Continuous and other finitely generated canonical cofinal maps on ultrafilters
aa r X i v : . [ m a t h . L O ] A p r CONTINUOUS AND OTHER FINITELY GENERATEDCANONICAL COFINAL MAPS ON ULTRAFILTERS
NATASHA DOBRINEN
Abstract.
This paper investigates conditions under which canonicalcofinal maps of the following three types exist: continuous, generatedby finitary end-extension preserving maps, and generated by finitarymaps. The main theorems prove that every monotone cofinal map on anultrafilter from a certain class of ultrafilters is actually canonical whenrestricted to some cofinal subset. These theorems are then applied to findconnections between Tukey, Rudin-Keisler, and Rudin-Blass reducibili-ties on large classes of ultrafilters.The main theorems on canonical cofinal maps are the following. Undera mild assumption, basic Tukey reductions are inherited under Tukeyreduction. In particular, every ultrafilter Tukey reducible to a p-pointhas continuous Tukey reductions. If U is a Fubini iterate of p-points,then each monotone cofinal map from U to some other ultrafilter isgenerated (on a cofinal subset of U ) by a finitary map on the base treefor U which is monotone and end-extension preserving - the analogueof continuous in this context. Further, every ultrafilter which is Tukeyreducible to some Fubini iterate of p-points has finitely generated cofinalmaps. Similar theorems also hold for some other classes of ultrafilters. Introduction
In this paper, ultrafilters on countable base sets are considered to bepartially ordered by reverse inclusion. A map from an ultrafilter U to anotherultrafilter V is cofinal if every image of a cofinal subset of U is a cofinal subsetof V . We say that V is Tukey reducible to U and write V ≤ T U if and onlyif there is a cofinal map from U to V . When U ≤ T V and V ≤ T U , then wesay that U is Tukey equivalent to V and write U ≡ T V . It is clear that ≡ T isan equivalence relation, and ≤ T on the equivalence classes forms a partialordering. The equivalence classes are called Tukey types . We point out thatsince ⊇ is a directed partial ordering on an ultrafilter, two ultrafilters areTukey equivalent if and only if they are cofinally similar ; that is, there isa partial ordering into which they both embed as cofinal subsets (see [18]). Mathematics Subject Classification.
Primary 03E04, 03E05, 03E35, 06A07; Sec-ondary 054A20, 054D.
Key words and phrases. ultrafilter, continuous cofinal map, finitely generated cofinalmap, Tukey, Rudin-Keisler, Rudin-Blass.This work was partially supported by National Science Foundation Grants DMS-142470 and DMS-1600781, Simons Foundation Collaboration Grant 245286, and a Uni-versity of Denver Faculty Research Grant.
Thus, for ultrafilters, Tukey equivalence is the same as cofinal similarity.An equivalent formulation of Tukey reducibility, noticed by Schmidt in [14],shows that
V ≤ T U if and only if there is a Tukey map from V to U ; thatis a map g : V → U such that every unbounded (with respect to the partialordering ⊇ ) subset of V is unbounded in U .This paper focuses on the existence of canonical cofinal maps of threetypes: continuous, approximated by basic maps (end-extension and levelpreserving finitary maps - see Definitions 2.2 and 4.2), and approximatedby monotone finitary maps (see Definition 6.2). In each of these cases, theoriginal cofinal map is generated by the approximating finitary map.The notion of Tukey reducibility between two directed partial orderingswas first introduced by Tukey in [18] to study the Moore-Smith theory ofnet convergence in topology. This naturally led to investigations of Tukeytypes of more general partial orderings, directed and later non-directed.These investigations often reveal useful information for the comparison ofdifferent partial orderings. For example, Tukey reducibility preserves calibre-like properties, such as the countable chain condition, property K, precalibre ℵ , σ -linked, and σ -centered (see [16]). For more on classification theories ofTukey types for certain classes of ordered sets, we refer the reader to [18],[3], [11], [15], and [16]. As the focus of this paper is canonical cofinal mapson ultrafilters, and as we have recently written a survey article giving anoverview of the motivation and the state of the art of the Tukey theory ofultrafilters (see [5]), we present here only the background and motivationsrelevant for this work.For ultrafilters, we may restrict our attention to monotone cofinal maps.A map f : U → V is monotone if for any X, Y ∈ U , X ⊇ Y implies f ( X ) ⊇ f ( Y ). It is not hard to show that whenever U ≥ T V , then there isa monotone cofinal map witnessing this (see Fact 6 of [8]).As cofinal maps between ultrafilters have domain and range of size con-tinuum, a priori, the Tukey type of an ultrafilter may have size 2 c . In-deed, this is the case for ultrafilters which have the maximum Tukey type([ c ] <ω , ⊆ ). However, if an ultrafilter has the property that every Tukey re-duction from it to another ultrafilter may be witnessed by a continuousmap, then it follows that its Tukey type, as well as the Tukey type of eachultrafilter Tukey reducible to it, has size at most continuum. This is thecase for p-points. ANONICAL COFINAL MAPS 3
Definition 1.1.
An ultrafilter U on ω is a p-point iff for each decreasingsequence X ⊇ X ⊇ . . . of elements of U , there is an U ∈ U such that U ⊆ ∗ X n , for all n < ω . Definition 1.2.
An ultrafilter U on ω has continuous Tukey reductions ifwhenever f : U → V is a monotone cofinal map, there is a cofinal subset
C ⊆ U such that f ↾ C is continuous.The following Theorem 20 in [8] has provided a fundamental tool for allsubsequent research on the classification of Tukey types of p-points. Theorem 1.3 (Dobrinen/Todorcevic [8]) . Suppose U is a p-point on ω .Given a monotone cofinal map f from U into another ultrafilter, there isan X ∈ U such that f is continuous on C = { Y ∈ U : Y ⊆ X } . Inparticular, U has continuous Tukey reductions. Moreover, these continuousTukey reductions are generated by monotone basic functions. Remark 1.4.
The proof of Theorem 20 in [8] shows that p-points have thestronger property of monotone basic Tukey reductions (see Definition 2.2).It was later proved by Raghavan in [13] that any ultrafilter Tukey re-ducible to some basically generated ultrafilter has Tukey type of cardinalityat most c . The notion of a basically generated ultrafilter first appeared asDefinition 15 in [8]. The slightly modified version from [13] will be used inthis paper, as it will simplify statements. In this paper, by a filter base foran ultrafilter U , we mean a cofinal subset of U which is closed under finiteintersections. Definition 1.5 (Definition 13, [13]) . An ultrafilter U on ω is basically gen-erated if it has a filter base B ⊆ U such that each sequence h A n : n < ω i ofmembers of B converging to another member of B has a subsequence whoseintersection is in U .It was shown in [8] that the class of basically generated ultrafilters con-tains all p-points and is closed under taking Fubini products. It is stillunknown whether the class of all Fubini iterates of p-points is the same asor strictly contained in the class of all basically generated ultrafilters.Continuous cofinal maps provide one of the main keys to the analysis ofthe structure of the Tukey types of p-points (see for instance [8], [13] and[12]). Moreover, continuous cofinal maps are crucial to providing a mech-anism for applying Ramsey-classification theorems on barriers to classifythe initial Tukey structures and Rudin-Keisler structures within these for N. DOBRINEN a large class of p-points: selective ultrafilters in [13]; weakly Ramsey ultra-filters and a large hierarchy of rapid p-points satisfying partition relationsin [9] and [10]; and k -arrow ultrafilters, hypercube ultrafilters, and a largeclass of p-points constructed using products of Fra¨ıss´e classes satisfying theRamsey property in [7]. Similar methods were developed for a hierarchy ofnon-p-points above a Ramsey ultrafilter in [6].Continuous cofinal maps are also used in the following theorem, whichreveals the surprising fact that the Tukey and Rudin-Blass orders sometimescoincide. Recall that V ≤ RB U if and only if there is a finite-to-one map f : ω → ω such that V = f ( U ). The following is Theorem 10 in [13]. Theorem 1.6 (Raghavan [13]) . Let U be any ultrafilter and let V be a q-point. If V ≤ T U and this is witnessed by a continuous, monotone cofinalmap from U to V , then V ≤ RB U . In Section 2, we prove in Theorem 2.5 that, under a mild assumption, theproperty of having basic cofinal maps is inherited under Tukey reduction.The proof uses the Extension Lemma 2.4 showing that any basic monotonemap on a cofinal subset of an ultrafilter may be extended to a basic mono-tone map on all of P ( ω ). In particular, p-points satisfy the mild assumption;hence we obtain the following theorem. Theorem 2.6.
Every ultrafilter Tukey reducible to a p-point has basic, andhence continuous, Tukey reductions.
Combined, Theorems 1.6 and 2.6 imply the following.
Theorem 2.7. If U is Tukey reducible to some p-point, then any q-pointTukey below U is actually Rudin-Blass below U . The rest of the paper involves finding the analogues of Theorems 1.3 and2.6 for countable iterations of Fubini products of p-points and applying themto connect Tukey reduction with Rudin-Keisler and Rudin-Blass reductions.We now delineate these results.Section 3 is a primer, explicitly showing how any countable iteration ofFubini products of p-points, which we also simply call a
Fubini iterate ofp-points , can be viewed as an ultrafilter generated by trees on a front on ω . This precise way of viewing Fubini iterates of p-points sets the stage forfinding the analogue of Theorem 1.3 for this more general class of ultrafil-ters. While it is not possible to show that Fubini iterates of p-points havecontinuous Tukey reductions (as that is simply not true), we do show thatthe key properties of continuous maps hold for this class of ultrafilters. ANONICAL COFINAL MAPS 5
In Section 4 we define the notion of a basic map for Fubini iterates, whichis in particular an end-extension preserving map from finite subsets of thetree ˆ B of initial segments of members of a front B into finite subsets of ω (see Definition 4.2). This is the analogue of continuity for Fubini iterates ofp-points. One of the main results of this paper is the following. Theorem 4.4.
Fubini iterates of p-points have basic Tukey reductions.
Thus, monotone cofinal maps on Fubini iterates of p-points are continu-ous, with respect to the product topology on the space 2 ˆ B . As basic mapson fronts have the key property (end-extension preserving) of continuousmaps used to convert Tukey reduction to Rudin-Keisler reduction in [13],[9], [10], [7], and [6], it seems likely that they will play a crucial role in ob-taining similar results for ultra-Ramsey spaces of Chapter 6 of Todorcevic’sbook [17].Sections 5 and 6 contain applications of Theorem 4.4 to a broad classof ultrafilters. In Section 5, we directly apply Theorem 4.4 to obtain ananalogue of Theorem 10 of Raghavan in [13]. In Theorem 5.1, we prove thatif U is a Fubini iterate of p-points and V is a q-point Tukey reducible to U , then there is a finite-to-one map on a large subset of ˆ B , where B is thefront base for U , such that its image on U generates a subfilter of V . Oneof the consequences of this is the following. Theorem 5.3.
Suppose U is a finite iteration of Fubini products of p-points.If V is a q-point and V ≤ T U , then V ≤ RK U . This improves one aspect of Corollary 56 of Raghavan in [13] as V is onlyrequired to be a q-point, not a selective ultrafilter. The improvement thoughcomes at the expense of limiting U to a finite Fubini iterate of p-points. Itis unknown whether this can be extended to all Fubini iterates of p-points.In Section 6 we prove the analogue of Theorem 2.5 for ultrafilters Tukeyreducible to a Fubini iterate of p-points. (See Definition 6.2 for finitelygenerated Tukey reductions .) Theorem 6.3.
Let U be any Fubini iterate of p-points. If V ≤ T U , then V has finitely generated Tukey reductions. These finitary maps are an improvement on the maps ψ ϕ used in [13](see Definition 7 in [13]) in the sense that our finitary maps are shown togenerate the original cofinal maps. Theorem 6.3 is used to extend Theorem17 of Raghavan in [13] to the class of all ultrafilters Tukey reducible to someFubini iterate of p-points, in contrast to his result where U is assumed to be N. DOBRINEN basically generated. It is still open whether every ultrafilter Tukey reducibleto a p-point is basically generated (see discussion around Problem 7.6), asthe class of basically generated ultrafilters and the class of ultrafilters Tukeyreducible to a Fubini iterate of p-points may be very different.
Theorem 6.4. If U is Tukey reducible to a Fubini iterate of p-points, thenfor each V ≤ T U , there is a filter U ( P ) ≡ T U such that V ≤ RK U ( P ) . The paper closes with a list of open problems in Section 7.The results in Sections 2, 3 and 4 were completed in 2010, presented atthe Logic Colloquium in Paris that year, and have appeared in the unpub-lished preprint [4]. The present paper includes much revised presentationsand proofs of those results, new extensions of them, and additional appli-cations.
Acknowledgement.
Profuse thanks go to the referee, whose thoroughreading and suggestions greatly improved this paper, especially the presen-tation in Section 4.2.
