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CHAINS, ANTICHAINS, AND COMPLEMENTS IN INFINITE PARTITION LATTICES JAMESEMILAVERY,JEAN-YVESMOYEN,ANDJAKOBGRUESIMONSEN Abstract. WeconsiderthepartitionlatticeΠκ onanysetoftransfinitecar- dinality κ and properties of Πκ whose analogues do not hold for finite cardi- 5 nalities. Assuming the Axiom of Choice we prove: (I) the cardinality of any 1 maximal well-ordered chain is between the cofinality cf(κ) and κ, and κ al- 0 waysoccursasthecardinalityofamaximalwell-orderedchain;(II)thereare 2 maximalchainsinΠκ ofcardinality>κ;(III)if,foreverycardinalλ<κ,we have 2λ <2κ, there exists a maximal chain of cardinality <2κ (but ≥κ) in c Π2κ;(IV)therearemaximalantichainsofcardinalityκand2κ inΠκ;(V)all e cardinalsoftheformκλ with0≤λ≤κoccurasthenumberofcomplements D tosomepartitionP ∈Πκ,andonlythesecardinalitiesappear. Moreover,we giveadirectformulaforthenumberofcomplementstoagivenpartition;(VI) 1 3 Πκ isnotorthocomplementedforany,finiteorinfinite,cardinalityκ>2. ] A Let κ be a cardinal and let S be a set of cardinality κ. The set of partitions R of S forms a lattice when endowed with the binary relation , called refinement, ≤ . defined by if and only if each block of is a subset of a block of . This h P ≤ Q P Q t lattice is called the partition lattice on S, and is denoted Π(S). By the standard a correspondence between partitions and equivalence relations, it follows that Π(S) m is isomorphic to the lattice Equ(S) of equivalence relations on S ordered by set [ inclusion on S S. × 2 As the particulars of S do not affect the order-theoretic properties of Π(S) we v shall without loss of generality restrict our attention to the lattice Π = Π(κ). κ 4 InitiatedbyaseminalpaperbyOre[Ore42],manyofthepropertiesofΠ thathold κ 8 for arbitrary cardinals κ are well-known. Indeed, it is known that Π is complete, 2 κ matroid(henceatomisticandsemimodular), non-modular(hencenon-distributive) 5 0 forκ 4,relativelycomplemented(hencecomplemented),andsimple[Bir40, 8-9], ≥ § . [RS92], [Grä03, Sec. IV.4]. 1 0 For properties depending on κ, only a few results exist in the literature for 5 infinite κ. Czédli has proved that if there is no inaccessible cardinal κ then ≤ 1 the following holds: If κ 4, Π is generated by four elements [Czé96a], and if κ : ≥ v κ 7, Πκ is (1+1+2)-generated [Czé99] (for κ= 0, slightly stronger results hold ≥ ℵ i [Czé96b]). It appears that no further results are known, beyond those holding for X allcardinalities, finiteorinfinite. Theaimofthepresentnoteistoproveanumber r a of results concerning Πκ that depend on κ being an infinite cardinal. JamesAvery: NielsBohrInstitute,UniversityofCopenhagen,Blegdamsvej17,2100 Copenhagen Ø, Denmark, E-Mail: [email protected] Jean-Yves Moyen: Laboratoire d’Informatique de Paris Nord, Universite´ Paris XIII, 99, avenue J.-B. Cle´ment, 93430 Villetaneuse, France, E-Mail: [email protected] Jakob Grue Simonsen: Department of Computer Science, University of Copenhagen (DIKU), Njalsgade 128-132, 2300 Copenhagen S, Denmark, E-Mail: [email protected] Date:June4,2015. 1991 Mathematics Subject Classification. Primary06B05;06C15. Key words and phrases. Partition lattice; order theory; antichain; chain; complement; ortho- complementedlattice;cardinal;ordinal. 