partially ordered set X has countable width if and only if every collection of pairwise incomparable elements of X is countable. It is order-separable if and only if there is a countable subset D of X such that whenever p, q ∈ X and p \u3c q, there is r ∈ D such that p ≤ r ≤ q. Can every order-separable poset of countable width be written as the union of a countable number of chains? We show that the answer to this question is  no  if there is a 2-entangled subset of IR, and  yes  under the Open Coloring Axiom

Gruenhage, Gary

Mashburn, Joe

English

University of Dayton

University of DaytoneCommonsMathematics Faculty Publications Department of Mathematics1999On The Decomposition of Order-separable Posetsof Countable Width into ChainsGary GruenhageAuburn University Main CampusJoe MashburnUniversity of Dayton, joe.mashburn@udayton.eduFollow this and additional works at: https://ecommons.udayton.edu/mth_fac_pubPart of the Applied Mathematics Commons, Mathematics Commons, and the Statistics andProbability CommonsThis Article is brought to you for free and open access by the Department of Mathematics at eCommons. It has been accepted for inclusion inMathematics Faculty Publications by an authorized administrator of eCommons. For more information, please contact frice1@udayton.edu,mschlangen1@udayton.edu.eCommons CitationGruenhage, Gary and Mashburn, Joe, "On The Decomposition of Order-separable Posets of Countable Width into Chains" (1999).Mathematics Faculty Publications. 25.https://ecommons.udayton.edu/mth_fac_pub/25On the decomposition of order-separable posets ofcountable width into chainsGary Gruenhage∗Department of MathematicsAuburn UniversityAuburn, AL 36849-5310(garyg@mail.auburn.edu)Joe MashburnDepartment of MathematicsUniversity of DaytonDayton, OH 45469-2316(joe.mashburn@udayton.edu)Abstract. A partially ordered set X has countable width if and only if everycollection of pairwise incomparable elements of X is countable. It is order-separableif and only if there is a countable subset D of X such that whenever p, q ∈ X andp < q, there is r ∈ D such that p ≤ r ≤ q. Can every order-separable poset ofcountable width be written as the union of a countable number of chains? We showthat the answer to this question is ”no” if there is a 2-entangled subset of IR, and”yes” under the Open Coloring Axiom.Keywords: countable width, order-separable, chain, k-entangled subset, Open Col-oring AxiomMathematics Subject Classification (1991): 06A06, 03E051. IntroductionThe decomposition of partially ordered sets into chains has been a sig-nificant part of the study of the structure of partially ordered sets. Thesuccess in this area has come primarily using posets with the propertythat there is n ∈ ω such that for every antichain (by which we meana set of incomparable elements) has cardinality ≤ n. See, for example,these references: [1], and [3]–[14]. In [13], Peles constructed an exampleof a poset P such that every antichain of P is finite, but P is not theunion of a countable number of chains. In [15] a problem was studiedwhich required posets to be order-separable and have countable width.These posets seemed to be good candidates for decomposition into acountable number of chains. This led to the question which we will∗ Research of the first author partially supported by National Science Foundationgrant DMS 970-4849c© 1999 Kluwer Academic Publishers. Printed in the Netherlands.cccos2.tex; 23/08/1999; 12:03; p.12answer in this paper. That is, can every order-separable poset havingcountable width be written as the union of a countable number ofchains? In Section 2, we use a type of subset of IR called a 2-entangledset to show that under certain axioms, such as the continuum hypoth-esis (CH), there are order-separable posets of countable width thatcannot be written as the union of a countable number of chains. Onthe other hand, we show in Section 3 that under the Open ColoringAxiom, every order-separable poset can indeed be written as the unionof a countable number of chains.Let us define some of the terminology that we have used, thenconsider a couple of related questions. The definition of entangled setsand the statement of the Open Coloring Axiom will be left to theappropriate sections.Definition. A poset X is said to have