arXiv:math/9805145v1 [math.LO] 15 May 1998 ( rpsto .) h iuto ssmlrfrmn cardina many for similar is situation The 1.6). Proposition e ..Rslnwk,Sea RS 534, [RoSh Ros Shelah lanowski, e.g. see ust)seMn M] M](n ento . eo) nSh ultrafilter In an below). 1.1 Definition (and § [M2] [M1], Monk see subsets) les(e P9].Nwteitnini S 4]wsfrLen for was 345] [Sh in f intention holds the but Now fails, [Pe97]). proof (see indicated filters the that noted Peterson D. ii cardinal limit a cardinal ieryodrdsbe fteBoenAgba(h nydi attained). only not (the is Algebra supremum Boolean the the case of the subset ordered linearly a hr Length where ain433. cation ∗ ]i ssi htKpebr n hlhntdta yteLo´ the by that noted Shelah and Koppelberg that said is it 1] ( ) ∗ fcus,i ( if course, Of 1991 Date ntelnt fBoenagba tecriaiyo linear of cardinality (the algebras Boolean of length the On eew rv httesaeet( statement the that prove we Here h eerhwsprilyspotdb IreiBscRese Basic “Israeli by supported partially was research The + ) µ etme ,2018. 8, September : i ahmtc ujc Classification. Subject Mathematics h egh” iial o oeohrcria invariants cardinal ultrap other the some than for algebras. smaller Similarly is lengths”. algebras the Boolean of ultraproduct Abstract. | | < i<κ i<κ λ Q Q Hence . Length( EGHO OLA LERSAND ALGEBRAS BOOLEAN OF LENGTH Length( Length D + { EAHMMGDRADSHRNSHELAH SAHARON AND MAGIDOR MENACHEM κ < i ( B on D ∈ } ∗ al hn(sn lrpout of ultraproducts (using then fails ) stefis adnlntrpeetda h adnlt of cardinality the as represented not cardinal first the is ) epoetecnitnywt F f“h egho an of length “the of ZFC with consistency the prove We B + κ B i ( Length : ) n ola algebras Boolean and B i and , ) i / ) |≤ D| | ⇒ / ULTRAPRODUCTS |≤ D| i<κ 0. Q + Length( ( i<κ Q Length Introduction Length B i sarglrcardinal regular a is ) µ i / 1 D| i<κ rmr:0Ex 33;Scnay 03E55. Secondary: 03E35; 06Exx, Primary: Q + + ( ∗ § ( < B i<κ ] ehave we 1]) a al(e hoe . and 1.3 Theorem (see fail may ) B Q i i Length( ) B / / B D D i i ,and ), / ( κ < i is D ), λ i<κ Q lk o oesuccessor some for –like { ehave we ) h rhFudto” Publi- Foundation”. arch κ < i ( B H D ∈ } i / ( χ D fBoolean of invariants. l ) rrglrultra- regular or Length : outof roduct ). B , eec being fference gth hoe for theorem s lh[h345, [Sh elah i : ) yordered ly + i.e. , κ < i + ( B i is ) or 2 MENACHEMMAGIDORANDSAHARONSHELAH
+ hence {i : Length (Bi) is an inaccessible cardinal}∈ D, so the example we produce is in some respect the only one possible. (Note that our convention is that “inaccessible” means regular limit (> ℵ0), not necessarily strong limit.) More results on cardinal invariants of ultraproducts of Boolean algebras can be found in [Sh 462], [Sh 479], [RoSh 534] and [Sh 620], [RoSh 651]. This paper is continued for other cardinal invariants (in particular spread) in [ShSi 677]. We thank Otmar Spinas and Todd Eisworth for corrections and com- ments.
1. The main result Definition 1.1. 1. For a Boolean algebra B, let its length, Length(B), be sup{|X| : X ⊆ B, and X is linearly ordered (in B) }. 2. For a Boolean algebra B, let its strict length, Length+(B), be sup{|X|+ : X ⊆ B, and X is linearly ordered (in B)}.