Basic Tukey reductions inherited under Tukey reducibility
One of the crucial tools used to determine the structure of the Tukeytypes of p-points is the existence of continuous cofinal maps (Theorem 20in [8]). Continuity contributes to the analysis of the structure of the Tukeytypes of p-points by essentially reducing the number of cofinal maps un-der consideration from 2 c to c , with the immediate consequence that thereare at most c many ultrafilters Tukey reducible to any p-point. Continu-ity further contributes to finding exact Tukey and Rudin-Keisler struc-tures below certain classes of p-points satisfying partition relations. Thefact that each monotone cofinal map on a p-point is approximated by afinitary end-extension preserving function is what allows for application ofRamsey-classification theorems to find the exact Tukey and Rudin-Keislerstructures below the p-points forced by certain topological Ramsey spaces(see [13], [9], [10], and [7]). Other applications of cofinal maps representedby finitary end-extension preserving maps appear in Dobrinen’s contribu-tions in [2], and further in [6] where the precise Tukey and Rudin-Keilserstructures below ultrafilters forced by P ( ω k ) / Fin ⊗ k were found.The notion of a basic map is a strengthening of continuity, and is thesame as continuity when the domain is a compact subset of 2 ω (see Definition2.2 below). The Extension Lemma 2.4 shows that all basic Tukey reductionson some cofinal subset of an ultrafilter extend to a basic map on P ( ω ). Thiswill be employed in the proof of the main theorem of this section, Theorem ANONICAL COFINAL MAPS 7 W is Tukey reducible to a p-pointand V is a q-point, then W ≥ T V implies W ≥ RB V (see Theorem 2.7).We begin with some basic definitions. The following standard notationis used: 2 <ω denotes the collection of finite sequences s : n →
2, for n < ω .We use s, t, u, . . . to denote members of 2 <ω . For s, t ∈ <ω , we write s ⊑ t to denote that s is an initial segment of t ; that is, dom ( s ) ≤ dom ( t ) and t ↾ dom ( s ) = s . We also use a ⊑ X for sets a, X ⊆ ω to denote that, giventheir strictly increasing enumerations, a is an initial segment of X . a ⊏ X denotes that a is a proper initial segment of X .We would like to identify subsets of ω with their characteristic functions.For X ⊆ ω , we let χ X denote the characteristic function of X with domain ω . If X is infinite, we shall often abuse notation and use X to denote boththe set X and its characteristic function χ X . Given m < ω , we let χ X ↾ m denote the characteristic function of X ∩ m with domain m . Whenever noambiguity arises, we shall abuse notation and use X ↾ m to denote both thecharacteristic function χ X ↾ m and the set X ∩ m . For precision throughout,for s ∈ <ω , we use the notation [[ s ]] to denote s − ( { } ), the set of i in thedomain of s for which s ( i ) = 1. For pointwise images of a function f on aset S , we shall use the notation f [ S ] to denote { f ( x ) : x ∈ S } . In Sections3 and following, for a tree T ⊆ [ ω ] <ω , we shall use [ T ] to denote the set ofall branches through T , meaning the collection of all maximal branches (ifthere are any finite maximal branches) and all cofinal branches (if there areany infinite branches) through T . This will not cause any ambiguity. Definition 2.1.
Given a subset C of 2 <ω , we shall call a map ˆ f : C → <ω level preserving if there is a strictly increasing sequence ( k m ) m<ω such thatˆ f takes each member of C ∩ k m to a member of 2 m . A level preservingmap ˆ f is end-extension preserving if whenever m < n , s ∈ C ∩ k m , and t ∈ C ∩ k n , then s ⊑ t implies ˆ f ( s ) ⊑ ˆ f ( t ). A level and end-extensionpreserving map shall be called basic . A level-preserving map ˆ f is monotone if for each m < ω and s, t ∈ C ∩ k m , [[ s ]] ⊆ [[ t ]] implies [[ ˆ f ( s )]] ⊆ [[ ˆ f ( t )]]. Definition 2.2.
Let f be a map from a subset C ⊆ ω into 2 ω . We saythat f is represented by a basic map if there is a strictly increasing sequence( k m ) m<ω such that, letting(2.1) C = { X ↾ k m : X ∈ C , m < ω } , N. DOBRINEN there is a basic map ˆ f : C → <ω such that for each X ∈ C ,(2.2) f ( X ) = [ m<ω ˆ f ( X ↾ k m ) . In this case, we say that ˆ f generates f . If each monotone cofinal functionfrom an ultrafilter U to another ultrafilter is represented by a basic map onsome cofinal subset of U , then we say that U has basic Tukey reductions .Note that if f ↾ C is generated by a basic map ˆ f , then for each X ∈ C and m < ω , f ( X ) ↾ m = ˆ f ( X ↾ k m ). This fact will be used throughout thepaper.Recall that for a set C ⊆ <ω , [ C ] denotes the set of all branches through C . If C = { X ↾ k m : X ∈ C , m < ω } , where C ⊆ ω and ( k m ) m<ω is a strictlyincreasing sequence, then [ C ] ⊆ ω . Lemma 2.3.
Let ˆ f be a monotone basic map with domain C ⊆ <ω . Then ˆ f induces a monotone continuous map f ∗ on [ C ] by f ∗ ( X ) = S m<ω ˆ f ( X ↾ k m ) , for X ∈ [ C ] . Further, if ˆ f generates f on C and C ⊇ { X ↾ k m : X ∈C , m < ω } , then f ∗ ↾ C = f ↾ C ; hence f ↾ C is continuous.Proof. That f ∗ is continuous on [ C ] is trivial, since ˆ f is basic. Since ˆ f ismonotone, it follows that f ∗ is monotone. If ˆ f generates f on C , then trivially f ∗ ↾ C is simply f ↾ C . (cid:3) Lemma 2.4 (Extension) . Suppose U and V are nonprincipal ultrafilters, f : U → V is a monotone cofinal map, and there is a cofinal subset
C ⊆ U such that f ↾ C is represented by a monotone basic map ˆ f . Let ( k m ) m<ω bethe strictly increasing sequence such that the domain, C , of ˆ f is containedin S m<ω k m . Then there is a continuous monotone map ˜ f : 2 ω → ω suchthat(1) ˜ f is represented by a monotone basic map ˆ g on S m<ω k m ;(2) ˜ f ↾ C = f ↾ C ; and(3) ˜ f ↾ U is a cofinal map from U to V .Proof. Let ˆ f be a monotone basic map generating f ↾ C , and let ( k m ) m<ω be the levels on which ˆ f is defined. Thus, the domain of ˆ f is C = { X ↾ k m : X ∈ C , m < ω } , and for each s ∈ C ∩ k m , ˆ f ( s ) ∈ m . Claim.
There is a monotone basic map ˆ g , with domain S m<ω k m , whichgenerates a function ˜ f : 2 ω → ω such that ˜ f ↾ C = f ↾ C . Proof.
Since C is cofinal in U and U is nonprincipal, the finite sequence ofzeros of length k m is in C , for each m < ω . Let D = S m<ω k m and define ANONICAL COFINAL MAPS 9 ˆ g on D as follows: For t ∈ k m , define ˆ g ( t ) to be the function from m into 2such that for i ∈ m ,(2.3) ˆ g ( t )( i ) = max { ˆ f ( s )( i ) : s ∈ C, | s | ≤ k m , and [[ s ]] ⊆ [[ t ]] } . That is, ˆ g ( t )( i ) = 1 if and only if there is some s ∈ C such that | s | ≤ k m ,[[ s ]] ⊆ [[ t ]], and ˆ f ( s )( i ) = 1. It follows from the definition that ˆ g is monotoneand level preserving. Since ˆ f is monotone, ˆ g ↾ C equals ˆ f .To see that ˆ g is end-extension preserving, suppose t ⊏ t ′ , where t ∈ k m and t ′ ∈ k m ′ for some m < m ′ . Fix i < m . Suppose that ˆ g ( t ′ )( i ) = 1.Then there is some s ′ ∈ C ∩ k n such that i < n ≤ m ′ , [[ s ′ ]] ⊆ [[ t ′ ]],and ˆ f ( s ′ )( i ) = 1. Letting j = min { m, n } and s = s ′ ↾ k j , we see that s ∈ C and [[ s ]] ⊆ [[ t ′ ↾ k m ]], where t ′ ↾ k m = t ; moreover, ˆ f ( s )( i ) = 1,since ˆ f ( s ) = ˆ f ( s ′ ) ↾ j . It follows that ˆ g ( t )( i ) = 1. On the other hand, ifˆ g ( t ′ )( i ) = 0, then by the definition of ˆ g , ˆ f ( s )( i ) = 0 for all s ∈ C ∩ S n ≤ m k n such that [[ s ]] ⊆ [[ t ]]. Hence, ˆ g ( t )( i ) = 0. Therefore, ˆ g ( t ′ ) ↾ m = ˆ g ( t ).Now define ˜ f : 2 ω → ω by(2.4) ˜ f ( Z ) = [ m<ω ˆ g ( Z ↾ k m ) . Then ˜ f is generated by the basic map ˆ g . It follows that ˜ f is monotone. Sinceˆ g ↾ C equals ˆ f , it follows that ˜ f ↾ C = f ↾ C . (cid:3) Thus, ˜ f is continuous on 2 ω and (1) and (2) of the Lemma hold. To show(3), it suffices to show that ˜ f ↾ U has range inside of V , since ˜ f ↾ C equals f ↾ C which is monotone and cofinal in V . Let U ∈ U be given. Fix S ∈ C such that U ⊇ S . Then ˜ f ( U ) ⊇ ˜ f ( S ) = f ( S ) ∈ V , which concludes theproof. (cid:3) Now, in what is the main theorem of this section, we show that the prop-erty of having basic Tukey reductions is inherited under Tukey reducibilitybelow any ultrafilter having monotone basic Tukey reductions and satisfyingthe property ( ∗ ) below. Theorem 2.5.
Suppose that the ultrafilter U has monotone basic Tukeyreductions. Suppose further that for each monotone cofinal map f from U to another ultrafilter, there is some cofinal subset C ⊆ U such that f ↾ C isrepresented by a monotone basic function on some levels ( k m ) m<ω satisfyingthe following property: ( ∗ ) For each X ∈ C and each m < ω , there is a Z ∈ C such that Z ⊇ X and Z ↾ k m = X ↾ k m . Then every ultrafilter V Tukey reducible to U also has basic Tukey reduc-tions.Proof. Suppose that U satisfies the hypotheses and let V ≤ T U . We mayassume that U is nonprincipal, for the result holds immediately if U isprincipal. By Lemma 2.4, there is a map ˜ f : 2 ω → ω generated by amonotone basic map ˆ f : S m<ω k m → <ω , for some increasing sequence( k m ) m<ω , such that ˜ f ↾ U : U → V is a cofinal map. We shall let f denotethe restricted map ˜ f ↾ U . Suppose W ≤ T V , and let h : V → W be amonotone cofinal map. Extend h to the map ˜ h : 2 ω → ω defined as follows:For each X ∈ ω , let(2.5) ˜ h ( X ) = \ { h ( V ) : V ∈ V and V ⊇ X } . Notice that ˜ h is monotone. Furthermore, it follows from h being monotonethat ˜ h ↾ V = h .Define ˜ g = ˜ h ◦ ˜ f . Then ˜ g : 2 ω → ω and is monotone. Letting g denote˜ g ↾ U , we see that g = h ◦ f ; hence g : U → W is a monotone cofinal map.By the hypotheses, there is a cofinal subset
C ⊆ U and a monotone basicmap ˆ g : C → <ω generating g ↾ C such that ( ∗ ) holds, where ( k m ) m<ω isthe strictly increasing sequence associated with ˆ g and C = { X ↾ k m : X ∈C and m < ω } .Without loss of generality, we may assume that ˆ f and ˆ g are defined on thesame levels ( k m ) m<ω and that k = 0: For if ˆ g is defined on { X ↾ j m : X ∈ C and m < ω } , we can take l m = max( k m , j m ) and define ˆ f ′ ( s ) = ˆ f ( s ↾ k m )for s ∈ l m and ˆ g ′ ( X ↾ l m ) = ˆ g ( X ↾ j m ) for X ∈ C and m < ω . Noticethat whenever s ∈ C ∩ k m and s ⊏ X ∈ C , then ˆ f ( s ) = f ( X ) ↾ m andˆ g ( s ) = g ( X ) ↾ m .Define(2.6) D = { ˆ f ( s ) : s ∈ C } and D = f [ C ] . Notice that in fact D = { Y ↾ m : Y ∈ D , m < ω } , and D is cofinal in V since f : U → V is monotone cofinal and C is a cofinal subset of U . Let C denote the closure of C in the topological space 2 ω . Since ˜ f is continuous onthe compact space 2 ω and f ↾ C = ˜ f ↾ C , it follows that D = f [ C ] = ˜ f [ C ]. Claim 1.
For each Y ∈ D and each m < ω , there is an ˜ m ≥ m satisfyingthe following: For each Z ∈ C such that ˜ f ( Z ) ↾ ˜ m = Y ↾ ˜ m , there is an X ∈ C such that ˜ f ( X ) = Y and X ↾ k m = Z ↾ k m . Proof.
Let Y ∈ D and suppose the claim fails. Then there is an m such thatfor each n ≥ m , there is a Z n ∈ C such that ˜ f ( Z n ) ↾ n = Y ↾ n , but for each ANONICAL COFINAL MAPS 11 X ∈ C such that ˜ f ( X ) = Y , Z n ↾ k m = X ↾ k m . C is compact, so there is asubsequence ( Z n i ) i<ω which converges to some X ∈ C . Since ˜ f is continuous,˜ f ( Z n i ) converges to ˜ f ( X ). Since for each i < ω , ˜ f ( Z n i ) ↾ n i = Y ↾ n i , itfollows that ˜ f ( Z n i ) converges to Y . Therefore, ˜ f ( X ) = Y . Further, since Z n i → X , there is a j such that for all i ≥ j , Z n i ↾ k m = X ↾ k m . But thisis a contradiction since X ∈ C and ˜ f ( X ) = Y . (cid:3) Claim 2.
There is a strictly increasing sequence ( j m ) m<ω such that for each m < ω , for all Y ∈ D and Z ∈ C with ˜ f ( Z ) ↾ j m = Y ↾ j m , there is an X ∈ C such that ˜ f ( X ) = Y and X ↾ k m = Z ↾ k m . Proof.
Let j = 0 and note that j vacuously satisfies the claim. Now sup-pose that m ≥ j < · · · < j m − satisfying theclaim. For each Y ∈ D , there is an ˜ m ( Y ) ≥ m satisfying Claim 1. The finitecharacteristic functions Y ↾ ˜ m ( Y ) determine basic open sets in 2 ω , and theunion of these open sets (over all Y ∈ D ) covers D . Since D is compact,there is a finite subcover, determined by some Y ↾ ˜ m ( Y ) , . . . , Y l ↾ ˜ m ( Y l ).Take j m > max { j m − , ˜ m ( Y ) , . . . , ˜ m ( Y l ) } . By this inductive construction,we obtain a sequence ( j m ) m<ω which satisfies the claim. (cid:3) Claim 3.