1 2 CHAINS, ANTICHAINS, AND COMPLEMENTS IN INFINITE PARTITION LATTICES 1. Preliminaries and notation We work in ZF with the full Axiom of Choice (AC). As usual, a set S is well- ordered iffitistotallyorderedandeverynon-emptysubsetofS hasaleastelement. Throughout the paper, we use von Neumann’s characterization of ordinals: a set S is an ordinal iff it is strictly well-ordered by (cid:40) and every element of S is a subset of S. The order type of a well-ordered set S is the (necessarily unique) ordinal α that is order-isomorphic to S. Cardinals and ordinals are denoted by Greek letters α,β,δ,γ,... for ordinals and κ,λ,... for cardinals. We denote by ω the initial κ ordinal of κ, and by α the cardinality of α. The cardinality of a set is denoted S and its powerset is d|en|oted P(S). | | Recall that a chain in a poset (P, ) is a subset of P that is totally ordered by ≤ . Similarly, an antichain in (P, ) is a subset of P such that any two distinct ≤ ≤ elements of the subset are -incomparable. A chain (respectively, antichain) in ≤ (P, ) is maximal if no element of P can be added to the chain without losing ≤ the property of being a chain (respectively, antichain). Observe that the bottom element, ,andthetopelement, ,areelementsofanymaximalchain. AchainC ⊥ (cid:62) in (P, ) is saturated if, for any two elements , of the chain with < , there ≤ Q S Q S is no element P C such that < < and C isl a chain. By the R ∈ \ Q R S ∪{R} MaximalChainTheorem[Hau14],everychaininaposetiscontainedinamaximal chain. We denote partitions (and equivalences) of κ by capital italic Roman letters , ,...,anddenotesubsetsofΠ suchaschainsandantichainsbycapitalboldface κ P Q letters C,D,...; If = B is a partition, we call each of its elements B a block. δ δ P { } Itiseasilyseenthat = γ γ κ and = κ ;thatis,thesetofallsingleton ⊥ {{ } | ∈ } (cid:62) { } subsetsofκ, respectivelythesingletonsetcontainingallelementsofκ. Asusual, if , Π , we write if < and no Π exists such that < < . κ κ P Q∈ P ≺Q P Q R∈ P R Q It follows that if and only if can be obtained by merging exactly two P ≺ Q Q distinct blocks of . P AblockBinducesanequivalencerelationonκ,definedbyδ γiffbothδ,γ B, ≡B ∈ andapartition naturallyinducesanequivalencerelationdefinedbyδ γiffthere P ≡P isablockB withδ γ. Conversely,anyequivalencerelationcorrespondstothe ∈P ≡B partitionwhoseblocksarethemaximalsetsofequivalentelements. Thisone-to-one correspondence allows us to consider a partition as its corresponding equivalence relation when convenient, and vice versa. If Π contains exactly one block B with B 2 and the remaining blocks κ P ∈ | | ≥ are all singletons, we call a singular partition, following Ore [Ore42]. P IfC Π ,thenitsgreatestlowerbound Cisthepartitionthatsatisfiesx y κ ⊆ ∧ ∧≡C iffx y forall C. Thatis,theblocksof Careallthenonemptyintersections ≡P P ∈ ∧ whosetermsareexactlyoneblockfromeverypartition C. Conversely,itsleast P ∈ upper bound C is the partition such that γ δ iff there exists a finite sequence ∨ ∨≡C of partitions 1,..., k C and elements β0,...,βk, κ such that γ = β0 P P ∈ ∈ P≡1 β1 β2 βk =δ. P≡2 P≡3 ···P≡k Finally, the cofinality cf(κ) of an infinite cardinal κ is the least cardinal λ such that a set of cardinality κ can be written as a union of λ sets of cardinality strictly (cid:8) (cid:12) (cid:12)(cid:83) (cid:12) (cid:9) s(cid:83)maller than κ: cf(κ) = min |I| (cid:12) κ=(cid:12) i∈IAi(cid:12)∧∀i∈I,|Ai|<κ . Since κ = i , we always have cf(κ) κ. i∈κ{ } ≤ If cf(κ)=κ, then the cardinal κ is called regular, otherwise it is called singular. UnderAC,whichisassumedthroughoutthispaper,everyinfinitesuccessorcardinal is regular. König’s Theorem [Kön05] implies cf(2κ)>κ, and we additionally have CHAINS, ANTICHAINS, AND COMPLEMENTS IN INFINITE PARTITION LATTICES 3 2κ 2λ whenever κ < λ. By Easton’s theorem [Eas70], these are the only two ≤ constraints on permissible values for 2κ when κ is regular and when only ZFC is assumed. In contrast, when the Generalized Continuum Hypothesis (GCH) is assumed, cardinal exponentiation is completely determined. For infinite κ, Π has cardinality 2κ which provides a weak upper bound on the κ cardinality of maximal chains and antichains. 