Remark 1.2. 1. In Definition 1.1, Length+(B) is (equivalently) the first λ such that for every linearly ordered X ⊆ B we have |X| < λ. 2. If Length+(B) is a limit cardinal then Length+(B) = Length(B); and if Length+(B) is a successor cardinal then Length+(B) = (Length(B))+.
Theorem 1.3. Suppose V satisfies GCH above µ (for simplicity), κ is mea- surable, κ<µ, µ is λ+-hypermeasurable (somewhat less will suffice), F is the function such that F (θ) = the first inaccessible > θ, and λ = F (µ) is κ well defined, and χ<µ, χ> 22 . Then for some forcing notion P not collapsing cardinals, except those in the interval (µ+, λ) [so in VP we have µ++ = λ = F V(µ)], and not adding subsets to χ, in VP, we have: (α) in VP the cardinal µ is a strong limit of cofinality κ, (β) for some strictly increasing continuous sequence hµi : i < κi of (strong V limit) singular cardinals > χ with limit µ, each λi =: F (µi) is still inaccessible and for any normal ultrafilter D∈ V on κ we have:
++ V Y λi/D has order type µ = F (µ). i<κ
Definition 1.4. A forcing notion Q is directed µ-complete if: for a directed quasi-order I (so (∀s0,s1 ∈ I)(∃t ∈ I)(s0 ≤I t & s1 ≤ t)) of cardinality <µ, andp ¯ = hpt : t ∈ Ii such that pt ∈ Q and s ≤I t ⇒ ps ≤Q pt, there is p ∈ Q such that t ∈ I ⇒ pt ≤Q p. LENGTH OF BOOLEAN ALGEBRAS AND ULTRAPRODUCTS 3
Proof Without loss of generality for every directed µ–complete forcing notion Q of cardinality at most λ satisfying the λ-c.c., in VQ the cardinal µ is still λ-hypermeasurable. [Why? If µ supercompact, use Laver [L], if µ is just λ-hypermeasurable see more in Gitik Shelah [GiSh 344].] Let Q be the forcing notion adding λ Cohen subsets to µ, i.e., {f : f a partial function from λ to {0, 1}, |Dom(f)| <µ}. In V, let R = Levy(µ+, < λ) = {f : f a partial two place function such that [f(α, i) defined ⇒ 0 < α < λ & i<µ+ & f(α, i) < α] and |Dom(f)| <µ+} (so R collapses all cardinals in (µ+, λ) and no others, so in VR the ordinal λ becomes µ++). Clearly R is µ+–complete and hence adds no sequence of length ≤ µ of members of V. Q In V , there is a sequence D¯ = hDi : i < κi of normal ultrafilters on µ µ as in [Mg4] and,g ¯ = hgi,j : i < j ≤ κi, gi,j ∈ H(µ) witness this (that is Q κ Di ∈ (V ) /Dj , in fact Di is equal to gi,j/Dj in the Mostowski collapse of Q κ ¯ (V ) /Dj ). Let D = hDi : i ≤ κi,g ¯ = hgi,j : i < j ≤ κi be Q–names of such sequences. Note˜ that˜ a Q–name˜A of˜ a subset of µ is an object of size ≤ µ, i.e., it consists of a µ–sequence of˜ µ–sequences of members of Q, say hhpi,j : j <µi : i<µi, and function f : µ × µ −→ {0, 1} such that each {pi,j : j<µ} is a maximal antichain of Q and pi,j Q “i ∈ A ⇔ f(i, j) = 1”. So the set of members of Q and the set of Q–names of subsets of µ are the same in V and in VR. So in VR×Q the sequence D¯ still gives a sequence of normal ultrafilters as required in [Mg4] as witnessed byg ¯ = hgi,j : i < j ≤ κi. Also the Magidor forcing P(D¯, g¯) (from there) for changing the cofinality of µ to κ (not collapsing cardinals not adding subsets to χ, the last is just by fixing the first element in the sequence) is the same in VQ and VR×Q and has the same set of names of subsets of µ. We now use the fact that P(D¯, g¯) satisfies the µ+-c.c. (see [Mg4]). Let P = (Q × R) ∗ P(D¯ , g¯), so again every ˜ ¯ Q∗P(D¯˜,¯g) (Q×R)∗P(D¯ ,¯g) Q ∗ P(D, g¯)-name involves only µ decisions so also V ˜ ˜ , V ˜ ˜ have the˜ ˜ same subsets of µ. So our only problem is to check conclusion (β) of Theorem 1.3. Let D∈ V be any normal ultrafilter on κ (so this holds also in VR, VR×Q, R×Q P(D¯ ) (V ) ˜ ).