countable width if and only ifevery antichain of X is countable.Definition. A poset X is order-separable if and only if there is acountable C ⊆ X such that for every p, q ∈ X with p < q there isr ∈ C such that p ≤ r ≤ q.Two questions related to the concepts investigated in this paperwhich were asked in [15] also have both positive and negative answers,depending on your set theory. A structure introduced in [15] is thecollection of nonoverlapping subsets of X.Definition.A collectionA of subsets of a posetX is called a collectionof nonoverlapping subsets of X if and only if A satisfies the followingconditions.1. A is a collection of pairwise disjoint sets, each having at least twoelements.2. The transitive closure of the relation{〈A,B〉 ∈ A2 : A 6= B ∧ ∃p ∈ A∃q ∈ B(p < q)}is a partial order.We use ν(X) to represent the supremum of the cardinalities of col-lections of nonoverlapping subsets of X. Obviously, if ν(X) ≤ ω thenX has countable width. To see why these kinds of collections are of anyinterest, we must make one more definition.cccos2.tex; 23/08/1999; 12:03; p.23Definition. A poset 〈X,<〉 is pliable if and only if for every linearextension ≺ of <, there is a strictly ≺-increasing function f : X → IR.In [15] it was shown that X is pliable if and only if ν(X) ≤ ω andX is order-separable, and the following two questions appeared.Question. If ν(X) ≤ ω, can X necessarily be written as the unionof a countable number of chains?Question. If X is pliable, can X necessarily be written as the unionof a countable number of chains?It will be shown in Section 2 that the answer to both questions is”no” if there is a 4-entangled set. An immediate corollary to Theorem2 in Section 3 is that the answer to the second question is ”yes” underOCA. Since it was shown in [15] that Souslin’s Hypothesis (SH) isequivalent to the statement that every poset X with ν(X) ≤ ω ispliable, it also follows that the answer to the first question is ”yes”under OCA+SH (and so, in particular, under the Proper Forcing Axiom(PFA)).2. Entangled SetsHow could one construct an order-separable poset of countable widththat is not the union of a countable number of chains? One approachwould be to find an uncountable poset in which all chains are countable.If one could then introduce a countable order-dense set, the result-ing poset would have the desired properties. This is precisely what2-entangled sets do. For k ∈ ω, k-entangled sets were introduced by She-lah and were shown in [2] to follow from CH and to be consistent withMAω1 . He defined them as follows.Definition. Let k ∈ ω. An uncountable subset A of IR is a k-entangledsubset of IR if and only if for every uncountable set A of k-tuples fromA such that α(i) 6= β(j) if α, β ∈ A and α 6= β or i 6= j, and everyσ : k → 2, there are α, β ∈ A such that α(i) < β(i) ⇐⇒ σ(i) = 1This means that for every uncountable collection of k-tuples fromA having distinct coordinates and disjoint as unordered sets, there arepairs illustrating every possible ordering between the coordinates. Anequivalent, and simpler, definition of 2-entangled sets is the following.cccos2.tex; 23/08/1999; 12:03; p.34Definition. An uncountable subset A of IR is a 2-entangled subsetof IR if and only if there is no uncountable monotone function from asubset of A to A with no fixed points.This means that if f is an uncountable function from a subset of Ato A, and the graph of f is given the usual product ordering (i.e.,〈a, b〉 ≤ 〈a′, b′〉 iff a ≤ a′ and b ≤ b′), then there are p, q, r, s ∈ A suchthat 〈p, f(p)〉 and 〈q, f(q)〉 are comparable, while 〈r, f(r)〉 and 〈s, f(s)〉are not. This gives us the desired properties that chains and antichainsare countable. If we also require that f is one-to-one, then the additionof Q| ×Q| will give us an order-separable poset.THEOREM 1. If there is a 2-entangled subset of IR, then there is anorder-separable poset of countable width which is not the union of acountable number of chains.Proof. Let A be a 2-entangled subset of IR and let f : A → A be aone-to-one function with no fixed points. Set X = f ∪ (Q| 2) with theusual product order. As was noted above, every chain and antichainof X is countable, so X has countable width and is not the union of acountable number of chains.Let 〈p, q〉 and 〈r, s〉 be elements of X with 〈p, q〉 < 〈r, s〉. Assumethat 〈p, q〉 /∈ Q| 2 and 〈r, s〉 /∈ Q| 2. Then 〈p, q〉, 〈r, s〉 ∈ f and, since fis a one-to-one function, p = r if and only if q = s. Therefore p < rand q < s. There are a, b ∈ Q| such that p < a < r and q < b < s. So〈p, q〉 < 〈a, b〉 < 〈r, s〉, and Q| 2 is order-dense in X.To obtain negative answers to the questions at the end of the pre-vious section, we need 4-entangled sets.THEOREM 2. If there is a 4-entangled subset of IR, then there is anorder-separable poset X with ν(X) ≤ ω which is not the union of acountable number of chains.Proof. Let X be as in the previous example, but with A 4-entangledinstead of 2-entangled. As before, X is order-separable and not a count-able union of chains.It remains to prove that ν(X) ≤ ω. Suppose on the contrary thatA = {Aα : α < ω1} is an uncountable collection of nonoverlappingsubsets of X. Then, by definition, each Aα has at least two points, saypα and qα. Applying the definition of 4-entangled to the 4-tuples pαfollowed by qα, we see that there are α, β < ω1 such that pα < pβ andqα > qβ. But then the transitive closure of the relation given in part2 of the definition of a nonoverlapping collection is not antisymmetric,hence not a partial order. That completes the proof.cccos2.tex; 23/08/1999; 12:03; p.453. Open Coloring AxiomIf there is to be a model of set theory in which order-separable posets ofcountable width will be the union of a countable number of chains, wewill have to use something that kills 2-entangled sets. One of the thingsthat does this is the Open Coloring Axiom (OCA). The statement ofthe axiom, in fact, indicates a strong connection with the problem athand. But the axiom applies to subsets of IR. Can we extend it togeneral posets? Here is a statement of OCA.[OCA] If [X]2 = K0 ∪K1 is a given partition where X ⊆ IR and whereK0 is open in [X]2, then either there is an uncountable 0-homogeneousset, or else X is the union of countably many 1-homogeneous sets.Here [X]2 is the set of all subsets of X having exactly two elements,and it is identified with {〈x, y〉 : x < y in X} where the topology isinherited from IR2. A subset Y of X is 0-homogeneous if [Y ]2 ⊆ K0and is 1-homogeneous is [Y ]2 ⊆ K1. OCA is a known consequence ofthe Proper Forcing Axiom (PFA). See [16] for more information.How does OCA help? Suppose that K0 consists of pairs of incompa-rable elements. Then a set that is 0-homogeneous must be an antichain.If there can’t be an uncountable antichain, then X is a countable unionof the other kinds of sets, chains. To use this, we must somehow rep-resent a poset as a subset of IR. It is well known that every separablezero-dimensional metric space is, in fact, a subset of IR. So the thingto do is to show that order-separable posets of countable width havea suitable separable zero-dimensional metric topology so that, whenembedded in IR, the set K0 is really open in IR2. This is not obvious,since the orders of the poset X and the order inherited from IR neednot have a lot in common.In the following theorem, we will use ↓ p to represent the set of allq ∈ X such that q ≤ p and ↑ p to represent the set of all q ∈ X suchthat p ≤ q. We will also use p ‖q to represent the fact that p and q areincomparable.THEOREM 3. (OCA). Every order-separable poset of countable widthis the union of a countable number of chains.Proof. Let X be an order-separable poset of countable width, andlet D be a countable order-dense subset of X. We are going to use Dto define a suitable 0-dimensional separable metric topology on X. Butin order for this to work, we first need to enlarge D and thin out X.Define an equivalence relation ∼ on X by setting p ∼ q if and onlyif for every r ∈ D, r ∈↓ p ⇔ r ∈↓ q and r ∈↑ p ⇔ r ∈↑ q. Then everycccos2.tex; 23/08/1999; 12:03; p.56equivalence class [p] is an antichain and is therefore countable. We maythus assume that [p] = {p} for every p ∈ X.For every p ∈ X letA+p (D) = {q ∈ X : q ‖p and ∀r ∈ D(r ≥ q ⇒ r ≥ p)}and letA−p (D) = {q ∈ X : q ‖p and ∀r ∈ D(r ≤ q ⇒ r ≤ p)}.Note that A+p (D) ∩D = ∅ and A−p (D) ∩D = ∅ for every p ∈ X.Claim 1. For every p ∈ X, |A+p (D)| ≤ ω and |A−p (D)| ≤ ω.We will show that A+p (D) is, in fact, an antichain, and thus it mustbe countable. If there are q, r ∈ A+p (D) such that q < r then there iss ∈ D such that q ≤ s ≤ r. But