A characterization of productive cellularity
Renan Maneli Mezabarba, Leandro Fiorini Aurichi, Lucia Renato Junqueira
aa r X i v : . [ m a t h . GN ] M a y THE CELLULAR SPECTRUM OF A POSET
RENAN M. MEZABARBA , LEANDRO F. AURICHI, AND L ´UCIA R. JUNQUEIRA Abstract.
We investigate the notion of productive cellularity on arbitrarypreorders by generalizing an intrinsic characterization of productively ccc pre-orders.
Introduction
Along this work, we shall be concerned with the behavior of the cellularity ofpreordered sets (posets for short) with respect to the product operation. Let usstart by recalling some basic concepts regarding posets. First of all, a preorder ≤ on a set P is just a reflexive and transitive binary relation – and it is called a partialorder if in addition ≤ is antisymmetric. Now, two elements p and q of a poset ( P , ≤ )are said to be compatible if there exists an r ∈ P such that r ≤ p, q . Naturally, wesay that p and q are incompatible if they are not compatible, what we abbreviatewith p ⊥ q . Finally, a subset A ⊂ P of pairwise incompatible elements is called an antichain of P .The cellularity of a poset P , denoted by c ( P ), is the supremum of all infinitecardinals of the form | A | for some antichain A ⊂ P . In this way, the so called countable chain condition (ccc for short) is achieved by the poset P precisely whenthe equality c ( P ) = ℵ holds. Finally, for a poset ( Q , (cid:22) ), we can consider the set P × Q preordered by the relation ⊑ , which is defined by the rule( p, q ) ⊑ ( p ′ , q ′ ) ⇔ p ≤ p ′ and q (cid:22) q ′ . Here is where the story gets interesting, at least from a foundational perspective:it may be the case that P × Q is ccc whenever P and Q are ccc posets, or it may not.Essentially, it all depends on the additional hypothesis we add to the ZFC axioms.Indeed, in the realm of Martin’s Axiom (MA) plus the negation of the ContinuumHypothesis (CH), one can prove that every product of ccc posets is a ccc poset ,while a Suslin line turns out to be a ccc poset whose square is not ccc . Since eachof the statements “MA + ¬ CH”and “there exists a Suslin line” are independent ofZFC , it follows that productivity of ccc posets is itself independent of ZFC.Since there are posets whose product with every ccc poset is ccc, e.g., the count-able posets, it seems natural to ask, in ZFC, when a poset P is productively ccc ,meaning that P × Q is ccc whenever Q is ccc. As it was shown in [6], a ZFC char-acterization of productively ccc posets can be obtained by analyzing the cellularityof some posets of antichains. The technique used there was an adaptation of the Mathematics Subject Classification.
Primary 54D20; Secondary 54G99, 54A10.
Key words and phrases. posets, cellularity, cellular productivity, countable chain condition. Supported by CNPq (2017/09252-3) and Capes (88882.315491/2019-01). Folklore, but possibly due to Juh´asz. Attributed to Kurepa. The reader may find the details in Kunen [4]. methods applied by Aurichi and Zdomskyy [2] to characterize productively Lindel¨ofspaces.Here, we generalize the results presented in [6], by characterizing what we call cellular spectrum of the poset P , denoted Sp ( P ): the class of those infinite cardinals κ such that for all posets Q , c ( Q ) ≤ κ ⇒ c ( P × Q ) ≤ κ. The paper is organized as follows. In the first section, we state and prove ourmain results concerning an intrinsic characterization of Sp ( P ). In the second sec-tion, we investigate cardinals invariants that belongs to the spectrum, while in thethird section we translate some classic theorems about ccc topological spaces tothis spectral context. Finally, the last section is dedicated to discussing new per-spectives for old (and open) problems concerning productively ccc posets. Alongthe text, κ and λ denote infinite cardinals.1. The main theorem
Let us start by fixing some notations that will be useful. For a poset P , weconsider the cellular spectrum of P , which is the classSp ( P ) := { κ ≥ ℵ : ∀ Q ( c ( Q ) ≤ κ ⇒ c ( P × Q ) ≤ κ ) } . Our main goal in this section is to determine necessary and sufficient conditionsin order to decide whether a given cardinal κ ≥ ℵ belongs to Sp ( P ). To thisend, we say that a family A of antichains of P is a κ - large family if | S A | ≥ κ + ,and we denote by L κ ( P ) the collection of all such families. Finally, for a κ -largefamily A ∈ L κ ( P ), we set F ( A ) = S A∈ A [ A ] < ℵ , partially ordered by the reverseinclusion relation.We shall use posets of the form F ( A ) in order to characterize the spectrumof P . This can be done because incompatibility conditions in F ( A ) translate tocompatibility conditions of P , as we show in the following lemma. Lemma 1.1.