¯ Claim 1.4.1. VQ∗P(D,¯g) In ˜ ˜ , the linear order Q F (µi)/D has true cofinality i<κ F (µ)= λ.
µ Proof of the Claim: Clearly Q “for i < κ we have µ /D is well ˜ i ordered” (as D is ℵ1–complete) and ˜ i µ µ Q “for i < κ we have µ /Di has cardinality 2 and even 2|i|/D has˜ cardinality 2µ” Qi<µ ˜ i 4 MENACHEMMAGIDORANDSAHARONSHELAH
<µ [Why? As µ = µ and Di is a uniform ultrafilter on µ. In details, let µ> µ µ h : 2 → µ be one-to-one, and for each η ∈ 2 define gη ∈ µ by gη(i) = h(η ↾ i). Then µ η 6= ν ∈ 2 ⇒ {i<µ : gη(i)= gν (i)} is a bounded subset of µ µ |i| and hence its complement belongs to Di but |{gη(i) : η ∈ 2}| = 2 ]. Consequently, for some F ∗ ∈ µµ we have
∗ Q “ Y F (ζ)/Di is isomorphic to λ”. ζ<µ ˜ If we look at the proof in [L] (or [GiSh 344]) which we use above, we see ∗ that w.l.o.g. F is the F above (and so does not depend on i). So let f i,α be Q-names such that ˜
Q “for i < κ and α < λ, f i,α ∈ Qζ<µ F (ζ) and ˜ fi,α/Di is the α-th function in Q F (ζ)/Di.” ˜ ζ<µ ˜
Q In V let Dκ = T Di and let Bi ∈ Di \ S Dj be as in [Mg4] (you can also i<κ j
fα ↾ Bi = fi,α ↾ Bi and fα ↾ (µ \ [ Bi) is constantly zero. i<κ
So hfα : α<µi is P(D¯,g¯) “for every A ∈ Dκ = T Di i<κ for every i < κ large enough we have µ ∈ A” ˜ i (where hµi : i < κi is the increasing continuous κ–sequence cofinal in µ which P(˜D¯, g¯) adds). Since for every p ∈ P(D¯, g¯), for some q ≥ p we have (recall from [Mg4] q that F (i) is the set which q “says” µi belongs to (when q does not forces a value to µi)) ˜ ˜ q [F (i) ⊆ Aα,β for every i < κ large enough], Q∗P(D¯ ,¯g) hence in V ˜ ˜ , for α < β < λ the set {i < κ : ¬(fα(µi) < fβ(µi))} is bounded, i.e., hhfα(µi) : i < κi : α < λi is < bd–increasing in F (µi). Jκ Q i<κ LENGTH OF BOOLEAN ALGEBRAS AND ULTRAPRODUCTS 5 Q On the other hand, in V , assume p P(D¯,g¯) “f ∈ Q F (µi)”. W.l.o.g. p ˜ i<κ ˜ ∗ forces that F (µi) <µi+1, and so by [Mg4] (possibly increasing p), we have, for some function˜ h, Dom(˜ h)= κ, h(i) ∈ [i]<ℵ0 , that above p, we know: f(i) depends on the value of µj for j ∈ {i}∪ h(i). So we can define a function˜ f ∗ ∈ µµ: ˜ ∗ f (ζ) = sup{γ : for some i < κ, possibly µi is ζ and γ is a possible value (above˜ p) of f(i)}. ˜ ∗ Q ∗ So f (ζ) < F (ζ), hence (in V ) for some α, f M |= “j(hαi : i < κi/D) is an ordinal smaller than the first inaccessible >µ”. But the property “not weakly inaccessible” is preserved by extending the universe (from M to V1). So we finish. 1.3 Remark 1.5. 1. The proof has little to do with our particular F . Assume F : µ −→ µ and we add (∗) F (µ)= λ for λ = j(F )(µ), j an appropriate elementary embedding. Then all the proof of 1.3 works except possibly the last sentence, which use some absoluteness of the definition of F . 