For each X ∈ C and m < ω ,(2.7) ˜ g ( X ) ↾ m ⊆ [[ˆ g ( X ↾ k m )]] . Proof.
Let X ∈ C and m < ω be given, and let s denote X ↾ k m . Then s ∈ C ,and for any Z ∈ C such that Z ↾ k m = s , we have that ˜ g ( Z ) ↾ m = ˆ g ( s ).Since the property ( ∗ ) on C implies there is a Z ∈ C such that Z ⊇ X and Z ↾ k m = s , and since ˜ g is monotone, it follows that˜ g ( X ) ↾ m ⊆ \ { ˜ g ( Z ) ↾ m : Z ∈ C and Z ⊇ X } = [[ˆ g ( s )]] . (cid:3) Claim 4.
Let Y ∈ D and m be given, and let t = Y ↾ j m . Then˜ h ( Y ) ↾ m ⊆ [[ˆ g ( s ) ↾ m ]] , for each s ∈ C ∩ k jm such that ˆ f ( s ) = t . Proof.
Let Y ∈ D and m be given, and let t = Y ↾ j m . Let s be any memberof C ∩ k jm such that ˆ f ( s ) = t . By Claim 2, there is an X ∈ C such that˜ f ( X ) = Y and X ↾ k m = s ↾ k m . To prove the claim, we shall show thatthe following holds:(2.8) ˜ h ( Y ) ↾ m = ˜ g ( X ) ↾ m ⊆ [[ˆ g ( X ↾ k m )]] = [[ˆ g ( s ) ↾ m ]] . The first equality holds since ˜ f ( X ) = Y and ˜ h ◦ ˜ f ( X ) = ˜ g ( X ). The lastequality holds since ˆ g ( X ↾ k m ) = ˆ g ( s ↾ k m ) = ˆ g ( s ) ↾ m . The inclusion holdsby Claim 3. (cid:3) Finally, we define the finitary function ˆ h which will represent h on D .Let D ′ = { t ∈ D : ∃ m < ω ( | t | = j m ) } . For t ∈ D ′ ∩ j m , define ˆ h ( t ) to bethe function from m into 2 such that for i ∈ m ,(2.9) ˆ h ( t )( i ) = min { ˆ g ( s )( i ) : s ∈ C ∩ k jm and ˆ f ( s ) = t } . In words, ˆ h ( t ) is the characteristic function with domain m of the intersec-tion of the subsets a of m for which there is some s ∈ C ∩ k jm with ˆ f ( s ) = t such that ˆ g ( s ) ↾ m is the characteristic function of a with domain m . Bydefinition, ˆ h is level preserving. Claim 5. ˆ h is basic and generates h ↾ D . Proof.
Let Y ∈ D , Z be a member of C such that ˜ f ( Z ) = Y , and m < ω begiven. Since Z ∈ C , ˜ f ( Z ) = f ( Z ). Let t = Y ↾ j m and u = Z ↾ k j m . Thenˆ f ( u ) = t , so [[ˆ g ( u ) ↾ m ]] ⊇ [[ˆ h ( t )]]. Since Z ∈ C , g ( Z ) ↾ m = [[ˆ g ( u ) ↾ m ]].Thus,(2.10) h ( Y ) ↾ m = h ◦ f ( Z ) ↾ m = g ( Z ) ↾ m = [[ˆ g ( u ) ↾ m ]] ⊇ [[ˆ h ( t )]] . Now suppose s ∈ C ∩ k jm and ˆ f ( s ) = t . By Claim 4, since ˜ h ↾ V = h ,we see that h ( Y ) ↾ m = ˜ h ( Y ) ↾ m ⊆ [[ˆ g ( s ) ↾ m ]]. Since s was arbitrary, itfollows that h ( Y ) ↾ m ⊆ [[ˆ h ( t )]]. Therefore, ˆ h ( Y ↾ j m ) = h ( Y ) ↾ m .Thus, ˆ h generates h on D . It follows that ˆ h is end-extension preserving:If t ⊏ t ′ are members of D ′ of lengths j m and j m ′ , respectively, then letting Y be any member of D such that t ′ ⊏ Y , we see that ˆ h ( t ) = h ( Y ) ↾ m =( h ( Y ) ↾ m ′ ) ↾ m = ˆ h ( t ′ ) ↾ m . Therefore, ˆ h is basic. (cid:3) Thus, h ↾ D is generated by the basic map ˆ h on D ′ . Thus, V has basicTukey reductions. (cid:3) Every p-point has monotone basic Tukey reductions satisfying the addi-tional property ( ∗ ) of Theorem 2.5, as was shown in the proof of Theorem20 of [8], the cofinal set C there being of the simple form P ( X ) ∩ U for some X ∈ U . Hence, the following theorem holds. Theorem 2.6.
Every ultrafilter Tukey reducible to a p-point has basic, andhence continuous, Tukey reductions.
Recall that an ultrafilter V is Rudin-Blass reducible to an ultrafilter W if there is a finite-to-one map h : ω → ω such that V = h ( W ). Thus, ANONICAL COFINAL MAPS 13
Rudin-Blass reducibility implies Rudin-Keisler reducibility. Our Theorem2.6 combines with Theorem 10 of Raghavan in [13] (see Theorem 1.6 for thestatement) to yield the following.
Theorem 2.7.
Suppose U is Tukey reducible to a p-point. Then for eachq-point V , V ≤ T U implies V ≤ RB U . Remark 2.8.
Stable ordered-union ultrafilters are the analogues of p-pointson the base set FIN = [ ω ] <ω \ {∅} (see [1]). In Theorems 71 and 72 of [8],it was shown that for each stable ordered union ultrafilter U , both U andits projection U min , max have continuous Tukey reductions, with respect tothe metric topology on the Milliken space of infinite block sequences. Itis of interest that the ultrafilter U min , max is rapid, but is neither a p-pointnor a q-point, and condition ( ∗ ) of Theorem 2.5 is satisfied. Further, allultrafilters selective for some topological Ramsey space have monotone basicTukey reductions, under a mild assumption which is satisfied in all knowntopological Ramsey spaces, Dobrinen and Trujillo showed in Theorem 56 ofin [7]. Many such ultrafilters are not p-points.It should be the case that by arguments similar to those in Theorem2.5, one can prove that every ultrafilter Tukey reducible to some stableordered union ultrafilter, or more generally, any ultrafilter selective for sometopological Ramsey space, also has continuous Tukey reductions. We leavethis as an open problem in Section 7.3. Iterated Fubini products of ultrafilters represented asultrafilters generated by ~ U -trees on fronts Fubini products of ultrafilters on base set ω are commonly viewed asultrafilters on base set ω × ω . As was pointed out to us by Todorcevic, Fubiniproducts of nonprincipal ultrafilters on base set ω may also be viewed asultrafilters on base set [ ω ] . This view leads well to precise investigations ofultrafilters constructed by iterating the Fubini product construction. In thissection, we review Fubini products of ultrafilters and countable iterationsof this construction. After reviewing the notion of front, we then show howevery ultrafilter obtained by iterating the Fubini product construction canbe viewed as an ultrafilter generated by certain subtrees of a base set whichis a tree, and in particular, a front. This section is a primer for the work inSection 4. Definition 3.1.
Let U and V n ( n < ω ) be ultrafilters. The Fubini product of V n over U , denoted lim n →U V n , is defined as follows:(3.1) lim n →U V n = { A ⊆ ω × ω : { n ∈ ω : { j ∈ ω : ( n, j ) ∈ A } ∈ V n } ∈ U } . When all V n = V , then we let U · V denote lim n →U V n .Let U and V be ultrafilters on countable base sets I and J , respectively.We say that U is isomorphic to V if there exists a bijection π : I → J suchthat V = { π [ U ] : U ∈ U } . Up to isomorphism, Definition 3.1 also definesFubini products of ultrafilters on arbitrary countable infinite sets.The Fubini product construction can be iterated countably many times,each time producing an ultrafilter. For example, given an ultrafilter V , let V denote V , and let V n +1 denote V · V n . Naturally, V ω denotes lim n →V V n .Continuing in this manner, we obtain V α , for all 2 ≤ α < ω ·
2. At this point,it is ambiguous what is meant by V ω · . It is standard practice for countablea limit ordinal α to let V α denote any ultrafilter constructed by choosing(arbitrarily) an increasing sequence ( α n ) <ω converging to α and defining V α to be lim n →V V α n , but this is ambiguous, since the choice of the sequence( α n ) n<ω is completely arbitrary.However, each countable iteration of Fubini products of ultrafilters (in-cluding the choice of sequence at limit stages) can be represented as anultrafilter generated by ~ U -trees (see Definition 3.3) on a base set which is afront. This representation is unambiguous at limit stages. For this reason,Theorem 4.4 in the next section, showing that iterations of Fubini prod-ucts of p-points have Tukey reductions which are as close to continuous aspossible, will be carried out in the setting of ~ U -trees.We now recall the definition of front. The reader desiring more back-ground on fronts and ~ U -trees than presented here is referred to [17], pages12 and 190, respectively. Definition 3.2.
A family B of finite subsets of some infinite subset I of ω is called a front on I if(1) a ⊏ b whenever a, b are in B ; and(2) For every infinite X ⊆ I there exists b ∈ B such that b ⊏ X .Recall the following standard set-theoretic notation: [ ω ] k denotes thecollection of k -element subsets of ω , [ ω ] Every front is lexicographically well-ordered, and hence has a uniquelexicographic rank associated with it, namely the ordinal length of its lexi-cographical well-ordering. For example, rank( {∅} ) = 1, rank([ ω ] ) = ω , andrank([ ω ] ) = ω · ω . We shall usually drop the adjective ‘lexicographic’ whentalking about ranks of fronts.Given a front B , for each n ∈ ω , we define B n = { b ∈ B : n = min( b ) } and B { n } = { b \{ n } : b ∈ B n } . Then B = S n ∈ ω B n , and each B n = {{ n }∪ a : a ∈ B { n } } . Note that for each n ∈ ω , B { n } is a front on [ n + 1 , ω ) with rankstrictly less than the rank of B . Conversely, given any collection of fronts B { n } on [ n + 1 , ω ), the union S n ∈ ω B n is a front on ω , where B n is definedas above to be {{ n } ∪ a : a ∈ B { n } } .Given any front B , we let ˆ B denote the collection of all initial segmentsof members of B . Let ˆ B − denote the collection of all proper initial segmentsof members of B ; that is, ˆ B − = ˆ B \ B . Both ˆ B and ˆ B − form trees underthe partial ordering ⊑ . Definition 3.3. Given a front B and a sequence ~ U = ( U c : c ∈ ˆ B − ) ofnonprincipal ultrafilters U c on ω , a ~ U -tree is a tree T ⊆ ˆ B such that ∅ ∈ T and for each c ∈ T ∩ ˆ B − , { n ∈ ω : c ∪ { n } ∈ T } ∈ U c . Notation. Given a front B and a sequence ~ U = ( U c : c ∈ ˆ B − ) of nonprin-cipal ultrafilters on ω , let T = T ( ~ U ) denote the collection of all ~ U -trees. Forany c ∈ ˆ B − and T ∈ T , let T c = { t ∈ T : t ⊑ c or t ⊐ c } , the tree with stem c consisting of all nodes in T comparable with c . If T is a U -tree, then theset of maximal branches through T , denoted [ T ], is exactly T ∩ B .Todorcevic pointed out to us the following correspondence between Fu-bini iterates and ultrafilters on fronts. Start by fixing a collection P ofnonprincipal ultrafilters on ω . Given α < ω , define P α +1 = { lim n →U V n : U ∈ P and V n ∈ P α } . For each limit ordinal α , define P α = S β<α P β . Then P <ω := S {P α : α < ω } is the collection of all iterated Fubini products ofnonprincipal ultrafilters on ω . Each W ∈ P <ω has a well-defined notion ofrank, namely rank( W ) is the least α < ω for which it is a member of P α .The following lemma will be used in the next section, with P being thecollection of all p-points. Lemma 3.4. If W ∈ P <ω , then there is a front B and ultrafilters U c , c ∈ ˆ B − such that W is isomorphic to the ultrafilter on B generated by the ( U c : c ∈ ˆ B − ) -trees. Proof. We prove by induction on α < ω that the fact holds for everyultrafilter in P α . If W ∈ P , then W is a nonprincipal ultrafilter and isrepresented on the front B = [ ω ] via the obvious isomorphism n 7→ { n } .Let 1 ≤ α < ω and assume the fact holds for each ultrafilter in S γ<α P γ .If α is a limit ordinal, then there is nothing to prove, so assume α = β +1 forsome β < ω . Suppose that W ∈ P α . Then W = lim n →U W n , where U is anonprincipal ultrafilter and for each n , W n ∈ P β . By the induction hypoth-esis, for each n < ω there is a front B { n } on ω and there are nonprincipalultrafilters U n,c , c ∈ d B { n }− , such that W n is isomorphic to the ultrafiltergenerated by ( U n,c : c ∈ d B { n }− )-trees on B { n } . Without loss of generality, wemay assume that B { n } is a front on [ n + 1 , ω ). In the standard way, we gluethe fronts together to obtain a new front: Let B = S n<ω {{ n }∪ b : b ∈ B { n } } .Then B is a front on ω .Let U ∅ = U and U { n }∪ c = U n,c , for each c ∈ B { n } . It is straightforwardto check that W is isomorphic to the ultrafilter on B generated by the hU b : b ∈ ˆ B − i -trees. (cid:3) The case when α = 2 of the proof of Lemma 3.4 yields the followingresult, which seems interesting in its own right. Lemma 3.5. The Fubini product lim n →U V n of nonprincipal ultrafilters on ω is isomorphic to the ultrafilter on B = [ ω ] generated by ~ U = ( U c : c ∈ [ ω ] ≤ ) -trees, where U ∅ = U and for each n ∈ ω , U { n } = V n . Basic cofinal maps on iterated Fubini products of p-points Fubini products of p-points do not in general have continuous Tukeyreductions. However, we will show that they do have canonical cofinal mapssatisfying many of the properties of continuous maps, which we call basic (see Definition 4.2 below). Making use of the natural representation of aFubini iterate of p-points as an ultrafilter generated by ~ U -trees on somefront B (recall Lemma 3.4), we show in Theorem 4.4 that countable iteratesof Fubini products of p-points have basic Tukey reductions . Such Tukeyreductions are continuous on the space 2 ˆ B with the Cantor topology, whereˆ B is the tree consisting of all initial segments of members of the front B .This extends a key property of p-points (recall Theorem 1.3) to a large classof ultrafilters. Theorem 4.4 will be applied in Sections 5 and 6.Corollary 11 of Raghavan in [13] shows that it is impossible for a Fu-bini product of two non-isomorphic selective ultrafilters to have continuouscofinal Tukey reductions. The next proposition shows that Fubini productsof nonprincipal ultrafilters do not have monotone basic Tukey reductions ANONICAL COFINAL MAPS 17 given by an approximating map on the finite subsets of the base for theultrafilter. Thus, it is impossible to attain Theorem 1.3 for Fubini productsof nonprincipal ultrafilters.Recall that the Cantor topology on 2 ω × ω is the topology generated bybasic open sets of the form { h ∈ ω × ω : s ⊂ h } , where s is a function fromsome finite subset of ω × ω into 2. Equivalently, letting h w i : i < ω i be anylinear order of the members of ω × ω in order type ω , the Cantor topologyon 2 ω × ω is generated by basic open sets of the form { h ∈ ω × ω : s ⊂ h } ,where s is a function from h w i : i < k i into 2 for some k < ω . For k < ω ,let W k denote { w i : i < k } , so that 2 W k denotes the set of all functions withdomain { w i : i < k } into 2. Given k < ω and X ⊆ ω × ω , let X ↾ W k denotethe characteristic function of { w i ∈ X : i < k } on domain W k . Proposition 4.1. Let U and V be any nonprincipal ultrafilters and let π : ω × ω → ω be defined by π ( m, n ) = m . Let f : 2 ω × ω → ω be defined by f ( A ) = π [ A ] . Then for any cofinal subset C of U · V , f ↾ C is not generatedby a monotone basic map: There is no strictly increasing sequence ( k m ) m<ω and monotone basic map ˆ f : S m<ω W km → <ω generating f ↾ C .Proof. Let g denote the restriction of f to the ultrafilter U · V . Notice that g is a monotone cofinal map onto U . Suppose toward a contradiction thatthere is a cofinal subset C of U · V for which g ↾ C is represented by amonotone basic function. In this context, using the Cantor topology on2 ω × ω in place of 2 ω , Definitions 2.1 and 2.2 are interpreted as follows: Thereis a strictly increasing sequence ( k m ) m<ω such that letting C = { X ↾ W k m : X ∈ C , m < ω } , there is a monotone basic map ˆ g : C → <ω such that foreach X ∈ C ,(4.1) g ( X ) = [ m<ω ˆ g ( X ↾ W k m ) . To be clear in this context, ˆ g is level preserving means that for each s ∈ C ∩ W km , ˆ g ( s ) is a member of 2 m . For two members s and t of C , t end-extends s , written s ⊑ t , if and only if the domain of s is W j and the domainof t is W l , for some j ≤ l , and for each i < j , s ( w i ) = t ( w i ).By the Extension Lemma 2.4, there is a monotone map ˜ f : 2 ω × ω → ω which is represented by a monotone basic map ˆ f on S m<ω W km such that˜ f ↾ C = g ↾ C . As seen in the proof of Lemma 2.4, modified to the currentcontext, the map ˆ f is defined by(4.2) ˆ f ( t )( w i ) = max { ˆ g ( s )( w i ) : s ∈ C, | s | ≤ k m and [[ s ]] ⊆ [[ t ]] } , for t ∈ W km and i < m . Claim. There is a j < ω and an infinite collection { s l : l < ω } ⊆ C , satisfy-ing the following: For each l < ω , letting d l denote [[ s l ]], j = min( π [ d l ]), andletting i l denote the least i such that both w i ∈ d l and j = π ( w i ), ( i l ) l<ω forms a strictly increasing sequence. Proof. For s ∈ C , let d s denote [[ s ]]. Suppose toward a contradiction thatfor each j < ω , there is an ˜ i j such that for each s ∈ C with j = min( π [ d s ]),there is an i < ˜ i j such that w i ∈ d s and j = π ( w i ). Fix X ∈ C . Since C is a cofinal subset of the Fubini product of two nonprincipal ultrafilters, itfollows that the set(4.3) J = { min( π [ d s ]) : s ∈ C and d s ⊆ X } is infinite. For j ∈ J let(4.4) S j = { s ∈ C : d s ⊆ X and min( π [ d s ]) = j } . Then for each j ∈ J and s ∈ S j , there is some w i ∈ d s with i < ˜ i j . Thus,every member of C has infinite intersection with { w i : ∃ j ( j = π ( w i ) and i < ˜ i j ) } . This contradicts the fact that V is nonprincipal.Thus, the negation of the supposition holds: There is a j < ω such thatfor each ˜ i < ω , there is an s ∈ C with min( π [ d s ]) = j such that, whenever w i ∈ d s and j = π ( w i ), then i ≥ ˜ i . Fix such a j . Take s ∈ C such thatmin( π [ d s ]) = j . Take i least such that w i ∈ d s and π ( w i ) = j . Using i as the next ˜ i , there is an s ∈ C with min( π [ d s ]) = j and i > i , where i is least such that w i ∈ d s and π ( i ) = j . In this way, one constructs acollection { s l : l < ω } satisfying the Claim. (cid:3) Take j < ω and { s l : l < ω } as in the Claim. Define Y l ∈ U · V by Y l = s l ∪ ( j, ω ) × ω . Then Y l → Y , where Y = ( j, ω ) × ω , which is a memberof U · V . Since ˜ f is generated by a basic map, ˜ f is continuous on 2 ω × ω .Hence, ˜ f ( Y l ) converges to ˜ f ( Y ).On the other hand, we shall show that for each l < ω , j is in ˜ f ( Y l ) while j is not in ˜ f ( Y ), contradicting continuity of f . Note that for each l < ω , s l ∈ C implies there is an X ∈ C whose characteristic function extends s l .Since g is generated by the basic map ˆ g , and j ∈ π [ X ] implies j ∈ g ( X ),it follows that j ∈ [[ˆ g ( s l )]]. By the definition of ˆ f in (4.2), it follows that j ∈ [[ ˆ f ( s l )]] ⊆ ˜ f ( Y l ). However, j is not in ˜ f ( Y ), since ˜ f is generated by ˆ f ,it follows from the definition of ˆ f in (4.2) that j [[ ˆ f ( Y ↾ k m )]] for any m < ω .Thus, there is no cofinal C ⊆ U · V for which g ↾ C is generated by amonotone basic map on the topological space 2 ω × ω . (cid:3) ANONICAL COFINAL MAPS 19 However, we will soon show that each ultrafilter W which is an iteratedFubini product of p-points has finitely generated Tukey reductions which,moreover, are basic, and hence continuous, with respect to the topologyon the appropriate tree space. Toward this end, we proceed to give thedefinition of basic for this context, and then prove the main results of thissection. Notation. For any subset A ⊆ [ ω ] <ω , recall that ˆ A denotes the set of allinitial segments of members of A . For any front B , we let ˆ B − denote ˆ B \ B .For any subset A ⊆ [ ω ] <ω and k < ω , let A ↾ k denote { a ∈ A : max( a ) < k } .For A ⊆ ˆ B and k < ω , we shall abuse notation and also use A ↾ k to denotethe characteristic function of the set A ↾ k on domain ˆ B ↾ k . For each k < ω ,let 2 ˆ B ↾ k denote the set of all functions from ˆ B ↾ k into { , } . Notice thatthis is exactly the collection of characteristic functions of subsets of ˆ B ↾ k on domain ˆ B ↾ k . Definition 4.2. Let B be a front on ω , ˜ T ⊆ ˆ B be a tree, and ( n k ) k<ω be anincreasing sequence. We say that a function ˆ f : S k<ω ˜ T ↾ n k → <ω is levelpreserving if ˆ f : 2 ˜ T ↾ n k → k , for each k < ω . ˆ f is end-extension preserving if whenever k < m and A ⊆ ˜ T then ˆ f ( A ↾ n k ) = ˆ f ( A ↾ n m ) ↾ k . ˆ f is basic if it is level and end-extensions preserving. ˆ f is monotone if whenever A ⊆ C ⊆ ˜ T and k < ω , then [[ ˆ f ( A ↾ n k )]] ⊆ [[ ˆ f ( C ↾ n k )]].Let U be an ultrafilter on B generated by ( U c : c ∈ ˆ B − )-trees, let f : U → V be a monotone cofinal map, where V is an ultrafilter on base ω , andlet ˜ T ∈ T ( ~ U ). Let T ↾ ˜ T denote the set of all ~ U -trees contained in ˜ T . Wesay that ˆ f : S k<ω ˜ T ↾ n k → <ω generates f on T ↾ ˜ T if for each T ∈ T ↾ ˜ T ,(4.5) f ([ T ]) = [ k<ω ˆ f ( T ↾ n k ) . We say that U has basic Tukey reductions if whenever f : U → V is amonotone cofinal map, then there is a ˜ T ∈ T ( ~ U ) and a basic map ˆ f whichgenerates f on T ↾ ˜ T . Remark 4.3. Note that if ˆ f witnesses that f is basic on T ↾ ˜ T , then ˆ f generates a continuous map on the collection of trees in T ↾ ˜ T , continuitybeing with respect to the Cantor topology on 2 ˆ B . Moreover, we may definea map ˆ g on B as follows: For each finite subset A ⊆ B , define ˆ g ( A ) = ˆ f ( ˆ A ),where ˆ A is the collection of all initial segments of members of A . Thenˆ g is finitary, but not necessarily continuous on 2 B , and ˆ g generates f on { [ T ] : T ∈ T ↾ ˜ T } which is a base for the ultrafilter. Thus, for ultrafil-ters generated by U -trees, basic Tukey reductions imply finitely representedTukey reductions on the original base set B .Now we prove the main theorem of this section. Fix a total order of [ ω ] <ω in order type ω such that max a < max b implies a ≺ b for all a, b ∈ [ ω ] <ω .Note that for each k < ω , the set { c ∈ [ ω ] <ω : max c = k } forms a finiteinterval in ([ ω ] <ω , ≺ ). Theorem 4.4. Let B be any front and ~ U = ( U c : c ∈ ˆ B − ) be a sequence ofp-points. Then the ultrafilter U on base B generated by the ~ U -trees has basicTukey reductions. Therefore, every countable iteration of Fubini products ofp-points has monotone basic Tukey reductions.Proof. Let V be some ultrafilter Tukey reducible to U , and let f : U → V bea monotone cofinal map. If V is a principal ultrafilter, say generated by thesingleton { r } , then we claim that the theorem trivially holds. Since the setconsisting of { r } is cofinal in V , there is some X ∈ U such that f ( X ) = { r } .Set n k = k and define ˆ f : S k<ω ˆ B ↾ k → <ω