Let A ∈ L κ ( P ) and let P, Q ∈ F ( A ) . Then P ⊥ Q in F ( A ) if, andonly if, P ∪ Q F ( A ) .Proof. Note that if P ∪ Q ∈ F ( A ), then P, Q ⊂ P ∪ Q , showing that P ⊥ Q doesnot hold. Conversely, if some R in F ( A ) contains both P and Q , then there isan A ∈ A such that R ⊂ A , showing that P ∪ Q is a finite subset of A , i.e., P ∪ Q ∈ F ( A ). (cid:3) Theorem 1.2.
Let P be a poset. Then κ ∈ Sp ( P ) if and only if c ( F ( A )) > κ forall A ∈ L κ ( P ) .Proof. If κ ∈ Sp ( P ) and A ∈ L κ ( P ) is such that c ( F ( A )) ≤ κ , then we have that c ( P × F ( A )) ≤ κ . Now, let T := n ( p, { p } ) : p ∈ [ A o . Since A is κ -large, the family T cannot be an antichain in P × F ( A ). Thus,there are p, p ′ ∈ S A with p = p ′ , r ∈ P and F ∈ F ( A ) such that( r, F ) ⊑ ( p, { p } ) , ( p ′ , { p ′ } ) The word “spectrum” references to the “frequency spectrum”, considered by Arhangel’skiiin [1], a class of cardinals related to the tightness of products of topological spaces.
HE CELLULAR SPECTRUM OF A POSET 3 implying p p ′ and { p, p ′ } ⊆ F ⊆ A for some A ∈ A , showing that A is not anantichain, a contradiction.Conversely, supposing κ Sp ( P ), we shall obtain a κ -large family A such that c ( F ( A )) ≤ κ . Let Q be a poset witnessing κ Sp ( P ), i.e., with c ( Q ) ≤ κ andsuch that there exists an antichain W ⊂ P × Q with |W| = κ + . For each r ∈ Q , let A r := { p ∈ P : ∃ q ∈ Q ( r ≤ q and ( p, q ) ∈ W ) } . We claim that A := {A r : r ∈ Q } is the desired κ -large family.Note that A r is clearly an antichain for each r ∈ Q , while | S A | = κ + holdsby the pigeonhole principle, showing that A ∈ L κ ( P ). It remains to show that c ( F ( A )) ≤ κ . Indeed, for if F ⊂ F ( A ) is such that |F| = κ + , for each F ∈ F wetake r F ∈ Q with F ⊂ A r F . Now we consider the set R := { r F : F ∈ F} ⊂ Q , thatwe shall use to obtain the desired inequality.There are two cases:(1) if |R| ≤ κ , then the pigeonhole principle gives F, G ∈ F with F = G and r ∈ R such that F ∪ G ⊂ A r , showing that F ∪ G ∈ F ( A );(2) if |R| = κ + , then c ( Q ) ≤ κ gives F, G ∈ F with F = G and r ∈ Q suchthat r ≤ r F , r G , showing that A r F ∪ A r G ⊂ A r , from which it follows that F ∪ G ∈ F ( A ).In both cases, we obtain F, G ∈ F with F = G , such that F ∪ G ∈ F ( A ), whichis equivalent to say that F ⊥ G by the previous lemma, showing that F is not anantichain of F ( A ), as desired. (cid:3) Corollary 1.3.