2. Results We summarize here the main contributions of the paper. Theorem (Well-ordered chains: Theorems 3.1, 3.2 and 3.3). Let κ be an infinite cardinal. The cardinal of any maximal well-ordered chain in Π is between cf(κ) κ and κ, and κ always occurs as the cardinal of a maximal well-ordered chain. Consequently, if κ is regular, the cardinal of any well-ordered maximal chain in Π is always exactly κ. κ Theorem (Long chains: Theorem 3.5). Let κ be an infinite cardinal. There exist chains of length >κ in Π . Under GCH, there exist chains of length 2κ. κ Theorem (Short chains: Theorem 4.11). Let κ be an infinite cardinal such that for every cardinal λ < κ we have 2λ < 2κ. Then there exists a maximal chain of cardinality <2κ (but κ) in Π2κ. ≥ Under GCH, for every infinite cardinal κ, there exists a maximal chain of cardinality κ in Π2κ. Theorem (Antichains: Theorem 5.1). Let κ be an infinite cardinal. There are maximal antichains in Π of cardinalities κ and 2κ κ Theorem (Complements: Theorems 6.7, 6.9, and 6.10). Let κ be an infinite cardinal. All cardinals of the form κλ with 0 λ κ occur as the number of comple- ≤ ≤ ments to some partition Π , and these are the only cardinalities the set of κ P ∈ complements can have. For non-trivial partitions / , , the number of complements is between κ P ∈ {⊥ (cid:62)} and 2κ, i.e. κλ with 1 λ κ. The number of complements to is 2κ unless (i) ≤ ≤ P contains exactly one block B of cardinality κ, and (ii) κ B <κ. If contains P | \ | P a block B of size κ, then has κ|κ\B| complements. P UnderGCH, / , hasκcomplementsifandonlyif(i) containsexactly P ∈{⊥ (cid:62)} P one block of cardinality κ, and (ii) κ B < cf(κ); otherwise, has 2κ comple- | \ | P ments. Theorem (Orthocomplements: Theorem 7.3). If κ>2, then Π is not orthocom- κ plemented. 3. Well-ordered chains and long chains in Π κ For finite κ = n, it is immediate that there is a maximal chain of cardinality n in Π , that n is the maximum cardinality of maximal chains, and that for n 3, n ≥ maximal chains are not unique. The following explores the possible lengths of chains when κ is an infinite cardinal. 3.1. Well-orderedchains. Letκbeaninfinitecardinal. Weconsiderwell-ordered maximalchainsinΠ . AnymaximalchaininΠ hasamaximalelement,namely . κ κ (cid:62) In contrast, a limit ordinal does not have a maximal element. Hence, if a maximal chain in Π is well-ordered of order type α, then α must be a successor ordinal. κ For clarity, we will write the order type of a maximal chain as α+1 to emphasize that the order type is that of a successor ordinal. Note that because α = α+1, | | | | 4 CHAINS, ANTICHAINS, AND COMPLEMENTS IN INFINITE PARTITION LATTICES this has no impact on the cardinality of the chain. Such chains can be written as C= β α — or as C= β <α+1 to emphasis the order type — β β {P | ≤ } {P | } with = and = . 0 α P ⊥ P (cid:62) Theorem 3.1. Let κ be an infinite cardinal. If a chain in Π is well-ordered of κ order type α, then α κ. | |≤ Note that as we are here speaking of any well-ordered chain, not necessarily a maximal one, its order type may be anything. Proof. AsΠ isisomorphictoEqu(κ),thesetofequivalencerelationsonκordered κ by , there is, for any chain of order type α in Π , a chain of order type α in κ the⊆poset (P(κ κ), ). Let C = β <α be such a chain and observe β × ⊆ {E | } that for every β with β + 1 < α there exists at least one γ . As β β+1 β ∈ E \ E C is totally ordered under , this implies γβ / β(cid:48) when β(cid:48) < β, i.e. the γβ ⊆ ∈ E do not repeat. Hence, γ β+1<α κ κ has cardinality α, implying β α κ κ =κ. { | } ⊆ × | | (cid:3) | |≤| × | Theorem3.2. Everywell-orderedmaximalchainofordertypeα+1inΠ satisfies κ α cf(κ). | |≥ Proof. Let C = β <α+1 be a maximal chain in Π of order type α+1. β κ {P | } That is, β γ implies and since C is maximal, . β γ β β+1 ≤ P ≤P P ≺P Considerthepartitionsinthischainthathaveatleastoneblockofcardinalityκ. Since = , there is at least one such partition. Let δ be the least ordinal with α (cid:62) P δ α such that contains a block of cardinality κ. It exists as every non-empty δ ≤ P set of ordinals has a minimal element. Let Bδ be a block of cardinality κ in . If δ were a successor ordinal, say δ P δ =δ(cid:48)+1, then δ(cid:48) δ, as otherwise the chain would not be maximal. Then Bδ P ≺P either occurs as a block of δ(cid:48), or it is the union of at most two blocks of δ(cid:48). As P P κ is infinite, this implies that δ(cid:48) contains a block of cardinality κ, contradicting P that δ is the least ordinal with this property. Hence, δ is not a successor ordinal, and is clearly not zero, whence it must be a limit ordinal. Definenowasequence Bβ whosefinalelementisBδ bypickinga Bδ and β≤δ { } ∈ for each β < δ letting Bβ be the block from that contains a. For any β(cid:48) < β, β the blocks of β(cid:48) are all subsets of blocks fromP β. Since a Bβ Bβ(cid:48), the block of that incPludes Bβ(cid:48) must be Bβ. Hence BβP(cid:48) Bβ for β∈(cid:48) < β∩ δ, and so the β P ⊆ ≤ sequence Bβ is a well-ordered chain of sets. By construction, Bβ Bδ. β≤δ β<δ { } ∪ ⊆ We will next show that Bβ =Bδ. β<δ Suppose that Bβ∪= Bδ and call B = Bβ, B¯ = Bδ B. Let be the β<δ β<δ ∪ (cid:54) ∪ \ Q partition defined by having the same blocks as except that Bδ is split into B δ and B¯. We have . P δ Q≺P Consider a β <δ and look at any block B of . Because < , B is a 0 0 Pβ0 Pβ0 Pδ 0 subset of a block A of . δ P If A is not Bδ, then it is also a block of . • If B B = then B is a subset of B¯, aQblock of . 0 0 • ∩ ∅ Q If B0 B x, then x Bβ1 for some β1 <δ. If β0 <β1, we get B0 Bβ1, • ∩ (cid:51) ∈ ⊆ because both blocks contain x and < . Conversely, if β β , then Pβ0 Pβ1 1 ≤ 0 Bβ1 Bβ0, and we must have B0 = Bβ0. In both cases, B0 B, a block ⊆ ⊆ of . Q Consequently,eachblockof isasubsetofablockof ,whereby < which Pβ0 Q Pβ0 Q contradicts maximality of the chain. By construction, for each β <δ, every block of has cardinality <κ, whereby β in particular (cid:12)(cid:12)Bβ(cid:12)(cid:12) < κ. But (cid:12)(cid:12)(cid:12)(cid:83)β<δBβ(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)Bδ(cid:12)(cid:12)P= κ, that is, we may write a CHAINS, ANTICHAINS, AND COMPLEMENTS IN INFINITE PARTITION LATTICES 5 set of cardinality κ as a union of δ sets of cardinality strictly less than κ. Thus, cf(κ) δ α. (cid:3) ≤| |≤| | Corollary 3.3. If κ is a regular cardinal, then every well-ordered maximal chain in Π indexed by an ordinal α satisfies α =κ. κ | | For singular cardinals we do not know whether a well-ordered maximal chain indexed by ω can always be realized. cf(κ) 3.2. Long chains in Π . For any nonempty S κ, we define the partition κ ⊆ diag(S) = S γ γ κ S . When S 2, diag(S) is the singular par- { }∪{{ } | ∈ \ } | | ≥ tition with non-singular block S. Remark 3.4. It is easy to verify that diag() is an order isomorphism between κ- subsetsoflength 2andthesingularpartiti·ons. Inparticular,chainsin(P(κ), ) ≥ ⊆ containing only sets of length 2 are mapped to chains with the same cardinality ≥ in (Π , ). κ ≤ Theorem 3.5. Let κ be an infinite cardinal. There is a chain of cardinality > κ in Π . The chain may be chosen to be maximal. κ Proof. By a result of Sierpin´ski [Sie22] (see also [Har05, Thm. 4.7.35]), there is a chain D(cid:48) in the poset (P(κ), ) of cardinality λ > κ. There is at most one ⊆ singleton element α D(cid:48), and as λ is infinite, D = D(cid:48) , α is still a chain { } ∈ \{∅ { }} of cardinality λ. Then, by Remark 3.4, the set C= diag(S) S D is a chain in Π of cardi- κ { | ∈ } nality λ > κ. By the Maximal Chain Theorem, any chain in a poset is contained in a maximal chain, and the result follows. (cid:3) If GCH is assumed, an