971), pp. 131–142.
[K] Ketonen J.,On the existence ofP-points in the Stone- ˇCech compactification of integers, Fund. Math.92(1976), 91–94.
[S] Simon P.,Points in extremally disconnected compact spaces, Rend. Circ. Mat. Palermo (2). Suppl. 24 (1990), 203–213.
[W] Wimmer E.L., The Shelah P-point independence theorem, Israel J. Math43(1982), no. 1, 28–48.
Czech Technical University, Prague
(Received April 10, 2014, revised June 10, 2014)