A poset P is productively ccc if, and only if, F ( A ) is not ccc forall A ∈ L κ ( P ) . The above characterizations become clearer in their contrapositve versions. Forinstance, Corollary 1.3 says that if a poset P is not productively ccc, then there isa witness of the form F ( A ) for some A ∈ L ℵ ( P ). Thus, if we have L ℵ ( P ) = ∅ ,then P is vacuously productively ccc, since there are no witnesses to the contrary.This gives a very clean proof for the well known fact that countable posets areproductively ccc. More generally, we have the following. Corollary 1.4. If | P | < κ , then κ ∈ Sp ( P ) . Remark 1.
Although the presentation of Theorem 1.2 has been order-theoreticflavored, we originally found it out in a topological context, closer to [2]. In thiscase, the cellularity of a space X is the cellularity of the poset O X of nonemptyopen sets of X , while the cellular spectrum of X , also denoted by Sp ( X ) , is theclass of those infinite cardinals κ such that c ( X × Y ) ≤ κ holds for all topologicalspaces Y with c ( Y ) ≤ κ .It seems quite clear that the arguments we used to settle Theorem 1.2 can becarried out to topological spaces. Still, this can be done indirectly, by using thecharacterization we already proved for posets. Theorem 1.5.
For a topological space X one has Sp ( X ) = Sp ( O X ) .Proof. Suppose κ ∈ Sp ( X ) and let Q be a poset with c ( Q ) ≤ κ . Denote by Y theset Q endowed with the topology generated by the sets of the form { s ∈ Q : s ≤ q } ,which clearly satisfies c ( Y ) = c ( Q ). Now, the hypothesis gives c ( X × Y ) ≤ κ , whilea straightforward calculation gives c ( X × Y ) = c ( O X × O Y ) = c ( O X × Q ) , R. M. MEZABARBA, L. F. AURICHI, AND L. R. JUNQUEIRA showing that κ ∈ Sp ( O X ). The converse is trivial. (cid:3) A few inhabitants of the spectrum
Besides showing that Sp ( P ) is nonempty for all posets P , Corollary 1.4 indicatesthat the possibly interesting cardinals in the cellular spectrum are smaller or equalto | P | . In particular, it makes sense to define the productive cellularity of P to bethe cardinal pc( P ) := min Sp ( P ) , in reference to the fact that P is productively ccc if, and only if, pc( P ) = ℵ .Since any poset T with a single element satisfies c ( T ) ≤ κ for all κ ≥ ℵ , itfollows that for every poset P one has c ( P ) ≤ pc( P ), from which it follows that(1) c ( P ) ≤ pc( P ) ≤ | P | + . We shall explore the gap between the cardinals pc( P ) and | P | + through the rest ofthis section.Recall that a subset D of a poset P is called dense if for all p ∈ P there is a d ∈ D such that d ≤ p . The density of P , denoted by d ( P ), is the least infinite cardinalof the form | D | with D ⊂ P dense, which is a generalization of the separability intopological spaces.Since the cardinality of an antichain of P is bounded by the cardinality of everydense subset of P , it follows immediately that c ( P ) ≤ d ( P ). This inequality canstrengthen in the following way. Theorem 2.1. If P is a poset, then d ( P ) ∈ Sp ( P ) .Proof. Let D ⊂ P be a dense subset and call κ := | D | . We shall prove that κ belongs to the cellular spectrum of P . By Theorem 1.2, we need to take A ∈ L κ ( P )and show that c ( F ( A )) > κ . Since D is dense, it follows that for each a ∈ S A there exists a δ ( a ) ∈ D such that δ ( a ) ≤ a . Hence there exists a d ∈ D such thatthe set A := { a ∈ S A : δ ( a ) = d } has cardinality at least κ + . Finally, since d ≤ a for all a ∈ A , one can readily sees that the family {{ a } : a ∈ A } witnesses theinequality c ( F ( A )) > κ , as desired. (cid:3) The arguments used above actually improve Corollary 1.4, allowing one to provethe following.