immediate consequence of Theorem 3.5 is that there is a chain of cardinality 2κ = Π in Π . κ κ | | For the special case of κ = , we may obtain existence of a maximal chain in 0 ℵ Πℵ0 of cardinality 2ℵ0 without use of (G)CH: Lemma 3.6. There is a maximal chain of cardinality 2ℵ0 in Πℵ0. Proof. For each r R, define the left Dedekind cut D = q Q q <r and r note that r < r(cid:48) ∈implies Dr (cid:40) Dr(cid:48) by density of Q in R{. H∈ence|, the }set of left Dedekind cuts is a chain in P(Q) of cardinality R = 2ℵ0, all elements of | | which have cardinality 2, and consequently, by Lemma 3.4, the set of partitions {diag(Dr) | r ∈R} is a≥chain in Πℵ0 of cardinality 2ℵ0. By the Maximal Chain Theorem,thischaincanbeextendedtoamaximalchain; notethat2ℵ0 isanupper bound on the cardinality of any chain in Π . (cid:3) ℵ0 For arbitrary infinite κ we do not know whether existence of a maximal chain in Π of cardinality 2κ can be proved without assuming GCH; we conjecture that κ GCH is necessary. In addition, we do not know whether the rather heavy-handed application of the Maximal Chain Theorem (equivalent to the Axiom of Choice) in the proof of Theorem 3.5 is necessary to establish existence of a maximal chain. 4. Short chains We now turn to the question of whether there are maximal chains of cardinality strictlylessthanκinΠ . Itisimmediatethattherearenomaximalchainsoffinite κ cardinalityinΠ . Indeed,eachstepinamaximalchainmergesexactlytwoblocks. ℵ0 Hence, starting from which has infinitely many blocks, after a finite number of ⊥ steps it is impossible to reach or any other partition that has only finitely many (cid:62) blocks. For larger cardinals, there is a general construction that proves existence 6 CHAINS, ANTICHAINS, AND COMPLEMENTS IN INFINITE PARTITION LATTICES of short chains. In the special case where GCH is assumed, it proves existence of a maximal chain of cardinality strictly less than κ for every successor cardinal κ. Because this construction is notationally cumbersome, we first prove the result for the particular case of κ=2ℵ0, the cardinality of the continuum, in order to aid the reader’s understanding. 4.1. Short chains in Π . 2ℵ0 Webuildamaximalchainofsize inΠ . Theproofboilsdowntothefactthat ℵ0 2ℵ0 there are countably many binary strings of finite length but 2ℵ0 binary strings of countably infinite length, hence an infinite binary tree of depth ω has uncountably many leaves but only countably many inner nodes. The construction proceeds in two steps: (1) Starting from , inductively construct a countable chain of “keyframe” (cid:62) partitions such that the nth keyframe has 2n blocks. The greatest lower bound of this chain will be with 2ℵ0 singleton blocks. ⊥ (2) Completethechainintoamaximalchainbyadding“inbetween”partitions between the keyframes. We’ll need 2n 1 extra partitions between the − (n 1)st and the nth keyframe.1 − Asafinitenumberofinbetweenpartitionsareaddedforeachn,onlyacountable number of partitions are added in total, and so the full chain has countable length. Keyframes. Consider partitions of the set 0,1 ω of countably infinite se- { } quences of bits; note that the cardinality of 0,1 ω is 2ℵ0. Let u 0,1 ω { } ∈ { } and let [u] be the set of elements of 0,1 ω with the same first k bits: [u] = k k { } u u ...u v v 0,1 ω . Definethekthkeyframetobe = [u] u 0,1 ω . 