as follows: For k < ω and s ∈ ˆ B ↾ k , ˆ f ( s ) is the sequence in 2 k such that for each i < k , ˆ f ( s )( i ) = 0if and only if i = r . Then ˆ f is monotone basic and generates f on U ↾ X .Thus, we shall assume from now on that V is nonprincipal.We let T denote T ( U c : c ∈ ˆ B − ), the set of all ~ U -trees. Recall that T isa base for the ultrafilter U . We make the convention that max ∅ = − 1. Foreach k < ω , let ˆ B ↾ k denote the collection of all b ∈ ˆ B with max b < k .Thus, ˆ B ↾ {∅} , ˆ B ↾ {∅ , { }} , and so forth. Fix an enumeration of thefinite, non-empty ⊑ -closed subsets of ˆ B as h A i : i < ω i so that for each i < j ,max S A i ≤ max S A j . Let ( p k ) k<ω denote the strictly increasing sequenceso that for each k , the sequence h A i : i < p k i lists all ⊑ -closed subsets ofˆ B ↾ k . (For example, if B = [ ω ] , then ˆ B = [ ω ] ≤ and ˆ B − = [ ω ] ≤ , and wemay let A = {∅} , A = {∅ , { }} , A = {∅ , { } , { }} , A = {∅ , { } , { , }} , A = {∅ , { } , { , } , { }} , A = {∅ , { }} . Note that p = 1, p = 2, and p = 6.)For k < ω and i < p k , define(4.6) ˆ B ki = A i ∪ { b ∈ ˆ B : ∃ a ∈ A i ( b ⊐ a and min( b \ a ) ≥ k ) } . Thus, ˆ B ki is the maximal tree in T for which T ↾ k = A i . For a tree T ⊆ ˆ B and c ∈ T ∩ ˆ B − , define the notation(4.7) U c ( T ) = { l > max( c ) : c ∪ { l } ∈ T } . ANONICAL COFINAL MAPS 21 We refer to U c ( T ) as the set of immediate extensions of c in T . Note thatif T ∈ T , then for each c ∈ T ∩ ˆ B − , U c ( T ) is a member of U c . For c ∈ ˆ B − ,recall that ˆ B c denotes the tree of all a ∈ ˆ B such that either a ⊑ c or else a ⊐ c .Our goal is to construct a tree ˜ T ∈ T and find a sequence ( n k ) k<ω ofgood cut-off points such that the following ( ⊛ ) holds.( ⊛ ) For each T ⊆ ˜ T in T , k < ω , and i < p n k such that A i = T ↾ n k , forevery j ≤ k , j ∈ f ([ T ]) ⇐⇒ j ∈ f ([ ˜ T ∩ ˆ B n k i ]) . Claim 1. The property ( ⊛ ) implies that f has monotone basic Tukey re-ductions. Proof. For k < ω and T ∈ T ↾ ˜ T , define(4.8) ˆ f ( T ↾ n k ) = f ([ ˜ T ∩ ˆ B n k i ]) ↾ k, where i is the integer below p n k such that T ↾ n k = A i . By definition, ˆ f is level preserving. Let l > k and let m be such that T ↾ n l = A m . Then A m ↾ n k = A i . For j < k , ( ⊛ ) implies that j ∈ f ([ ˜ T ∩ ˆ B n k i ]) if and onlyif j ∈ f ([ T ]) if and only if j ∈ f ([ ˜ T ∩ ˆ B n l m ]). Thus, j ∈ [[ ˆ f ( T ↾ n k )]] ifand only if j ∈ [[ ˆ f ( T ↾ n l )]]. Therefore, ˆ f is end-extension preserving; thatis, ˆ f ( T ↾ n l ) ↾ k = ˆ f ( T ↾ n k ). Furthermore, f ([ T ]) = S k<ω ˆ f ( T ↾ n k );thus, ˆ f generates f on T ↾ ˜ T . To see that ˆ f is monotone, suppose that S ↾ n k ⊆ T ↾ n k for some S, T ∈ T ↾ ˜ T . Let i, j < p k be such that A i = S ↾ n k and A j = T ↾ n k . Then(4.9) ˆ f ( S ↾ n k ) = f ([ ˜ T ∩ ˆ B n k i ]) ↾ k ⊆ f ([ ˜ T ∩ ˆ B n k j ]) ↾ k = ˆ f ( T ↾ n k ) , where ⊆ holds because of ( ⊛ ) and the fact that f is monotone and ˜ T ∩ ˆ B n k i ⊆ ˜ T ∩ ˆ B n k j . Therefore, ˆ f is a monotone basic map generating f on T ↾ ˜ T . (cid:3) The construction of ˜ T and ( n k ) k<ω takes place in three stages. Stage 1. In the first stage toward the construction of ˜ T , we will choosesome R ki ∈ T with A i ⊆ R ki such that for all k < ω , the following holds:( ∗ ) k For all i < p k and T ⊆ R ki in T with T ↾ k = A i , for each j ≤ k , j ∈ f ([ T ]) ⇐⇒ j ∈ f ([ R ki ]).We point out that for any front, A is always {∅} and p = 1. Since weare assuming V is nonprincipal, choose an R in T so that 0 f ([ R ]). Nowlet k > 0, and suppose we have chosen R lj for all l < k and j < p l . For i < p k − , if there is an R ⊆ R k − i in T such that R ↾ k = A i and k f ([ R ]),then let R ki be such an R ; if not, let R ki = R k − i ∩ ˆ B ki . Now suppose that p k − ≤ i < p k . If there is an R ∈ T such that R ↾ k = A i and 0 f ([ R ]), let R ki, be such an R ; if not, let R ki, = ˆ B ki . Given R ki,j for j < k , if there is an R ∈ T such that R ⊆ R ki,j , R ↾ k = A i , and j + 1 f ([ R ]), then let R ki,j +1 be such an R ; if not, then let R ki,j +1 = R ki,j . Finally, let R ki = R ki,k .It follows from the construction that for all 1 ≤ l ≤ k and p l − ≤ i < p l ,(4.10) R li, ⊇ R li, ⊇ . . . ⊇ R li,l = R li ⊇ R ki , and moreover, for any j ≤ l ,(4.11) R li,j ↾ l = R li ↾ l = R ki ↾ k = A i . Fix k < ω : we check that ( ∗ ) k holds. Let i < p k , T ⊆ R ki in T with T ↾ k = A i , and j ≤ k be given. If j ∈ f ([ T ]), then j must be in f ([ R ki ]),since T ⊆ R ki and f is monotone. Now suppose that j f ([ T ]); we willshow that j f ([ R ki ]). Let p − = 0, and let l ≤ k be the integer satisfying p l − ≤ i < p l . Note that max S A i = l − 1. Thus, T ↾ k = A i = T ↾ l . Wenow have two cases to check.Case 1: j ≤ l . Notice that T ⊆ R ki ⊆ R li ⊆ R li,j . If j = 0, then R li,j ⊆ ˆ B li and we let R ′ denote ˆ B li ; if j > 0, then R li,j ⊆ R li,j − and we let R ′ denote R li,j − . Since j is not in f ([ T ]) and T ↾ l = A i , T is a witness that thereis an R ⊆ R ′ with R ↾ l = A i such that j f ([ R ]). Thus, R li,j was chosenso that j f ([ R li,j ]). It follows that j f ([ R ki ]), since R ki ⊆ R li,j and f ismonotone.Case 2: l < j ≤ k . In this case, T ⊆ R ki ⊆ R ji ⊆ R j − i . Since T is awitness that there is an R ⊆ R j − i with R ↾ l = A i and j f ([ R ]), R ji was chosen so that j f ([ R ji ]). Thus, j f ([ R ki ]), since R ki ⊆ R ji and f ismonotone.Therefore, j ∈ f ([ T ]) if and only if j ∈ f ([ R ki ]); hence ( ∗ ) k holds. Thisconcludes Stage 1 of our construction.Given k < ω and c ∈ ˆ B − ↾ k , define(4.12) S kc = \ { R li : l ≤ k, i < p l , and c ∈ R li } . We claim that S kc is a member of T and that c ∈ S kc . To see this, notice thatfor c ∈ ˆ B − ↾ k , letting l ≤ k be least such that l > max c , then c is in A i forat least one i < p l . Since A i ⊆ R li , the set { ( l, i ) : l ≤ k, i < p l , and c ∈ R li } is nonempty; hence, c ∈ S kc . Moreover, S kc is a member of T , since it is afinite intersection of members of T . It follows that for each a ∈ S kc ∩ ˆ B − , theset of { l > max( a ) : a ∪ { l } ∈ S kc } is a member of the ultrafilter U c . Define(4.13) U kc := U c ( S kc ) = { l > max c : c ∪ { l } ∈ S kc } . ANONICAL COFINAL MAPS 23 Thus, for each c ∈ ˆ B − and j = max( c ) + 1, we have S jc ⊇ S j +1 c ⊇ . . . , eachof which is a member of T ; and U jc ⊇ U j +1 c ⊇ . . . , each of which is a memberof the p-point U c . Stage 2. In this stage we construct a tree T ∗ in T which will be thinneddown one more time in Stage 3 to obtain a subtree ˜ T ⊆ T ∗ in T such that f ↾ T ↾ ˜ T is basic. The tree T ∗ which we construct in this stage will havesets of immediate successors(4.14) U c := U c ( T ∗ ) = { l ≥ max( c ) + 1 : c ∪ { l } ∈ T ∗ } , for c ∈ T ∗ ∩ ˆ B − . The sets U c will have interval gaps which have rightendpoints which line up often and in a useful way (meshing). This willaid in finding good cut-off points n k needed in Stage 3 to thin T ∗ downto ˜ T . Toward obtaining these interval gaps, we will construct a family offunctions which we call meshing functions m ( c, · ) : ω → ω satisfying thefollowing ‘meshing property’:( † ) For each c ∈ ˆ B − and j < ω , if a ∈ ˆ B − is such that a ≺ c , then thereexists i < ω such that m ( a, i ) = m ( c, j ).The meshing functions of ( † ) will aid in obtaining a tree T ∗ ∈ T with thefollowing properties:( ‡ ) For all c ∈ T ∗ ∩ ˆ B − ,(a) U c ⊆ U max( c )+1 c ; and(b) For all i < ω , U c \ m ( c, i ) = U c \ m ( c, i + 1) ⊆ U m ( c, i ) c .We now begin the construction of the meshing functions m ( c, · ) and thesets U c , proceeding by recursion on the well-ordering ( ˆ B − , ≺ ). Since ∅ is ≺ -minimal in ˆ B − , we start by choosing g ∅ , m ( ∅ , · ), and Y ∅ as follows. Since U ∅ is a p-point, we may choose a U ∗∅ ∈ U ∅ such that U ∗∅ ⊆ ∗ U k ∅ for each k . (Recall the definition of U kc from equation (4.13).) Let g ∅ : ω → ω be astrictly increasing function such that for each k , U ∗∅ \ g ∅ ( k + 1) ⊆ U g ∅ ( k ) ∅ , and g ∅ (0) > 0. If S i ∈ ω [ g ∅ (2 i ) , g ∅ (2 i + 1)) ∈ U ∅ , then define m ( ∅ , k ) = g ∅ ( k + 1);otherwise, S i ∈ ω [ g ∅ (2 i + 1) , g ∅ (2 i + 2)) ∈ U ∅ , and we define m ( ∅ , k ) = g ∅ ( k ).Let Y ∅ = S i ∈ ω [ m ( ∅ , i + 1) , m ( ∅ , i + 2)) and define(4.15) U ∅ = U ∅ ∩ U ∗∅ ∩ Y ∅ . Note that for each k , U ∅ \ m ( ∅ , k + 1) ⊆ U m ( ∅ ,k ) ∅ .Now suppose c ∈ ˆ B − and for all b ≺ c in ˆ B − , g b and m ( b, · ) have beendefined. Since U c is a p-point, there is a U ∗ c ∈ U c for which U ∗ c ⊆ ∗ U kc , for all k > max c . Let a denote the immediate ≺ -predecessor of c in ˆ B − . Let g c : ω → ω be a strictly increasing function such that g c (0) > max { max c, g a (2) } ,and(1 g c ) For each i < ω , U ∗ c \ g c ( i + 1) ⊆ U g c ( i ) c ; and(2 g c ) For each j < ω , there is an i > g c ( j ) = m ( a, i ).Let Y c denote the one of the two sets S i ∈ ω [ g c (2 i ) , g c (2 i + 1)) or S i ∈ ω [ g c (2 i +1) , g c (2 i + 2)) which is in U c . In the first case define define m ( c, i ) = g c ( i + 1);in the second case m ( c, i ) = g c ( i ). Then(4.16) Y c = [ i<ω [ m ( c, i + 1) , m ( c, i + 2))and is in U c . Let(4.17) U c = U max( c )+1 c ∩ U ∗ c ∩ Y c . This concludes the recursive definition.We check that ( † ) holds. Let c ∈ ˆ B − and let a ≺ · · · ≺ a l ≺ c be theenumeration of all ≺ -predecessors of c in ˆ B − . Let j < ω be given. Either m ( c, j ) = g c (2 j ) or m ( c, j ) = g c (2 j + 1). By (2 g c ), g c (2 j ) = m ( a l , i ) forsome i , and g c (2 j + 1) = m ( a l , i ) for some i . Thus, there is an i such that m ( a l , i ) = m ( c, j ). Let i l denote this i . Likewise, either m ( a l , i l ) = g a l (2 i l )or m ( a l , i l ) = g a l (2 i l + 1). By (2 g al ), g a l (2 i l ) = m ( a l − , i ) for some i ,and g a l (2 i l + 1) = m ( a l − , i ) for some i . Let i l − denote the i such that m ( a l − , i l − ) = m ( a l , i l ). Continuing in this manner, we obtain numbers i k , k ≤ l , such that(4.18) m ( c, j ) = m ( a l , i l ) = m ( a l − , i l − ) = · · · = m ( a , i ) . Hence, ( † ) holds. Claim 2. There exists a strictly increasing sequence ( m k ) k<ω such that, forall k ,(4.19) ∀ c ∈ ˆ B − ↾ m k ∃ r ( m ( c, r ) = m k +1 ) . Proof. Let m be arbitrary, and let c be the ≺ -maximum of ˆ B − ↾ m . Since m ( c , · ) is strictly increasing, we can fix j ∈ ω such that m ( c , j ) > m .Let m = m ( c , j ). In general, given m k , let c k be the ≺ -maximum of ˆ B − ↾ m k . Since m ( c k , · ) is strictly increasing, we can fix some j k < ω such that m ( c k , j k ) > m k , and let m k +1 = m ( c k , j k ). In this manner, we inductivelyconstruct the sequence ( m k ) k<ω . To check that this sequence has the desiredproperty, let k < ω be given and fix c ∈ ˆ B − ↾ m k . Since c c k , it followsfrom ( † ) that there exists an r such that m ( c, r ) = m ( c k , j k ) = m k +1 . (cid:3) ANONICAL COFINAL MAPS 25 Let T ∗ be the tree in T defined by declaring for each c ∈ ˆ B − ∩ T ∗ , U c ( T ∗ ) = U c . If the reader is not satisfied with this top-down construction(which is precise as ∅ is in every member of T and this completely determinesthe rest of T ∗ ), we point out that T ∗ can also be seen as being constructedlevel by level as follows. Let ∅ ∈ T ∗ , and for each l ∈ U ∅ , put { l } in T ∗ ,so that the first level of T ∗ is exactly {{ l } : l ∈ U ∅ } . Suppose we haveconstructed the tree T ∗ up to level k , meaning that we know exactly what T ∗ ∩ ˆ B ∩ [ ω ] ≤ k is. For each c ∈ ˆ B − ∩ T ∗ ∩ [ ω ] k , let the immediate successorsof c in T ∗ be exactly the set U c ; in other words, for each l > max c , put c ∪ { l } ∈ T ∗ if and only if l ∈ U c . Recalling that max c < g c (0) ≤ m ( c, U c ⊆ Y c = Y c \ m ( c, U c is strictly greaterthan max c . Hence, by constructing T ∗ in this manner, we obtain a memberof T such that for each c ∈ T ∗ ∩ ˆ B − , U c ( T ∗ ) is exactly U c .We now check that ( ‡ ) holds. Let c ∈ T ∗ ∩ ˆ B − be given. By (4.17), U c ⊆ U max( c )+1 c , so ( ‡ ) (a) holds. By equation (4.16), we see that Y c ∩ [ m ( c, i ) , m ( c, i + 1)) = ∅ for each i . Thus,(4.20) U c ∩ [ m ( c, i ) , m ( c, i + 1)) = ∅ , since U c ⊆ Y c by (4.17). Recall that U ∗ c diagonalizes the collection of sets U kc for all k > max c , and the function g c was chosen to witness this diagonal-ization so that (1 g c ) holds. Either m ( c, i ) = g c ( i ) and m ( c, i + 1) = g c ( i + 1),or else m ( c, i ) = g c ( i + 1) and m ( c, i + 1) = g c ( i + 2). In either case, (1 g c )implies that U ∗ c \ m ( c, i + 1) ⊆ U m ( c,i ) c . Thus(4.21) U c \ m ( c, i + 1) ⊆ U m ( c,i ) c , since U c ⊆ U ∗ c by (4.17). ( ‡ ) (b) follows from (4.20) and (4.21). This finishesStage 2 of our construction. Stage 3. We will show that there is a strictly increasing sequence ( n k ) k<ω and a subtree ˜ T ⊆ T ∗ in T so that for all k ,(4.22) ∀ c ∈ ˜ T ∩ ( ˆ B − ↾ n k ) ∃ r ( m ( c, r ) = n k ) . The following lemma uses induction on the rank of the front. Lemma 4.5. Let B be a front on ω , and let T ∗ be a ~ U -tree, where ~ U = hU c : c ∈ ˆ B − i is a sequence of non-principal ultrafilters on ω . Suppose thatstrictly increasing functions m ( c, · ) : ω → ω for every c ∈ ˆ B − and a strictlyincreasing sequence ( m k ) k<ω are given such that, for all k , (4.23) ∀ c ∈ ˆ B − ↾ m k ∃ r ( m ( c, r ) = m k +1 ) . Then there exist ˜ T ⊆ T ∗ such that ˜ T is a ~ U -tree and a subsequence ( n k ) k<ω of ( m k ) k<ω such that, for all k , (4.24) ∀ c ∈ ˜ T ∩ ( ˆ B − ↾ n k ) ∃ r ( m ( c, r ) = n k ) . Proof. The proof will be by induction on the rank of B . First assumerank( B ) = 1. In this case, B = ˆ B = {∅} and ˆ B − = {} . Also notice that theonly hi -tree is {∅} . Therefore, setting ˜ T = {∅} and ( n k ) k<ω = ( m k ) k<ω willwork.Now assume that rank( B ) > 1. Then, for every l < ω , B { l } is a fronton [ l + 1 , ω ) of rank strictly less than rank( B ). Given l < ω , set m l ( c, · ) = m ( { l } ∪ c, · ) for c ∈ ˆ B −{ l } . For l ∈ U ∅ ( T ∗ ), define(4.25) T ∗ l = { c ∈ ˆ B { l } : { l } ∪ c ∈ T ∗ } , and observe that T ∗ l is a ~ U l -tree, where ~ U l = hU { l }∪ c : c ∈ ˆ B −{ l } i .Next, we will inductively define subsequences ( n jk ) k<ω of ( m k ) k<ω forevery j < ω . We will also make sure that ( n hk ) k<ω is a subsequence of( n jk ) k<ω whenever h ≥ j . Furthermore, we will obtain a ~ U l -tree ˜ T l ⊆ T ∗ l forevery l ∈ U ∅ ( T ∗ ), such that, for all k ,(4.26) ∀ c ∈ ˜ T l ∩ ( ˆ B −{ l } ↾ n lk ) ∃ r ( m l ( c, r ) = n lk ) . Let ( m k ) k<ω = ( m k ) k<ω . If 0 U ∅ ( T ∗ ), simply let ( n k ) k<ω = ( m k ) k<ω . If0 ∈ U ∅ ( T ∗ ), apply the induction hypothesis to B { } and T ∗ , with respect tothe functions m ( c, · ) and the sequence ( m k ) k<ω . This will yield a ~ U -tree˜ T ⊆ T ∗ and a subsequence ( n k ) k<ω of ( m k ) k<ω .Let ( m k ) k<ω = ( n k ) k<ω . If 1 U ∅ ( T ∗ ), simply let ( n k ) k<ω = ( m k ) k<ω . If1 ∈ U ∅ ( T ∗ ), apply the induction hypothesis to B { } and T ∗ , with repsect tothe functions m ( c, · ) and the sequence ( m k ) k<ω . Continue as in these firsttwo steps to complete the inductive construction.Set h (0) = n . Given h ( k ), fix q ( k ) such that n h ( k ) q ( k ) > h ( k ), then set h ( k + 1) = n h ( k ) q ( k ) . Notice that exactly one of S k<ω [ h (2 k ) , h (2 k + 1)) and S k<ω [ h (2 k +1) , h (2 k +2)) will belong to U ∅ , and denote it by Z . Set U ∅ ( ˜ T ) = U ∅ ( T ∗ ) ∩ Z , then define(4.27) ˜ T = {∅} ∪ [ l ∈ U ∅ ( ˜ T ) {{ l } ∪ c : c ∈ ˜ T l } . It is straightforward to check that ˜ T is a ~ U -tree contained in T ∗ . If Z = S k<ω [ h (2 k ) , h (2 k + 1)), then define n k = h (2 k + 2), otherwise define n k = h (2 k + 1). We will only complete the proof in the first case, as the othercase is similar. ANONICAL COFINAL MAPS 27 Fix k < ω and c ∈ ˜ T ∩ ( ˆ B − ↾ n k ). If c = ∅ , then c ∈ ˆ B − ↾ m j − ,where j > m j = n k . Therefore, there exists r such that m ( c, r ) = m j = n k . Now assume that c = ∅ , and let l = min c . Since l ∈ Z and l < h (2 k + 2), we must have l < h (2 k + 1). In particular, ( n h (2 k +1) ) k<ω isa subsequence of ( n lk ) k<ω . Therefore, there exists q such that n h (2 k +1) q (2 k +1) = n lq .Finally, since c \ { l } ∈ ˜ T l ∩ ( ˆ B −{ l } ↾ n lq ), we see that there exists r such that(4.28) m ( c, r ) = m l ( c \ { l } , r ) = n lq = n h (2 k +1) q (2 k +1) = h (2 k + 2) = n k , which is what we needed to show. (cid:3) Taking ˜ T as in Lemma 4.5 concludes Stage 3 of the construction.Finally, we check that ( ⊛ ) holds. Toward this, we first show that for all k < ω and i < p n k , ˜ T ∩ ˆ B n k i ⊆ R n k i . It will follow that for each T ∈ T ↾ ˜ T with T ↾ n k = A i , we in fact have T ⊆ R n k i . This along with ( ∗ ) n k for all k < ω will yield ( ⊛ ). Claim 3. Let k < ω and i < p n k be given, and suppose that A i ⊆ ˜ T . Then˜ T ∩ ˆ B n k i ⊆ R n k i . Proof. Let Q denote ˜ T ∩ ˆ B n k i . Since A i ⊆ ˜ T , we see that Q ↾ n k = A i whichequals R n k i ↾ n k . Thus, to prove the claim it is enough to show that for each c ∈ Q ∩ ˆ B − ,(4.29) U c ( Q ) \ n k ⊆ U c ( R n k i ) . Since Q ⊆ ˜ T ⊆ T ∗ , we see that for all c ∈ Q ∩ ˆ B − ,(4.30) U c ( Q ) \ n k ⊆ U c ( ˜ T ) \ n k ⊆ U c ( T ∗ ) \ n k . We have two cases for c .Case 1: c ∈ Q ∩ ( ˆ B − ↾ n k ). Then by Lemma 4.5, there is an r such that m ( c, r ) = n k . By ( ‡ ) (b), we have that(4.31) U c ( T ∗ ) \ n k = U c ( T ∗ ) \ m ( c, r + 1) ⊆ U m ( c, r ) c = U n k c . Since Q ↾ n k = A i ⊆ R n k i , c is in R n k i , and hence, S n k c ⊆ R n k i , recalling(4.12). Therefore, recalling (4.13),(4.32) U n k c = U c ( S n k c ) ⊆ U c ( R n k i ) . Hence, by (4.30), (4.31), and (4.32), we see that(4.33) U c ( Q ) \ n k ⊆ U c ( T ∗ ) \ n k ⊆ U n k c ⊆ U c ( R n k i ) . Case 2: For c ∈ Q ∩ ˆ B − such that max c ≥ n k , it follows from ( ‡ ) (a)that U c ( T ∗ ) ⊆ U c ( S max( c )+1 c ). The proof will proceed by induction on thecardinality of c \ n k . Suppose | c \ n k | = 1. Let l = max c , and let a denote c \ { l } . Then l ∈ U a ( Q ) \ n k . Further, a is a member of R n k i , since a ∈ Q ↾ n k = A i ⊆ R n k i .Since by Case 1, U a ( Q ) \ n k is contained in U a ( R n k i ), we have that c ∈ R n k i .Further,(4.34) U c ( Q ) ⊆ U c ( T ∗ ) ⊆ U c ( S l +1 c ) ⊆ U c ( R n k i ) . since l + 1 > n k , i < p n k and c ∈ R n k i imply that S l +1 c ⊆ R n k i , by thedefinition (4.12) of S l +1 c . Thus, Case 2 holds for the basis of our inductionscheme.Now assume that Case 2 holds for all c ∈ Q ∩ ˆ B − with 1 ≤ | c \ n k | ≤ m .Suppose c ∈ Q ∩ ˆ B − with | c \ n k | = m + 1. Letting l = max c and a = c \ { l } ,the induction hypothesis applied to a yields that a ∈ R n k i and U a ( Q ) ⊆ U a ( R n k i ). Thus, c ∈ R n k i . Again, as in (4.34), we find that U c ( Q ) ⊆ U c ( R n k i ),which finishes the proof of Case 2. (cid:3) To finish the proof of the theorem, we prove that ( ⊛ ) holds. Let T ∈ T ↾ ˜ T , k < ω , and suppose i < p n k is the integer such that T ↾ n k = A i . Letting Q denote ˜ T ∩ ˆ B n k i , we see that T ⊆ Q . By Claim 3, Q ⊆ R n k i ; so for all j ≤ k ,(4.35) j ∈ f ([ T ]) ⇐⇒ j ∈ f ([ R n k i ])by ( ∗ ) n k . Since Q ↾ n k = A i , it follows from ( ∗ ) n k that for all j ≤ k ,(4.36) j ∈ f ([ Q ]) ⇐⇒ j ∈ f ([ R n k i ]) . Equations (4.35) and (4.36) complete the proof of ( ⊛ ). By Claim 1, f ismonotone basic on T ↾ ˜ T . The concludes the proof of the theorem. (cid:3) We conclude this section pointing out how f may be basic on 2 ˆ B whilebeing only finitely generated on 2 B . Given a front B and a set C ⊆ B ,let ˆ C denote { ˆ X : X ∈ C} , a subset of 2 ˆ B . Letting C denote { ˆ X ↾ k m : X ∈ C and m < ω } , we point out that any finitary function ˆ f : C → <ω determines functions f ′ : C → ω and f ∗ : ˆ C → ω by setting f ′ ( X ) = f ∗ ( ˆ X ) = S m<ω ˆ f ( ˆ X ↾ k m ). In particular, f ′ ( X ) = f ∗ ( ˆ X ) for each X ∈ C .The following is straightforward to prove. Proposition 4.6. Suppose B is a front, C is a subset of B , and C = { ˆ X ↾ k m : X ∈ C , m < ω } . If ˆ f : C → <ω is a basic map, then f ∗ is continuouson ˆ C as a subspace of ˆ B . In particular, given a map f : U → V in the setting of Theorem 4.4, themap f ∗ : T ↾ ˜ T → V defined by f ∗ ( T ) = S m<ω ˆ f ( T ↾ k m ) is a continuousmap on its domain T ↾ ˜ T . This map f ∗ is equivalent to f in the following ANONICAL COFINAL MAPS 29 sense: For each T ∈ T ↾ ˜ T , f ∗ ( T ) = f ([ T ]). In contrast, letting C = { [ T ] : T ∈ T ↾ ˜ T } , the map f : C → V is not necessarily continuous. However, f ↾ C is still represented by the monotone finitary map ˆ f ; given X ∈ C , f ( X ) = S m<ω [[ ˆ f ( ˆ X ↾ k m )]].5. Further connections between Tukey, Rudin-Blass, andRudin-Keisler reductions In Lemma 9 of [13], Raghavan distilled properties of cofinal maps which,when satisfied, yield that Tukey reducibility above a q-point implies Rudin-Blass reducibility. He then showed that continuous cofinal maps satisfy theseproperties, thus yielding Theorem 10 in [13], (see Theorem 1.6 in Section 1).The proof of the next theorem follows the general structure of Raghavan’sproofs. The key differences are that we start with a weaker assumption,basic maps on Fubini iterates of p-points, and obtain a finite-to-one mapon the base tree ˆ B for the ultrafilter rather than the base B itself. Whilethe following theorem may be of interest in itself, we will apply it to proveTheorems 5.3 and 5.4 showing that for finite Fubini iterates of p-points, andfor generic ultrafilters forced by P ( ω k ) / Fin ⊗ k , Tukey reducibility above aq-point is equivalent to Rudin-Keisler reducibility. Theorem 5.1. Suppose U is a Fubini iterate of p-points and V is a q-point.If V ≤ T U , then