Corollary 2.2. If κ ≥ d ( P ) , then κ ∈ Sp ( P ) . In particular, (1) can be replaced by(2) c ( P ) ≤ pc( P ) ≤ d ( P ) . Note that in order to finish the proof of Theorem 2.1, we used a strong propertyof the family A , namely the existence of d ∈ P such that d ≤ a for all a ∈ A . Aswe shall see below, this condition can be relaxed.For a natural number n ≥
2, a subset A ⊂ P is called n - linked if for all F ∈ [ A ] n there exists p A ∈ P such that p A ≤ p for each p ∈ F ; A is called centered if A is n -linked for all n ≥
2. Then we have the following.
Lemma 2.3.
Let P be a poset and A ∈ L κ ( P ) . If a subset A ⊂ S A is n -linkedfor some natural number n ≥ , then the family {{ a } : a ∈ A } is an antichain in F ( A ) . HE CELLULAR SPECTRUM OF A POSET 5
Proof.
For a, b ∈ A with a = b , there is an r ∈ P such that r ≤ a, b . Then wehave { a } , { b } ∈ F ( A ) while { a } ∪ { b } = { a, b } 6∈ F ( A ), yielding { a } ⊥ { b } , byLemma 1.1. (cid:3) Although the above lemma seems to be innocuous, it has some interesting con-sequences. Let us recall a few more concepts in order to apply Lemma 2.3. Follow-ing [7], we say that a poset P has the K n -property if for each A ∈ [ P ] ℵ there existsan n -linked subset B ∈ [ A ] ℵ . By replacing the occurrence of the term “ n -linked”with “centered”, we obtain the property usually called ℵ -precaliber , but for sakeof brevity we shall refer to it simply by K ω -property . The letter “K” is a referenceto Knaster, who first considered this type of property, for n = 2.In the same way cellularity generalizes the countable chain condition, we definebelow the Knaster invariants of P in order to generalize K n and K σ properties.More precisely, for each natural number n ≥ K n ( P ) := min { κ ≥ ℵ : ∀ A ∈ [ P ] κ + ∃ B ∈ [ A ] κ + ( B is n -linked) } , and we let(4) K ω ( P ) := min { κ ≥ ℵ : ∀ A ∈ [ P ] κ + ∃ B ∈ [ A ] κ + ( B is centered) } . Note that for a poset P and an ordinal α ∈ [2 , ω ], P has the K α -property if,and only if, K α ( P ) = ℵ . The relations between the Knaster properties with thecountable chain condition are in some sense preserved in the spectral context.For a dense subset D ⊂ P with | D | = κ , the same reasoning applied in The-orem 2.1 allows one to prove that for every A ∈ [ P ] κ + there is a centered subset B ∈ [ A ] κ + , showing that K ω ( P ) ≤ d ( P ). Since we clearly have K α ( P ) ≤ K β ( P ) for α ≤ β ≤ ω , it follows that K ( P ) ≤ K n ( P ) ≤ K n +1 ( P ) ≤ K ω ( P ) ≤ d ( P )holds for every poset P . We now put pc( P ) in one extreme of the above inequalities. Theorem 2.4. If P is a poset, then K ( P ) ∈ Sp ( P ) .Proof. Let κ := K ( P ) and let A ∈ L κ ( P ). Since | S A | > κ , there exists an A ⊆ S A such that | A | = κ + . Now, there exists a 2-linked subset B ∈ [ A ] κ + , sothe conclusion follows from Lemma 2.3. (cid:3) Differently of what happened in Theorem 2.1, we are not able to adapt the aboveargument to show that every κ ≥ K ( P ) belongs to Sp ( P ). Still, Lemma 2.3 canbe used similarly to prove that the cardinal invariants K α ( P ) belongs to Sp ( P ) forall α ∈ [2 , ω ]. In summary, for every poset P and every natural number n ≥
