Cascades, Order and Ultrafilters

0.19 MB·English

by  Andrzej Starosolski

Cascades, Order and Ultrafilters
Andrzej Starosolski
Politechnika S´la¸ska, Wydzia l Matematyki Stosowanej ul. Kaszubska 23, 44-101 Gliwice,
Poland
2
1
0
2

n
a
Abstract
J

0 Weinvestigatemutualbehaviorofcascades,contoursofwhicharecontained
1
ina fixed ultrafilter. Using thatrelationwe prove(ZFC) that the class ofstrict

J -ultrafilters,introducedbyJ.E.BaumgartnerinUltrafilters on ω,is empty.
] ωω
O We translate the result to the language of < -sequences under an ultrafilter,
∞
L investigated by C. Laflamme in A few special ordinal ultrafilters, to show that
h. if there is an arbitrarylong finite <∞-sequence under u than u is at leaststrict
J - ultrafilter.
t ωω+1
a
m Keywords: ordinal ultrafilters, monotone sequential cascades
2010 MSC: 03E04,03E05
[

1
v
6
4 1. Introduction
1
2 Baumgartner in the article Ultrafilters on ω ([1]) introduced a notion of I-
. ultrafilters: Let I be an ideal on X, an ultrafilter (on ω) is an I-ultrafilter, if
1
0 andonlyif,foreveryfunctionf :ω →X thereisasetU ∈usuchthatf[U]∈I.
2 This kind of ultrafilters wasstudied by largegroupofmathematician. We shall
1 mentiononlythe mostimportantpapersinthis subjectfromourpointofview:
:
v J. Brendle [3], C. Laflamme [13], Shelah [14] ,[15], Bl aszczyk [2]. Among other
i types of ultrafilters J. E. Baumgartner introduced ordinal ultrafilters, precisely
X
ω sequenceofclassesofultrafilters. We saythatuisJ ultrafilter(onω)iffor
1 α
r
a each function f : ω → ω1 there is U ∈ u such that ot(f(U)) < α, where ot(·)
denotes the ordre type. For additional information about ordinal ultrafilters
a look at [1], [3], [18] is recommended. In [1] J. E. Baumgartner proved (in
Theorems 4.2 and 4.6) that for each successor ordinal α<ω the class of strict
1
J -ultrafilters (see below) is nonempty if P-points exist, he also pointed out
ωα
that: ”Ingeneralwedonotknow,whether,ifαislimit,thereisaJωα-ultrafilter
that is no J -ultrafilter, for some β < ωα, even if CH or MA assumed”. Here,
β
suchultrafilterswecallstrictJ -ultrafilters,andwepartiallysolvetheproblem,
ωα
showing (ZFC) that the class of strict J -ultrafilters is empty.
ωω
Email address: [email protected] (AndrzejStarosolski)
Preprint submitted toElsevier January 11, 2012
If u is a filter(base) on A ⊂ B, then we identify u with the filter on B for
whichuis a filter-base. Letu, v be ultrafiltersonω,recallthat v < u ifthere
∞
isafunctionf :ω →ω suchthatf(u)=v andf isnotfinite-to-oneorconstant
on any set U ∈ u. In [13] C. Laflamme proved (reformulation of Lemma 3.2)
that if an ultrafilter u has an infinite decreasing < - sequence below, then u is
∞
at least strict J -ultrafilter. He also stated the following
ωω+1
[13, Open Problem1] What about the correspondinginfluence of increasing
< -chains below u? Given such an ultrafilter u with an increasing infinite
∞
< -sequence u > ... > u > u below, fix maps g and f witnessing
∞ RK ∞ 1 ∞ 0 i i
u> u andu > u respectively. Theproblemisreallyaboutthepossible
RK i i+1 ∞ i
connections between g and f ◦g even relative to members of u.
i i i+1
[13, Open Problem2]Can we haveanultrafilter u with arbitrarylong finite
< -chains below u without infinite one? This looks like the most promising
∞
way to build a strict J -ultrafilter.
ωω
We find affirmative answer to the first problem and negative answer to the
second one.
2. Prelimineries
In [6] S. Dolecki and F. Mynard introduced monotone sequential cascades -
special kind of trees - as a tool to describe topological sequential spaces. Cas-
cadesandtheir contoursappearedto be alsoa useful toolto investigatecertain
types of ultrafilters on ω, namely ordinal ultrafilters and P-hierarchy (see [18],
[17]), here we focus on the first of them.
The cascade isa treeV, orderedby ”⊑”,without infinite branchesandwith
a least element ∅ . A cascade is sequential if for each non-maximal element
V
of V (v ∈ V \ maxV) the set v+V of immediate successors of v (in V) is
countably infinite. We write v+ instead of v+W if it is known in which cascade
thesuccessorsofv areconsidered. Ifv ∈V \maxV,thenthe setv+ (ifinfinite)
may be endowed with an order of the type ω, and then by (v ) we denote
n n∈ω
the sequenceof elements ofv+, andby v - the n-th elementof v+W. We say
nW
that v is a predecessor of v′ (we write v =pred(v′)) if v′ ∈v+.
The rankof v ∈V (r (v) or r(v)) is defined inductively asfollows: r(v)=0
V