0 1 k−1 k k { | ∈{ } } K { | ∈{ } } Similarly, define = . ω K ⊥ Thus, the keyframes are as in Figure 1. Each level of the picture corresponds to one partition, with its blocks depicted. It is instructive to note that the set of blocks of all partitions form an infinite binary tree and that the number of nodes in each keyframe is finite (whence the corresponding partition has only finitely many blocks). The blocks of constitute the bottom most level of the tree, and ω K hencearetheleavesoftheinfinitebinarytree, ofwhichthereare2|ω|,andthetotal number of keyframe partitions corresponds to the number of levels in the infinite tree, and is thus clearly countable. The total number of internal nodes in the tree is (cid:80) 2|α| = . α<ω ℵ0 Completing the chain. The set of keyframe partitions clearly form a chain, but just as clearly this chain is not maximal. Inbetween partitions must be added between the keyframes to obtain a maximal chain. As seen in Figure 1, between the kth and the (k+1)st keyframe, 2k blocks have been split. In order to locally saturate the chain, it suffices to add 2k 1 new − partitions, each one with one more of the 2k blocks cut in two. This will ensure that each partition is a successor to the previous one. Afteraddingthesenewpartitions, thepictureisnowasshowninFigure2. This chain is easily seen to be maximal. At each (non-keyframe) partition, the new blocks, resulting from splitting one of the blocks of the parent partition, are shown with hatching. Sincethereare2k inbetweenpartitionsbetweenthekth and(k+1)st keyframes, the total number of inbetween partitions is (cid:80) 2k, which is countable. Since k<ω there are also only countably many keyframes, the total number of partitions in the chain is countable. 1“Keyframe”and“Inbetween”aretermsfromanimation: A“keyframe”isadrawingthatde- finesthestartorendofamovement. Theanimationframesbetweenkeyframes–the“inbetweens”– aredrawntomakethetransitionsbetweenkeyframessmooth. CHAINS, ANTICHAINS, AND COMPLEMENTS IN INFINITE PARTITION LATTICES 7 ... 0 K 000......... 111......... 1 K 0000...... 0011...... 1... 1 K1 0000...... 0011...... 111000......... 111111......... 2 K 000000...... 000011...... 01... 10... 11... 1 K2 000... 001... 001100...... 001111...... 10... 11... 2 K2 000... 001... 010... 011... 110000...... 110011...... 11... 3 K2 000000...... 000011...... 001100...... 001111...... 110000...... 110011...... 111111000......... 111111111......... 3 K . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ω • • K Figure 1. Keyframe partitions 4.2. Short chains: general construction for large cardinals. Let λ be any infinite cardinal and κ = 2λ. We will build a maximal chain of cardinality λ in Π =Π . κ 2λ Underlying set. Let S be the set of binary sequences indexed by the initial ordinal of λ. That is S = 0,1 ωλ. S contains κ elements; we shall use Π(S) as a { } representative of Π . κ Keyframes. Let f S and δ ω . We denote by [f] the set of sequences λ δ ∈ ≤ that agree with f on the initial δ elements: [f] = g S β <δ,f(β)=g(β) δ { ∈ | ∀ } Observe that [f] is uniquely defined by a binary sequence indexed by δ. The δ bracketed notation [f] is used to emphasize that [f] is truly the equivalence class of f in the corresponding equivalence, even though everything is written in terms of partitions. Let = [f] f S . Then, is a partition of S which we call a keyframe δ δ δ K { | ∈ } K partition. From the definition, it follows that = and that = . Note K0 (cid:62) Kωλ ⊥ that consists of 2|δ| blocks and is in bijective correspondence with 0,1 δ. δ K { } Lemma4.1. Ifα<β then[f] (cid:40)[f] and < . Furthermore, δ ω β α β α δ λ K K {K | ≤ } is a chain in Π . κ Proof. The first point is immediate by definition of the refinement ordering: if f and g agree on their initial β elements, then they trivially agree on their initial α elements, so [f] (cid:40)[f] . Since this holds for every f S, we further get < . β α β α The second point is an immediate consequence of the∈first. K K(cid:3) 8 CHAINS, ANTICHAINS, AND COMPLEMENTS IN INFINITE PARTITION LATTICES ... 0 K 000......... 111......... 1 K 0000...... 0011...... 1... 1 K1 0000...... 0011...... 111000......... 111111......... 2 K 000000...... 000011...... 