2, wehave(5) c ( P ) ≤ pc( P ) ≤ K ( P ) ≤ K n ( P ) ≤ K n +1 ( P ) ≤ K σ ( P ) ≤ d ( P ) . The spectrum of products and some topological translations
Although the (productive) cellularity of posets may be interesting by itself, thetopological interpretations of the previous results deserve some attention. For awarming up example, the topological counterpart of Theorem 2.1 says that sepa-rable spaces are productively ccc. Indeed, similar to what occurs with cellularity,the density of a topological space X , denoted by d ( X ), is the density of the poset O X . Thus, in this context, Theorems 1.5 and 2.1 together say that the density of a R. M. MEZABARBA, L. F. AURICHI, AND L. R. JUNQUEIRA space X belongs to Sp ( X ) and, since X is separable if and only if d ( X ) = ℵ , ourclaim follows. However, even more is known to be true: Proposition 3.1 (Fremlin [3], Corollary 12J) . Every product of separable spacesis productively ccc.
The above proposition make us wonder about the behavior of the cellular spec-trum of a poset with respect to products. The very definition of the cellular spec-trum implies that for posets P and Q one has κ ∈ Sp ( P ) ∩ Sp ( Q ) ⇒ κ ∈ Sp ( P × Q ) , showing that Sp ( P ) ∩ Sp ( Q ) ⊂ Sp ( P × Q ). The reverse inclusion follows from thenext easy lemma, whose proof we left for the reader. Lemma 3.2. If ϕ : P → Q is an increasing function from the poset P onto the poset Q , then Sp ( P ) ⊂ Sp ( Q ) . Theorem 3.3. If P and Q are posets, then Sp ( P × Q ) = Sp ( P ) ∩ Sp ( Q ) .Proof. We already have Sp ( P ) ∩ Sp ( Q ) ⊂ Sp ( P × Q ). Now, since the projections P × Q → P and P × Q → Q are both increasing and surjective, the reverse inclusionfollows from the previous lemma. (cid:3) In order to extend this result for arbitrary products of posets, we need to con-sider a slightly different product, closer to the topological counterpart of arbitraryproducts. We follow the definitions presented by Kunen in [4], where the readermay find more details.Let { P i : i ∈ I } be a nonempty family of posets such that for each i ∈ I thereis a largest element 1 i ∈ P i . Such posets are called forcing posets in [4]. The finitesupport product of the forcing posets P i , denoted by Q fin i ∈ I P i , is the subset of Q i ∈ I P i whose elements are those I -tuples f such that |{ i ∈ I : f i = 1 i }| < ℵ , endowedwith the coordinate-wise preordering. In some sense, this is the order-theoreticversion of the standard topology on arbitrary products of topological spaces. Theorem 3.4.
For a nonempty family { P i : i ∈ I } of forcing posets one has Sp (cid:16)Q fin i ∈ I P i (cid:17) = T i ∈ I Sp ( P i ) .Proof. The inclusion Sp (cid:16)Q fin i ∈I P i (cid:17) ⊂ T i ∈I Sp ( P i ) follows from Lemma 3.2. On theother hand, the reverse inclusion can be proved with a straightforward applicationof the ∆-system lemma. (cid:3) With the previous theorem established, Fremlin’s result about separable spacesbecomes the topological counterpart of the following.