2. We can also vary R. ′ (Q×R)∗P(D¯ ,¯g) <κ 3. Let λi = F (µi). In V = V ˜ ˜ we have µ = µ , moreover (∀α < λ)[| α |<κ< λ] and µ is strong limit. Hence if in V′, P is a forcing notion satisfying the κ-c.c. of cardinality <µ, and D∗ is an ultrafilter 6 MENACHEMMAGIDORANDSAHARONSHELAH ′ P ∗ on κ extending D then (in (V ) ) we have: the ultraproduct Q λi/D i<κ is λ–like. Proposition 1.6. Suppose hλi : i < κi is a sequence of (weakly) inaccessible cardinals > κ, D an ultrafilter on κ, and the linear order Q (λi,<)/D is i<κ λ–like, λ regular. 1. There are Boolean algebras Bi (for i < κ) such that: + (a) Length(Bi) = Length (Bi)= λi, + (b) Length ( Q Bi/D)= λ, i<κ + (c) if λ = µ then Length( Q Bi/D)= µ. i<κ 2. Assume λi is regular > κ, |Bi| = λi, Bi = S Bi,α, Bi,α increasing α<λ continuous in α (the Bi, Bi,α are Boolean algebras of course) and we have (∀α < λi)(|Bi,α| < λi) and + (∗)0 if i < κ, δ < λi, cf(δ) = κ then: Bi,δ <◦ Bi (<◦ is complete subalgebra sign) i.e. (∗)1 if x ∈ B \ B , x 6= 0 then for some y∗ we have: Bi,Bi,δ i i,δ ∗ ∗ (a) y ∈ Bi,δ and y 6= 0 ∗ (b) for any y ∈ Bi,α such that Bi |=“ y ≤ y & y 6= 0”: Bi |= “ y ∩ x 6=0& x − y 6= 0 ” + Then Length ( Q Bi/D) ≤ λ. i<κ i Proof (1) Let for i < κ and α < λi, Bα be the Boolean algebra i generated freely by {xζ : ζ < α} except i,α i,α ⊗ xζ ≤ xξ for ζ<ξ<α. i Let Bi be the free product of {Bα : α < λi} so Bi is freely generated by i,α {xζ : α < λi, ζ<α} except for ⊗. i,α Let Bi,β be the subalgebra of Bi generated by {xζ : α<β,ζ <α}. Now clause (a) holds immediately, and the inequality ≥ in clause (b) holds by theLo´stheorem, and the other inequality follows by part (2) of the proposition. Lastly clause (c) follows. (2) W.l.o.g. the set of members of Bi is λi, and the set of elements of Bi,α is an initial segment. + Let Si = {δ : δ < λi, the set of members of Bi,δ is δ and cf(δ)= κ } LENGTH OF BOOLEAN ALGEBRAS AND ULTRAPRODUCTS 7 ∗ Let (B,< ,S)= Q (Bi, Remark 1.8. 1. We can say more on when ultraproducts of free products of Boolean algebras has not too large length. 2. We can use the disjoint sum of hBi,α : α < λii instead. + + 3. In the proof of 1.6 we actually have Depth (Bi) = Length (Bi) and Depth+(B) = Length+(B), and so similarly without +; where Depth+(B) = sup{|X|+ : X ⊆ B is well ordered} Depth(B) = sup{|X| : X ⊆ B is well ordered}. So the parallel of 1.6 holds for Depth instead Length. 4. Recal c(B) is the cellularity of a Boolean algebra b, i.e., sup{|X| : X is a set of pairwise disjoint non zero elements}. If in the proof of 1.6 i defining Bα we replace i,α i,α ⊗ xζ ≤ xξ by 4 i,α i,α ⊗ xζ ∩ xξ = 0 8 MENACHEMMAGIDORANDSAHARONSHELAH + + + then c(Bi)= λi = c (Bi), c(B)= µ , λ = c (B). (Same proof.) 