01... 10... 11... 1 K2 000... 001... 001100...... 001111...... 10... 11... 2 K2 000... 001... 010... 011... 110000...... 110011...... 11... 3 K2 000000...... 000011...... 001100...... 001111...... 110000...... 110011...... 111111000......... 111111111......... 3 K . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ω • • K Figure 2. A short maximal chain Locally saturating the chain. Wefixδ <ω andconstructasaturatedchain λ bounded below by and above by . δ+1 δ K K has cardinality 2|δ|. Fix, by the Axiom of Choice, a well-ordering < of δ δ δ K K with order type ω . For 0 α < ω , let the block S be the αth element of 2|δ| ≤ 2|δ| δ,α according to this ordering. δ K For each 0 α<ω , define the set [f]α as ≤ 2|δ| δ (cid:26) [f] if [f] < S [f]α = δ+1 δ δ δ,α δ [f] if S [f] δ δ,α δ δ ≤ and let α = [f]α f S . Observe that the sets [f]α are pairwise disjoint and Kδ { δ | ∈ } δ that their union is S. Thus, α is a partition of S (an “inbetween” partition), Kδ obtained by splitting the initial α blocks of . By construction there are 2|δ| δ K inbetween partitions between and . δ δ+1 K K Lemma 4.2. For all f S, [f]0 =[f] . Thus 0 = . ∈ δ δ Kδ Kδ Proof. By construction, S is minimal for < , thus it is impossible for [f] to be δ,0 δ δ strictly smaller. (cid:3) Lemma 4.3. Let α<β. Then for each f, [f]β [f]α. δ ⊆ δ Proof. By cases depending on < . First note that α < β implies S < S by δ δ,α δ δ,β construction. (1) If [f] < S < S then [f]β =[f] =[f]α. δ δ δ,α δ δ,β δ δ+1 δ (2) If S [f] < S then [f]β =[f] (cid:40)[f] =[f]α. δ,α ≤δ δ δ δ,β δ δ+1 δ δ (3) If S < S [f] then [f]β =[f] =[f]α. δ,α δ δ,β ≤δ δ δ δ δ CHAINS, ANTICHAINS, AND COMPLEMENTS IN INFINITE PARTITION LATTICES 9 (cid:3) Lemma 4.4. Let α<β. Then β < α. Kδ Kδ Proof. ByLemma4.3,eachblockof β iscontainedinablockof α,thus β α. Kδ Kδ Kδ ≤Kδ Since α < β, there exists f such that S [f] < S . Hence there are δ,α δ δ δ δ,β ≤ blocks of β that are strictly contained in blocks of α, and thus the inequality is indeed strKicδt. Kδ (cid:3) Corollary 4.5. Let C(cid:48) = α 0 α<ω and C = C(cid:48) . C is a δ {Kδ | ≤ 2|δ|} δ δ ∪{Kδ+1} δ chain with minimum and maximum = 0. Kδ+1 Kδ Kδ Note that including in C(cid:48) (that is, defining only C ) appears to be more Kδ+1 δ δ natural, but because it is both an element of C and of C , it becomes notation- δ δ+1 ally awkward when taking unions of chains and computing the cardinality of such unions. For any f S let f [f] (resp. f [f] ) be the binary sequence such 0 δ 1 δ ∈ ∈ ∈ that f (δ(cid:48)) = 0 (resp. f (δ(cid:48)) = 1) for all δ(cid:48) δ. Notice that [f ] = [f ] but 0 1 0 δ 1 δ ≥ [f ] =[f ] . 0 δ+1 1 δ+1 (cid:54) For any g [f] , either g(δ) = 0 and [g] = [f ] , or g(δ) = 1 and δ δ+1 0 δ+1 ∈ [g] =[f ] . Thus, [f] =[f ] [f ] . δ+1 1 δ+1 δ 0 δ+1 1 δ+1 ∪ Lemma 4.6. For every α<ω , α+1 α. 2|δ| Kδ ≺Kδ Proof. Fix δ and α. Let f S and consider the block [f]α of α. There are three ∈ δ Kδ cases: (1) If [f] < S < S then [f]α =[f]α+1 =[f] δ δ δ,α δ δ,α+1 δ δ δ+1 (2) If S < S [f] then [f]α =[f] =[f]α+1. δ,α δ δ,α+1 ≤δ δ δ δ δ (3) If S < [f] =S then either [f]α+1 =[f ] or [f]α+1 =[f ] . δ,α+1 δ δ δ,α δ 0 δ+1 δ 1 δ+1 (1) and (2) follow immediately from the definition. For case (3), we first obtain [f]α =[f] and [f]α+1 =[f] by definition. But since [f] =[f ] [f ] , we δ δ δ δ+1 δ 0 δ+1∪ 1 δ+1 must have either f [f ] or f [f ] . Hence [f]α+1 =[f] is either [f ] ∈ 0 δ+1 ∈ 1 δ+1 δ δ+1 0 δ+1 or [f ] , as desired. By the above, every block of α is a block of α+1 except 1 δ+1 Kδ Kδ for a single block (the unique block for which [f] = S ), which is obtained by δ δ,α taking the union of exactly two blocks of α+1. Hence, α+1 α. (cid:3) Kδ Kδ ≺Kδ Proposition 4.7. Let C(cid:48) = α 0 α<ω and C =C(cid:48) . C is δ {Kδ | ≤ 2|δ|} δ δ ∪{Kδ+1} δ a saturated chain in