there is a finite-to-one function τ : ˜ T → ω , for some ˜ T ∈ T ,such that { τ [ T ] : T ∈ T ↾ ˜ T } ⊆ V .Proof. Let B be the front which is a base for U , and as usual, let T denotethe set of all ~ U -trees on ˆ B . We begin by establishing some useful notation:Given m < n and T ⊆ [ ω ] <ω , let T ↾ [ m, n ) denote the set of all a ∈ T suchthat m ≤ max a < n .Let f : U → V be a monotone cofinal map. Let ˜ T ∈ T be given byTheorem 4.4, so that f is generated on T ↾ ˜ T by some monotone basic mapˆ f : S m<ω ˜ T ↾ k m → <ω . Let ψ : P ( ˜ T ) → P ( ω ) be defined by(5.1) ψ ( A ) = { k ∈ ω : ∀ T ∈ T ↾ ˜ T ( A ⊆ T → k ∈ f ([ T ])) } , for each A ⊆ ˜ T . Note that ψ is monotone and further that for each T ∈ T ↾ ˜ T and each m < ω , [[ ˆ f ( T ↾ k m )]] = ψ ( T ↾ k m ) ∩ m . By the same argumentas in Lemma 8 in [13], we may assume that for each finite A ⊆ ˜ T , ψ ( A ) isfinite.Notice that for any pair m, j < ω , if j ψ ( ˜ T ↾ m ), then there is a T ∈ T ↾ ˜ T such that T ⊐ ˜ T ↾ m and j f ([ T ]); hence, j [[ ˆ f ( T ↾ k j )]]. Itfollows that for all S ∈ T ↾ ˜ T satisfying S ⊒ T ↾ k j , j f ([ S ]) and hence also j ψ ( S ). Without loss of generality, assume that T ↾ [ m, k j ) = ∅ ; thatis, each a ∈ T has max a [ m, k j ). Now if A is a finite ⊏ -closed subset of ˜ T and A ↾ [ m, k j ) = ∅ , then there is an S ∈ T ↾ ˜ T such that ˜ T ↾ m ∪ A ⊆ S , S ↾ [ m, k j ) = ∅ , and S ⊐ T ↾ k j . Then j f ([ S ]), so j ψ ( ˜ T ↾ m ∪ A ).Define g : ω → ω by g (0) = 0; and given g ( n ), choose g ( n + 1) > g ( n ) sothat(5.2) g ( n + 1) > max { k g ( n ) , max( ψ ( ˜ T ↾ k g ( n ) )) } . Since V is a q-point, there is a V ∈ V such that for each n < ω , | V ∩ [ g ( n ) , g ( n + 1)) | = 1. We may, without loss of generality, assume that V = S n ∈ ω [ g (2 n ) , g (2 n + 1)) is in V , and let V = V ∩ V . Enumerate V as { v i : i < ω } . Notice that for each i < ω ,(5.3) g (2 i ) ≤ v i < g (2 i + 1) . Without loss of generality, assume that v > 0. Then v ψ ( ∅ ), sinceassuming V is nonprincipal, ψ ( ∅ ) must be empty.Our construction ensures the following properties: For all i < ω ,(1) g ( i + 1) > max( k g ( i ) , max( ψ ( ˜ T ↾ k g ( i ) )));(2) g (2 i ) ≤ v i < g (2 i + 1);(3) k v i < g (2 i + 2);(4) ψ ( ˜ T ↾ k v i ) ⊆ g (2 i + 2).We will now define a strictly increasing function h : ω → ω so that foreach i < ω , the following hold:(a) h ( i ) < k v i ;(b) v i ψ ( ˜ T ↾ h ( i ));(c) For each finite, ⊏ -closed set A ⊆ ˜ T , if v i ∈ ψ ( A ) then A ↾ [ h ( i ) , h ( i +1)) = ∅ .Define h (0) = 0. Then (a) - (c) trivially hold. Suppose h ( i ) has beendefined so that (a) - (c) hold. Define h ( i +1) = k v i . Then h ( i +1) > h ( i ), since k v i > h ( i ) by (a) of the induction hypothesis. (a) holds, since k v i < k v i +1 .To see that (b) holds, note that(5.4) ψ ( ˜ T ↾ h ( i + 1)) = ψ ( ˜ T ↾ k v i ) ⊆ ψ ( ˜ T ↾ k g (2 i +1) ) ⊆ g (2 i + 2) ≤ v i +1 , where the inclusions hold by (3) and (1), and the inequality holds by (2).Thus, v i +1 ψ ( ˜ T ↾ h ( i + 1)).To check (c), fix a finite ⊏ -closed set A ⊆ ˜ T such that A ↾ [ h ( i ) , h ( i +1)) = ∅ ; that is, for all a ∈ A , max a [ h ( i ) , h ( i + 1)). Let A ′ = A \ ( ˜ T ↾ h ( i + 1)). We claim that v i ψ ( ˜ T ↾ h ( i ) ∪ A ′ ). By (b), v i ψ ( ˜ T ↾ h ( i )).Therefore, v i [[ ˆ f ( ˜ T ↾ h ( i ))]]. Now ( ˜ T ↾ h ( i ) ∪ A ′ ) ↾ h ( i +1) = ˜ T ↾ h ( i ), since ANONICAL COFINAL MAPS 31 A ′ ↾ [ h ( i ) , h ( i + 1)) = ∅ . So v i [[ ˆ f (( ˜ T ↾ h ( i ) ∪ A ′ ) ↾ h ( i + 1))]]. Therefore,for each S ∈ T ↾ ˜ T such that S ↾ h ( i + 1) = ( ˜ T ↾ h ( i ) ∪ A ′ ) ↾ h ( i + 1),which we point out is the same as ˜ T ↾ h ( i ), we have v i f ([ S ]). Thus, if S ⊒ ˜ T ↾ h ( i ) ∪ A ′ and satisfies S ↾ [ h ( i ) , h ( i + 1)) = ∅ , then v i f ([ S ]),since this gets decided by height k v i which equals h ( i + 1). Therefore, v i ψ ( ˜ T ↾ h ( i ) ∪ A ′ ), which proves (c).Now we define a function τ : ˜ T → ω as follows: For i < ω and a ∈ ˜ T , ifmax a ∈ [ h ( i ) , h ( i + 1)), then τ ( a ) = v i . Claim. For each T ∈ T ↾ ˜ T , τ [ T ] ∈ V . Proof. Suppose not. Then there is a T ∈ T ↾ ˜ T such that τ [ T ] 6∈ V ; so τ [ T ] ∈ V ∗ . Then there is an S ∈ T ↾ T such that f ([ S ]) ⊆ ( ω \ τ [ T ]) ∩ V ∈V . Let j be least such that ˆ f ( χ S ↾ k j ) = ∅ and let s = S ↾ k j . Then ∅ 6 = [[ ˆ f ( χ S ↾ k j )]] ⊆ ψ ( s ). Fix some v i ∈ ψ ( s ). Then v i τ [ T ] since v i ∈ ψ ( s ) ⊆ f ([ S ]) ⊆ ω \ τ [ T ]. However, v i ∈ ψ ( s ) implies s ↾ [ h ( i ) , h ( i + 1)) = ∅ ,by (c). For each a ∈ s ↾ [ h ( i ) , h ( i + 1)), τ ( a ) = v i . Since s ⊆ T , we have v i ∈ τ [ T ], a contradiction. Thus, τ [ T ] ∈ V . (cid:3) Therefore, τ is a finite-to-one map from ˜ T into ω , and the set of τ -imagesof members of T ↾ ˜ T generate a filter contained inside V . (cid:3) The previous theorem does not necessarily imply that the τ -image of T ↾ ˜ T generates V . However, under certain conditions, it does. In the casethat T ↾ ˜ T generates an ultrafilter on base set the tree ˜ T , as is the casewhen all the p-points are the same selective ultrafilter, then the τ -image of U is V . The following is a Rudin-Blass analogue, but on ˆ B instead of B . Corollary 5.2. If U is a Fubini power of some selective ultrafilter V , wherethe base set for U is the front B , then there is a finite-to-one map τ : ˆ B → ω such that { τ [ T ] : T ∈ T } generates V . It is useful to point out the connection and contrast between this corol-lary and the following previously known results. Every Fubini power of someselective ultrafilter is Tukey equivalent to that selective ultrafilter (Corol-lary 37 in [8]). Thus, if U is a Fubini iterate of one selective ultrafilter,then the q-point V in Theorem 5.1 must be that selective ultrafilter. On theother hand, the only ultrafilters Tukey reducible to a selective ultrafilter arethose ultrafilters isomorphic to some Fubini power of the selective ultrafilter(Theorem 24 of Todorcevic in [13]). Thus, the collection of ultrafilters Tukeyequivalent to a given selective ultrafilter is exactly the collection of Fubini powers of that selective ultrafilter; and any Fubini power of a selective ul-trafilter is trivially Rudin-Keisler above that same selective ultrafilter.In Corollary 56 of [13], Raghavan showed that if U is some Fubini iterateof p-points and V is selective, then V ≤ T U implies V ≤ RK U . We nowgeneralize this to q-points, though at the cost of assuming U is only a finiteFubini iterate of p-points. Theorem 5.3. Suppose U is a finite iterate of Fubini products of p-points.If V is a q-point and V ≤ T U , then V ≤ RK U .Proof. Let k denote the length of the Fubini iteration, so [ ω ] k is the frontwhich is a base for U . Let τ be the finite-to-one map from Theorem 5.1,and without loss of generality, assume τ is defined on all of [ ω ] ≤ k . For each T ∈ T , we notice that S ≤ l ≤ k τ [ T ∩ [ ω ] l ] = τ [ T ] ∈ V . For each T ∈ T , let L ( T ) = { ≤ l ≤ k : τ [ T ∩ [ ω ] l ] ∈ V} . Then there is a T ∈ T such that forall S ∈ T ↾ T , L ( S ) = L ( T ). Let l = max( L ( T )).Now { S ∩ [ ω ] l : S ∈ T ↾ T } generates an ultrafilter on base set [ ω ] l ∩ T ;further, for each S ∈ T ↾ T , τ [ S ∩ [ ω ] l ] is a member of V (since l ∈ L ( S )).Thus, { τ [ S ∩ [ ω ] l ] : S ∈ T ↾ T } generates an ultrafilter, and each of these τ -images is in V . It follows that { τ [ S ∩ [ ω ] l ] : S ∈ T ↾ T } generates V . If l = k ,we are done, and in fact we have a Rudin-Blass map from U to V . If l < k ,then define σ : [ ω ] k → ω by σ ( a ) = τ ( π l ( a )). Then σ is a Rudin-Keisler mapfrom U into V . (cid:3) We point out that the basic maps on the generic ultrafilters G k forced by P ( ω k ) / Fin ⊗ k , 2 ≤ k < ω , in [6] have exactly the same properties as basicmaps on [ ω ] k in this paper. (See Definition 37 and Theorem 38 in [6].) Hence,Theorem 5.1 also applies to these ultrafilters. Since for each 1 ≤ l ≤ k , theprojection of G k to [ ω ] l yields the generic ultrafilter G l , the same proof as inTheorem 5.3 yields the following theorem. Theorem 5.4. Suppose G k is a generic ultrafilter forced by P ( ω k ) / Fin ⊗ k ,for any ≤ k < ω . If V is a q-point and V ≤ T G k , then V ≤ RK G k . Remark 5.5. We cannot in general weaken the requirement of q-point torapid in Theorem 5.1. In [10], it is shown that there are Tukey equivalentrapid p-points, and hence a Fubini iterate of such p-points Tukey equivalentto a rapid p-point, which are Rudin-Keisler incomparable. ANONICAL COFINAL MAPS 33 Ultrafilters Tukey reducible to Fubini iterates ofp-points have finitely generated Tukey reductions In this section, we prove the analogue of Theorem 2.6 for the class ofall ultrafilters which are Tukey reducible to some Fubini iterate of p-points.Namely, in Theorem 6.3, we prove that every ultrafilter Tukey reducibleto some Fubini iterate of p-points has finitely generated Tukey reductions(Definition 6.2). This sharpens a result of Raghavan (Lemma 16 in [13])by obtaining finitary maps which generate the original cofinal map on somefilter base rather than some possibly different cofinal map. Also, the class onwhich we obtain finitely generated Tukey reductions is closed under Tukeyreduction, whereas the class where his result applies (basically generatedultrafilters) is not known to be closed under Tukey reduction. Theorem 6.3allows us to extend Theorem 17 of Raghavan in [13] relating Tukey reductionto Rudin-Keisler reduction for basically generated ultrafilters to the classof all ultrafilters Tukey reducible to some Fubini iterate of p-points (seeTheorem 6.4 and the discussion preceding it).The next lemma is the analogue of Lemma 2.4 for the space 2 ˆ B in placeof 2 ω . As the proof is almost verbatim by making the obvious changes, weomit it. Lemma 6.1 (Extension lemma for fronts) . Suppose U is a nonprincipalultrafilter with base set a front B . Suppose f : U → V is a monotone cofinalmap, and there is a cofinal subset C ⊆ U such that f ↾ C is represented bya monotone basic function, in the sense of Definition 4.2. Then there is acontinuous, monotone ˜ f : 2 ˆ B → ω such that(1) ˜ f is represented by a monotone basic map ˆ f : S m<ω ˆ B ↾ k m → <ω ,in the sense of Definition 4.2.Moreover, defining the function f ′ : U → ω by f ′ ( U ) = ˜ f ( ˆ U ) , for U ∈ U ,(2) f ′ ↾ C = f ↾ C ; and(3) f ′ ↾ U is a cofinal map from U to V . Definition 6.2 (Finitely generated Tukey reduction) . We say that an ul-trafilter V on base set ω has finitely generated Tukey reductions if for eachmonotone cofinal map f : V → W , there is a cofinal subset C ⊆ V , astrictly increasing sequence ( k m ) m<ω , and a function ˆ f : C → <ω , where C = { X ↾ k m : X ∈ C , m < ω } , such that(1) ˆ f is level preserving: For each m < ω and s ∈ C , | s | = k m implies | ˆ f ( s ) | = m ;(2) ˆ f generates f on C : For each X ∈ C , (6.1) f ( X ) = [ m<ω [[ ˆ f ( X ↾ k m )]] . The difference between a basic Tukey reduction and a finitely generatedTukey reduction is that the map ˆ f in Definition 6.2 is not required to beend-extension preserving.Now we prove the main theorem of this section. This is the extension ofTheorem 2.5 (which holds for ultrafilters Tukey reducible to some p-point)to the setting of all ultrafilters Tukey reducible to some Fubini iterate ofp-points. Theorem 6.3. Let U be any Fubini iterate of p-points. If V ≤ T U , then V has finitely generated Tukey reductions.Proof. Suppose that U is an iteration of Fubini products of p-points andthat V ≤ T U . Let B be a front which is a base for U , and without lossof generality, assume that ω is the base set for the ultrafilter V . By Theo-rem 4.4, U has basic Tukey reductions. Applying Lemma 6.1, we obtain acontinuous monotone map ˜ f : 2 ˆ B → ω which is generated by a monotonebasic map ˆ f : S m<ω ˆ B ↾ k m → <ω , for some increasing sequence ( k m ) m<ω .Hence, for each A ⊆ ˆ B , ˜ f ( A ) = S m<ω [[ ˆ f ( A ↾ k m )]]. Furthermore, defining f ( U ) = ˜ f ( ˆ U ) for U ∈ U , we see that f : U → V is a monotone cofinal map.Suppose W ≤ T V , and let h : V → W be a monotone cofinal map.Extend h to the map ˜ h : 2 ω → ω defined as follows: For each X ∈ ω , let(6.2) ˜ h ( X ) = \ { h ( V ) : V ∈ V and V ⊇ X } . It follows from h being monotone that ˜ h is monotone and that ˜ h ↾ V = h .Letting ˜ g denote ˜ h ◦ ˜ f , we see that the map ˜ g : 2 ˆ B → ω is monotone.For U ∈ U , ˜ g ( ˆ U ) = ˜ h ( ˜ f ( ˆ U )) = ˜ h ( f ( U )) = h ◦ f ( U ). Thus, letting g denote h ◦ f , we see that g : U → W is a monotone cofinal map with the propertythat for each U ∈ U , g ( U ) = ˜ g ( ˆ U ). By Theorem 4.4, there is a ~ U -tree ˜ T andan increasing sequence ( k m ) m<ω such that the restriction of g to C = { [ T ] : T ∈ T ↾ ˜ T } is generated by some monotone basic map ˆ g : C → <ω , where C = { T ↾ k m : T ∈ T ↾ ˜ T and m < ω } . Notice that C = S m<ω ˜ T ↾ k m . Without loss of generality, we may assumethat the levels k m are the same for ˆ f and ˆ g , by taking the minimum of thetwo m -th levels.Let g ∗ be the function on 2 ˜ T into 2 ω determined by ˆ g as follows: For A ∈ ˜ T , define(6.3) g ∗ ( A ) = [ m<ω ˆ g ( A ↾ k m ) . ANONICAL COFINAL MAPS 35 Since ˆ g is end-extension preserving, it follows that for each A ∈ ˜ T and m < ω , g ∗ ( A ) ↾ m = ˆ g ( A ↾ k m ). We point out that the restriction of g ∗ to T ↾ ˜ T yields exactly ˜ g , since ˆ g generates ˜ g on T ↾ ˜ T . Claim 1. For each A ∈ ˜ T ,(6.4) g ∗ ( A ) = \ { g ( X ) : X ∈ C and ˆ X ⊇ A } ⊇ ˜ g ( A ) . Proof. Let A be any member of 2 ˜ T . Let m be given and note that A ↾ k m ∈ C . For any X ∈ C such that ˆ X ↾ k m = A ↾ k m , we have(6.5) ˆ g ( A ↾ k m ) = ˆ g ( ˆ X ↾ k m ) = ˜ g ( ˆ X ) ↾ m = g ( X ) ↾ m. Since C = { [ T ] : T ∈ T ↾ ˜ T } , there is an X ∈ C such that ˆ X ⊇ A andˆ X ↾ k m = A ↾ k m . (This is the key property of C needed for this proof.)Thus,(6.6) [[ˆ g ( A ↾ k m )]] = Y ↾ m ⊇ ˜ g ( A ) ↾ m, where Y = T { g ( X ) : X ∈ C and ˆ X ⊇ A } . Taking the union over all m < ω in (6.6) yields the claim. (cid:3) Define(6.7) D = { ˆ f ( s ) : s ∈ C } and D = f [ C ] . Then D is cofinal in V , and every member of D is a limit of members of D .We point out that T ↾ ˜ T is a subspace of 2 ˆ B , and the closure of T ↾ ˜ T in2 ˆ B is the compact space 2 ˜ T . Define a function ˆ h : D → <ω as follows: For t ∈ D ∩ m and i ∈ m , define(6.8) ˆ h ( t )( i ) = min { ˆ g ( s )( i ) : s ∈ C ∩ ˆ B ↾ k m and ˆ f ( s ) = t } . That is, ˆ h ( t ) is the function from m into 2 such that for i ∈ m , ˆ h ( t )( i ) = 1if and only if ˆ g ( s )( i ) = 1 for all s ∈ C ∩ ˆ B ↾ k m satisfying ˆ f ( s ) = t . Bydefinition, ˆ h is level preserving.We proceed to prove that ˆ h represents h on D . Fix Y ∈ D . Then thereis an X ∈ C such that f ( X ) = Y . For each m < ω , ˆ f ( ˆ X ↾ k m ) = Y ↾ m , so[[ˆ h ( Y ↾ m )]] ⊆ [[ˆ g ( ˆ X ↾ k m )]] = ˜ g ( ˆ X ) ↾ m = g ( X ) ↾ m = h ◦ f ( X ) ↾ m = h ( Y ) ↾ m. (6.9)Thus, S m<ω [[ˆ h ( Y ↾ m )]] ⊆ h ( Y ).Next we show that for each l ∈ h ( Y ), there is some n such that l ∈ [[ˆ h ( Y ↾ n )]]. Let m = l + 1 and let t = Y ↾ m . Let(6.10) S l = { s ∈ C ∩ ˆ B ↾ k m : ∀ A ∈ ˜ T with s ⊏ A, ˜ f ( A ) = Y } . Claim 2. For each s ∈ S l , there is an n s such that each s ′ ∈ ˆ B ↾ n s with s ′ ⊐ s satisfies ˆ f ( s ′ ) ⊏ Y . Proof. If not, then for some s ∈ S l , for each n there is an s ′ ∈ C of length n with s ′ ⊒ s and some A ∈ ˜ T with A ⊐ s ′ such that ˜ f ( A ) = Y . By K¨onig’sLemma, there is a sequence s ⊏ s ⊏ s ⊏ . . . of strictly increasing lengthssuch that for each i < ω , there is some A i ∈ ˜ T such that A i ⊐ s i and˜ f ( A i ) = Y . Letting A ′ = S i<ω s i , we see that A ′ ∈ ˜ T and that ˜ f ( A ′ ) = Y ,by continuity of f and the fact that ˆ f generates ˜ f on 2 ˜ T . But this contradictsthe fact that s ∈ S l . (cid:3) Since S l is finite, we may take n = max { n s : s ∈ S l } . Then for all s ′ oflength k n , if ˆ f ( s ′ ) ⊏ Y then s ′ does not end-extend any s in S l . Claim 3. For all s ′ ∈ C of length k n such that ˆ f ( s ′ ) ⊏ Y , l ∈ [[ˆ g ( s ′ )]]. Proof. For each s ′ ∈ C ∩ ˆ B ↾ k n satisfying ˆ f ( s ′ ) = Y ↾ n , we see that s ′ ↾ k m is not in S l . So there is some A ∈ ˜ T such that A ⊐ s ′ ↾ k m and ˜ f ( A ) = Y .It follows that(6.11) [[ˆ g ( s ′ ) ↾ m ]] = [[ˆ g ( s ′ ↾ k m )]] ⊇ ˜ g ( A ) ↾ m = ˜ h ◦ ˜ f ( A ) ↾ m = h ( Y ) ↾ m, where the ⊇ follows from Claim 1, since [[ˆ g ( s ′ ↾ k m )]] = g ∗ ( A ) ↾ m ⊇ ˜ g ( A ) ↾ m . Thus, l ∈ [[ˆ g ( s ′ )]], for each s ′ ∈ ˜ T ↾ k n satisfying ˆ f ( s ′ ) = Y ↾ n . (cid:3) Therefore, l ∈ [[ˆ h ( Y ↾ n )]]. Thus, for each l ∈ h ( Y ), there is an n l suchthat l ∈ [[ˆ h ( Y ↾ n l )]]. It follows that, for any j < ω , there is an n such thatˆ h ( χ Y ↾ n ) ↾ j = h ( Y ) ↾ j . This n may be obtained by taking the maximumof the n s over all s ∈ S { S l : l < j } , Hence, S m<ω [[ˆ h ( Y ↾ m )]] = h ( Y ).Thus, h ↾ D is finitely represented by ˆ h on D . (cid:3) Theorem 6.3 is now applied to extend Theorem 17 of Raghavan in [13] toall ultrafilters Tukey reducible to some Fubini iterate of p-points. Raghavanshowed that for any basically generated ultrafilter U , whenever V ≤ T U there is a filter U ( P ) which is Tukey equivalent to U such that V ≤ RK U ( P ).It is routine to check that the maps in Theorem 6.3 satisfy the conditions ofthe maps in Theorem 17 of Raghavan in [13]. Thus, we obtain the following. Theorem 6.4. If U is Tukey reducible to a Fubini iterate of p-points, thenfor each V ≤ T U , there is a filter U ( P ) ≡ T U such that V ≤ RK U ( P ) . Here, assuming without loss of generality that the base set for U is ω , P is the collection of ⊏ -minimal finite subsets s of ω for which ˆ h ( s ) = ∅ , whereˆ h witnesses that a given monotone cofinal h : U → V is finitely generated. U ( P ) is the collection of all sets of the form { s ∈ P : s ⊆ U } , for U ∈ U . ANONICAL COFINAL MAPS 37 Remark 6.5. The same proof of Theorem 6.3 works for the basic cofinalmaps for the generic ultrafilters G k forced by P ( ω k ) / Fin ⊗ k , 2 ≤ k < ω ,in [6]. Thus, Theorem 6.4 also holds when U is an ultrafilter forced by P ( ω k ) / Fin ⊗ k . 7. Open problems We conclude this paper by highlighting some of the more importantopen problems in this area. Theorem 2.6 showed that every ultrafilter Tukeybelow a p-point has continuous Tukey reductions. Problem 7.1. Determine the class of all ultrafilters that have continuousTukey reductions.In particular, are there ultrafilters not Tukey reducible to a p-point whichsatisfy the conditions of Theorem 2.5?By Theorem 56 of Dobrinen and Trujillo in [7], under very mild condi-tions, any ultrafilter selective for some topological Ramsey space has basic,an hence, continuous (with respect to its metric topology) Tukey reduc-tions. This is especially of interest when the ultrafilter associated with thetopological Ramsey space is not a p-point. It should be the case that byarguments similar to those in this paper one can prove the following. Problem 7.2. Prove the analogues of Theorems 2.5, 4.4, and 6.3 for sta-ble ordered union ultrafilters and their iterated Fubini products, and moregenerally for ultrafilters selective for some topological Ramsey space, withrespect to the correct topologies.More generally, we would like to know the following. Problem 7.3. Determine the class of all ultrafilters which have finitelygenerated Tukey reductions. Is this the same as the class of all ultrafilterswith Tukey type strictly below the maximum Tukey type?In Section 5, we applied Theorem 5.1 to obtain more examples whenTukey reducibility implies Rudin-Keisler reducibility. Theorem 5.3 improveson one aspect of Corollary 56 in[13] of Raghavan provided that there areq-points which are not selective and which are Tukey below some Fubiniiterate of p-points. Do such ultrafilters ever exist? Problem 7.4. Is there a q-point which is not selective which is Tukeyreducible to some finite Fubini iterate of p-points? Or does V ≤ T U with U a Fubini iterate of p-points and V a q-point imply that V is actuallyselective? Problem 7.5. Can Theorem 5.3 be extended to all countable iterates ofFubini products of p-points? Are similar results true for all ultrafilters Tukeyreducible to some Fubini iterate of p-points?Question 25 in [8] asks whether asks whether every ultrafilter Tukeyreducible to a p-point is basically generated. Question 26 in [8] asks whetherthe classes of basically generated and Fubini iterates of p-points the same, orwhether the former is strictly larger than the latter? Though these questionsin general are still open, we ask the even more general questions. Problem 7.6. Is the property of being basically generated inherited underTukey reduction? That is, if V is Tukey reducible to a basically generatedultrafilter, is V necessarily basically generated?Or the possibly weaker problem: If V is Tukey reducible to some Fubiniiterate of p-points, is V necessarily basically generated?There are certain collections of p-points, in particular those associatedwith the topological Ramsey spaces in [9], [10], and [7], for which everyultrafilter Tukey below some Fubini iterate of these p-points is again aFubini iterate of these p-points and hence basically generated. However, theabove questions are in general still open.Work in this paper and work in [13] found conditions when Tukey re-ducibility implies Rudin-Keisler or even Rudin-Blass reducibility. Problem 7.7. When in general does U ≥ T V imply U ≥ RK V or U ≥ RB V ?Finally, how closely related are the properties of having finitely generatedTukey reductions and having Tukey type below the maximum? Problem 7.8. 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