Corollary 3.5. If { P i : i ∈ I } is a nonempty family of forcing posets, then sup i ∈ I d ( P i ) ∈ Sp (cid:16)Q fin i ∈ I P i (cid:17) .Proof. Since d ( P j ) ≤ sup i ∈ I d ( P i ) and d ( P j ) ∈ Sp ( P j ), it follows from Corollary 2.2that sup i ∈ I d ( P i ) ∈ Sp ( P j ) for all j ∈ I , showing thatsup i ∈ I d ( P i ) ∈ \ i ∈ I Sp ( P i ) = Sp fin Y i ∈ I P i ! . (cid:3) HE CELLULAR SPECTRUM OF A POSET 7 Further questions and comments
Corollary 2.2 and the absence of a similar result for the Knaster invariants suggesta natural question about the behavior of the cardinals in the cellular spectrum.More precisely:
Question 4.1.
Let P be a poset. Does every cardinal κ such that pc( P ) < κ < d ( P ) belongs to Sp ( P ) ? Concerning the Knaster invariants, we still do not know if they are consistentlydifferent from each other. On the other hand, they all coincide under a standardassumption.
Example 4.2.
Assuming the existence of a Suslin Line R , one has c ( R ) = ℵ ,while pc( R ) > ℵ since R is not productively ccc. On the other hand, the inequality d ( X ) ≤ c ( X ) + holds for every LOTS X . Thus, we have d ( R ) = ℵ , from which itfollows that all the Knaster invariants of R collapse to ℵ . It may also be interesting to explore the connections of the cellular spectrumwith Martin’s Axiom related topics. As we mentioned earlier, the standard strategyto show that MA+ ¬ CH implies that every ccc poset is productively ccc starts byshowing that every ccc poset has the K ω -property . Note that we can restate bothassertions, respectively, as the following implications: ∀ P c ( P ) = ℵ ⇒ pc( P ) = ℵ ;(6) ∀ P c ( P ) = ℵ ⇒ K ω ( P ) = ℵ . (7)Now, since pc( P ) ≤ K ω ( P ), it follows immediately that (7) ⇒ (6). Althoughit is not completely well known, (7) is equivalent to MA ℵ , thanks to the nextproposition, due to Todorˇcevi´c and Veliˇckovi´c [7]. Proposition 4.3 (Todorˇcevi´c and Veliˇckovi´c [7], Theorem 3.4) . MA ℵ holds if,and only if, every uncountable ccc poset has an uncountable centered subset. In [5], up to terminology, Larson and Todorˇcevi´c asks whether any of the as-sumptions(1) ∀ P c ( P ) = ℵ ⇒ K ( P ) = ℵ or(2) ∀ P c ( P ) = ℵ ⇒ pc( P ) = ℵ imply MA ℵ . Thus, after all we have done so far, Larson and Todorˇcevi´c’s ques-tions suggest the following. Question 4.4.
Does MA ℵ implies c ( P ) = pc( P ) for every poset P ? Does theconverse hold? Question 4.5.
Does MA ℵ implies c ( P ) = K ( P ) for every poset P ? Does theconverse hold? References [1] A. Arhangel’skii. The frequency spectrum of a topological space and the product operation.
Trans. Moscow Math. Soc. , 2:163–200, 1981.[2] L. F. Aurichi and L. Zdomskyy. Internal characterizations of productively Lindel¨of spaces.
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R. M. MEZABARBA, L. F. AURICHI, AND L. R. JUNQUEIRA [3] D. Fremlin.
Consequences of Martin’s axiom . Cambridge tracts in mathematics 84. CambridgeUniversity Press, 1984.[4] K. Kunen.
Set Theory . College Publications, London, 2011.[5] P. Larson and S. Todorˇcevi´c. Chain conditions in maximal models.
Fundamenta Mathematicae ,168(1):77–104, 2001.[6] R. M. Mezabarba.
Selection principles in hyperspaces . PhD thesis, ICMC-USP, 2018.[7] S. Todorˇcevi´c and B. Veliˇckovi´c. Martin’s axiom and partitions.
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Centro de Ciˆencias Exatas, Universidade Federal do Esp´ırito Santo, Vit´oria, ES,29075-910, Brazil
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