5. We can also get the parallel to 1.6 for the independence number. Let i i,α B be the Boolean algebra generated by {xζ : ζ < α and α < λi} freely except i,α i,β (⊗5) xζ ∩ xξ =0 if α<β<κ, ζ < α, ξ < β. i i,α Let Ii be the ideal of B which {xζ : ζ < α, α < λi} generates. i Clearly it is a maximal ideal. Let Bi,α be the ideal of B generated by i,β {xζ : ζ < α and β < α}. Again w.l.o.g. the universe of Bi is λi and let Ci = {δ < λi : for x ∈ Bi we have: x < δ iff x ∈ Bi,δ ∨−x ∈ Bi,α}. It is a club of λi. The Bi,β are not Boolean subalgebras of Bi, just Boolean subrings; Bi now (∗)0 in proposition 1.6 is changed somewhat. We will have P = ∗ ∗ ∗ Bi Ii and (B,< , P )= Q (Bi, References [GiSh 344] Moti Gitik and Saharon Shelah. On certain indestructibility of strong cardinals and a question of Hajnal. Archive for Mathematical Logic, 28:35–42, 1989. [L] Richard Laver. Making supercompact indestructible under κ-directed forcing. Israel J. of Math., 29:385–388, 1978. [Mg4] Menachem Magidor. Changing cofinality of cardinals. Fund. Math., XCIX:61–71, 1978. [M1] Donald Monk. Cardinal Invariants of Boolean Algebras. Lectures in Mathe- matics. ETH Zurich, Birkhauser Verlag, Basel Boston Berlin, 1990. [M2] Donald Monk. Cardinal Invariants of Boolean Algebras, volume 142 of Progress in Mathematics. Birkh¨auser Verlag, Basel–Boston–Berlin, 1996. [Pe97] Douglas Peterson. Cardinal functions on ultraproducts of Boolean algebras. Journal of Symbolic Logic, 62:43–59, 1997. [RoSh 651] Andrzej Ros lanowski and Saharon Shelah. Forcing for hL, hd and Depth. Colloquium Mathematicum, submitted. [RoSh 534] Andrzej Roslanowski and Saharon Shelah. Cardinal invariants of ultrapoducts of Boolean algebras. Fundamenta Mathematicae, 155:101–151, 1998. [Sh 620] Saharon Shelah. Special Subsets of cf(µ)µ, Boolean Algebras and Maharam measure Algebras. General Topology and its Applications - Proc. of Prague Topological Symposium 1996, accepted. LENGTH OF BOOLEAN ALGEBRAS AND ULTRAPRODUCTS 9 [Sh:92] Saharon Shelah. Remarks on Boolean algebras. Algebra Universalis, 11:77–89, 1980. [Sh 111] Saharon Shelah. On power of singular cardinals. Notre Dame Journal of For- mal Logic, 27:263–299, 1986. [Sh 345] Saharon Shelah. Products of regular cardinals and cardinal invariants of prod- ucts of Boolean algebras. Israel Journal of Mathematics, 70:129–187, 1990. [Sh 420] Saharon Shelah. Advances in Cardinal Arithmetic. In Finite and Infinite Com- binatorics in Sets and Logic, pages 355–383. Kluwer Academic Publishers, 1993. N.W. Sauer et al (eds.). [Sh 479] Saharon Shelah. On Monk’s questions. Fundamenta Mathematicae, 151:1–19, 1996. [Sh 462] Saharon Shelah. On σ-entangled linear orders. Fundamenta Mathematicae, 153:199–275, 1997. [ShSi 677] Saharon Shelah and Otmar Spinas. On incomparability and related cardinal characteristics of ultrapowers of Boolean algebras. in preparation. Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel E-mail address: [email protected] Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel, and Rutgers University, Mathematics Department, New Brunswick, NJ 08854, USA E-mail address: [email protected]