q ≤ s implies that p ≤ s, so p ≤ r, acontradiction. A similar argument shows that |A−p (D)| ≤ ω.Let D0 = D, and for n < ω, let Dn+1 = Dn ∪ [⋃p∈Dn(A+p (D) ∪A−p (D)). Let Dω =⋃n∈ωDn. Then A+p (Dω) ⊂ A+p (D) ⊂ Dω for everyp ∈ Dω, and since A+p (Dω) ∩Dω = ∅, it follows that A+p (Dω) = ∅ (andsimilarly A−p (Dω) = ∅) for every p ∈ Dω.We will henceforth assume, then, without loss of generality, that Ditself has the property that A+p (D) = A−p (D) = ∅ for every p ∈ D.Now for every p ∈ X letB+p = {q ∈ X : q ‖p and ∀r ∈ D(r ≥ p⇒ r ≥ q)}and letB−p = {q ∈ X : q ‖p and ∀r ∈ D(r ≤ p⇒ r ≤ q)}Claim 2. For every p ∈ X, |B+p | ≤ ω and |B−p | ≤ ω.We will again show that these sets are antichains. If there are q,s ∈ B+p such that q < s, then there is t ∈ D such that q ≤ t ≤ s.If r ∈ D and r ≥ p, then r ≥ s ≥ t. Also, p ‖t, so p ∈ A+t (D); butA+t (D) = ∅ since t ∈ D, a contradiction. The proof that |B−p | ≤ ω issimilar.Now we can write X as a union of countable sets Mα, α < κ, suchthat p ∈ Mα implies that B−p ∪ B+p ⊂ Mα. Let M ′α = Mα \⋃β<αMβ.Without loss of generality, each M ′α 6= ∅. Note that X is the union ofcountable many sets Y such that |Y ∩M ′α| = 1 for each α < κ. Thus itcccos2.tex; 23/08/1999; 12:03; p.67will suffice to prove that if Y = {pα : α < κ}, where pα ∈ M ′α, then Yis a countable union of chains, and that is what we now do.It is not hard to show that the topology on X generated by thesubbase consisting of all ↓p and ↑p for p ∈ D and their complements isa zero-dimensional topology with a countable base. It is also T1 underthe assumption that [p] = {p} for all p ∈ X. Therefore X is metrizableand is homeomorphic to a subset of IR. Let Y be as above, and give Ythe subspace topology; this is the space to which we apply OCA.Let K0 ={{pα, pβ} ∈ [Y ]2 : pα‖pβ}. We’ll be done if we show thatK0 is open, for then OCA implies that Y is a countable union of chains.To this end, let {pα, pβ} ∈ K0. We may assume that α ∈ β. Thenpβ /∈ B−pα ∪B+pα , so there are q, r ∈ D such that q ≤ pα, q  pβ, pα ≤ r,and pβ  r. Since q ‖pβ and pβ /∈ A+q (D), there is s ∈ D such thats ≥ pβ and s  q. Similarly, since pβ /∈ A−r (D), there is t ∈ D suchthat t ≤ pβ and t  r. Let 〈x, y〉 ∈ (↑ q ∩ ↓ r) × (↑ t∩ ↓ s). If x ≤ ythen q ≤ x ≤ y ≤ s, a contradiction. If y ≤ x then t ≤ y ≤ x ≤ r,another contradiction. Therefore, [Y ]2 ∩ [(↑ q ∩ ↓ r) × (↑ t∩ ↓ s)] is aneighborhood of {pα, pβ} that is contained in K0, so K0 is open. Thatcompletes the proof.References1. U. Abraham, A note on Dilworth’s Theorem in the infinite case, Order 4 (1987),107–1252. U. Abraham and S. Shelah, Martin’s Axiom does not imply that every twoℵ1-dense sets of reals are isomorphic, Israel J. Math. 38 (1981), 161–1763. B. Dushnik and E.W. Miller, Partially ordered sets, Amer. J. of Math. 63(1941), 600–6104. D. Kurepa, Sur la puissance des ensembles partiellement ordonne´s, Sprawozd.Towarz. Nauk Warsaw. Mat.-Fiz. 32 (1939), no. 1/3, 61–675. D. Kurepa, Une propriete´ des familles d’ensembles bien ordonne´s line´aires,Studia Math. 9 (1940), 23–426. D. Kurepa, Transformations monotones des ensembles partiellement ordonne´s,Revista de Ciencas 42 (1940), 827–846; 43 (1941), 483–5007. D. Kurepa, On two problems concerning ordered sets, Glasnik Mat.-Fiz. I Astr.13 (1958), no. 4, 229–2348. D. Kurepa, On the cardinal number of ordered sets and of symmetrical struc-tures in dependence on the cardinal numbers of its chains and antichains,Glasnik Mat.-Fiz. I Astr. 14 (1959), no. 3, 183–2039. D. Kurepa, Star number and antistar number of ordered sets and graphs,Glasnik Mat.-Fiz. I Astr. 18 (1963), no. 1–2, 27–3710. D. Kurepa, Monotone mappings between some kinds of ordered sets, GlasnikMat.-Fiz. I Astr. 19 (1964), no. 3–4, 175–18611. D. Kurepa, A link between ordered sets and trees on the rectangle treehypothesis, Publ. Inst. Math. 31 (45) (1982), 121–128cccos2.tex; 23/08/1999; 12:03; p.7812. E.C. Milner, Z.S. Wang, and B.Y. Li, Some inequalities for partial orders, Order3 (1987), 369–38213. 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On The Decomposition of Order-separable Posets of Countable Width into Chains

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On The Decomposition of Order-separable Posets of Countable Width into Chains

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