Π , with minimum and maximum . κ δ+1 δ K K Proof. By Corollary 4.5, C is a chain with minimum and maximum . δ δ+1 δ K K Suppose that there exists a partition / C with < < and such that δ δ+1 δ P ∈ K P K C(cid:48) is still a chain. δ∪{P} Let C+ = α C < α be the set of partitions larger than . δ {Kδ ∈ δ | P Kδ } P If C+ has a minimal element β, then by Lemma 4.6, β+1 β. But • δ Kδ Kδ ≺ Kδ as < β (by construction of C+), either = β+1 which contradicts P Kδ P Kδ / C ; or < β+1 which contradicts the minimality of β in C+. P ∈ δ P Kδ Kδ δ If C+ has no minimal element, let β be the smallest ordinal such that • δ β / C+ and note that β < . β is a limit ordinal, otherwise C+ would Kδ ∈ δ Kδ P δ have a minimal element ( β−1). As each block of is divided into two Kδ Kδ blocksof ,allblocksof mustbeblocksfromeitherofthetwo. Since δ+1 K P β < ,thereexistf S forwhich[f] isablockof and[f ] ,[f ] Kδ P ∈ δ P 0 δ+1 1 δ+1 are two distinct blocks of β. Kδ 10 CHAINS, ANTICHAINS, AND COMPLEMENTS IN INFINITE PARTITION LATTICES Letαbetheordinalsuchthat[f] =S . Theblocks[f ] and[f ] δ δ,α 0 δ+1 1 δ+1 aredistinctin β,soα<β. Becauseβisalimitordinal,α+1<β,whereby Kδ α+1 C+ and < α+1. However, [f ] and [f ] are distinct in Kδ ∈ δ P Kδ 0 δ+1 1 δ+1 α+1 which contradicts the fact that they are the same in . Kδ P (cid:3) (cid:16) (cid:17) Completing the chain. In the following, let C= (cid:83) C(cid:48) . {⊥}∪ 0≤δ<ωλ δ Theorem 4.8. C is a maximal chain in Π . κ Proof. By Proposition 4.7, C =C(cid:48) is a saturated chain with minimum δ δ ∪{Kδ+1} and maximum , whereby each C(cid:48) is a chain. Noting that δ < γ implies Kδ+1 Kδ δ < by Lemma 4.1, every element of C(cid:48) is comparable to every element of Kγ Kδ δ C(cid:48). Hence, C is a chain. γ To prove maximality, suppose that there exists a partition / C such that P ∈ C is still a chain. Let C+ = C < be the set of keyframes α α ∪{P} {K ∈ | P K } larger than . P IfC+ hasaminimalelement ,then < < ,butbyProposition β β+1 β • K K P K 4.7, the chain is saturated between these two. If C+ has no minimal element, let β be the smallest ordinal such that • / C+ and note that < is the largest keyframe partition smaller β β K ∈ K P than in C. β is a limit ordinal, otherwise C+ would have a minimal P element ( ). β−1 K Since < , there exist two sequences f and g that are in the same β K P block of but in different blocks of . Thus, [f] =[g] and there exists β β β P K (cid:54) α<β such that f(α)=g(α). (cid:54) Because β is a limit ordinal, α+1<β, and C+ (by minimality α+1 K ∈ of outside of C+) and < . However, f(α) = g(α) implies that β α+1 K P K (cid:54) [f] and [g] are distinct, which contradicts the fact that f and g are α+1 α+1 in the same block in . P (cid:3) Length of the chain. Recall that C(cid:48) has cardinality 2|δ|. δ Lemma 4.9. C has cardinality (cid:80) 2|δ|. δ<ωλ Proof. |C|=(cid:12)(cid:12)(cid:83)δ<ωλC(cid:48)δ∪{⊥}(cid:12)(cid:12)=(cid:12)(cid:12)(cid:83)δ<ωλC(cid:48)δ(cid:12)(cid:12)=(cid:80)δ<ωλ|Cδ|=(cid:80)δ<ωλ2|δ| (cid:3) Lemma 4.10. Let λ be a cardinal such that for any µ<λ, we have 2µ <2λ. Then (cid:80) 2|δ| <2λ. δ<ωλ Proof. If δ <ω , then δ <λ, thus by hypothesis, 2|δ| <2λ. λ Assume now, for con|tr|adiction, that (cid:80) 2|δ| 2λ. Recall that δ<ωλ ≥ cf(2λ)=inf(cid:8) I (cid:12)(cid:12) 2λ = i∈IAi i I, Ai <2λ(cid:9), | | |∪ |∧∀ ∈ | | which is the same as cf(2λ) = inf(cid:8)|I| (cid:12)(cid:12) 2λ =(cid:80)i∈Iµi∧∀i∈I,µi <2λ(cid:9). It then followsthatcf(2λ) ω =λ. ButbyastandardconsequenceofKönig’sTheorem λ [Kön05] we always≤ha|ve c|f(2λ)>λ under the Axiom of Choice. (cid:3) We can now prove our main result on short chains. Theorem 4.11. Let λ be an infinite cardinal such that for every cardinal µ < λ we have 2µ <2λ. Then there exists a maximal chain of cardinality <2λ (but λ) ≥ in Π . 2λ