if v ∈ maxV, and otherwise r(v) is the least ordinal greater than the ranks of
allimmediate successorsof v. The rank r(V) of the cascadeV is, by definition,
the rank of ∅ . If it is possible to order all sets v+ (for v ∈ V \maxV) so
V
that for each v ∈ V \maxV the sequence (r(v ) ) is non-decreasing (other
n n<ω
words if for each v ∈ V \∅ the set {v′ ∈ (pred(v))+ : r(v′) < α} is finite for
V
each α < r(v)), then the cascade V is monotone, and we fix such an order on
V without indication. Thus we introduce lexicographic order < on V in the
lex
following way: v′ < v′′ if v′ ⊐v′′ or if there exist v, v˜′ ⊑v′ andv˜′′ ⊑v′′ such
lex
that v˜′ ∈v+ and v˜′′ ∈v+ and v˜′ =v , v˜′′ =v and n<m.
n m
Let W be a cascade, and let {V :w ∈maxW} be a set of pairwise disjoint
w
cascades such that V ∩W = ∅ for all w ∈ maxW. Then, the confluence of
w
cascades V with respect to the cascade W (we write W "V ) is defined as a
w w
2
cascadeconstructedbytheidentificationofw∈maxW with∅ andaccording
Vw
tothefollowingrules: ∅W =∅W"Vw;ifw ∈W\maxW,thenw+W"Vw =w+W;
if w ∈ Vw0 (for a certain w0 ∈ maxW), then w+W"Vw = w+Vw0; in each case
we also assume that the order on the set of successors remains unchanged. By
(n)"V we denote W "V if W is a sequential cascade of rank 1.
n w
AlsowelabelelementsofacascadeV bysequencesofnaturalsoflengthr(V)
orless,bythe function whichpreservesthe lexicographicorder,v isa resulting
l
nameforanelementofV,wherelisthementionedsequence(i.e. v =(v ) ∈
l⌢n l n
v+); by V we denote v↑ and by L we understand {l ∈ ω<ω : r (v ) = α}.
l l α,V V l
Let v, v′ ∈ V, we say that v′ is a predecessor of v (in V) if v ∈ v′+, we write
v′ =pred (v). For a finite sequence l =(n ,...,n ) of natural numbers by l−
V 0 k
we denote a sequence l with the last element removed,i.e. l− =(n ,...,n );
0 k−1
by l+ we denote a set of all sequences l′ such that l′− =l.
If U = {u : s ∈S} is a family of filters on X and if p is a filter on S, then
s
the contour of {u } along p is defined by
s
U= u = u .
Z Z s s
p p P[∈ps\∈P
Such a construction has been used by many authors ([8], [9], [10]) and is
also known as a sum (or as a limit) of filters. On the sequential cascade, we
consider the finest topology such that for all but the maximal elements v of V,
the co-finite filter on the set v+V converges to v. For the sequential cascade
V we define the contour of V (we write V) as the trace on maxV of the
neighborhood filter of ∅ (the trace of a Rfilter u on a set A is the family of
V
intersections of elements of u with A). Similar filters were considered in [11],
[12], [4]. Let V be a monotone sequential cascade and let u = V. Then
the rank r(u) of u is, by definition, the rank of V. It was shown inR [7] that if
V = W, then r(V)=r(W).
R LetRS beacountableset. Afamily{u } offiltersisreferredtoasdiscrete
s s∈S
if there exists a pairwise disjoint family {U } of sets such that U ∈ u for
s s∈S s s
each s ∈ S. For v ∈ V we denote by v↑ a subcascade of V built by v and all
successorsofv. IfU ⊂maxV andU ∈ V,thenbyU↓V wedenote the biggest
(intheset-theoreticalorder)monotonesRequentialsubcascadeofcascadeV built
ofsomev ∈V suchthatU∩maxv↑ 6=∅. Wewritev↑ andU↓ insteadofv↑V and
U↓V ifweknowinwhichcascadethesubcascadeisconsidered. Thereadermay
find more information about monotone sequential cascades and their contours
in [5], [6], [7], [16], [17], [18].
In the remainder of this paper each filter is considered to be on ω, unless
indicated otherwise.
3. Existence of ordinal ultrafilters
ForamonotonesequentialcascadeV byf wedenoteanlexicographic order
V
respectingfunctionmaxV →ω ,i.e.,suchafunctionthatv′ < v′′ ifff (v′)<
1 lex V
3
f (v′′) for each v′, v′′ ∈ maxV. If f : ω → ω and f = f for some monotone
V 1 V
sequential cascade V then we say that V corresponds with an order of f.
Let V and W be monotone sequential cascades such that maxV ⊃maxW.
We say that W increases the order of V (we write) W ⇛ V if ot(f (U)) ≥
W
indec(ot(f (U)))foreachU ⊂maxW,whereindec(α)isthebiggestindecom-
V
posable ordinal less then, or equal to α; by Cantor normal form theorem such
a number exists and is defined uniquely. Clearly this relation is idempotent
and transitive. Although relation of increasing of order says that one cascade
is somehow bigger then another, this relation is quite independent with the
containment of contours.
Example 3.1. ( T ⊃ V 6⇒ T ⇛ V) Let (V ) be a sequence of pairwise
n n∈ω
disjoint monotoneR sequeRntial cascades of rank 2. For each n < ω choose v -
n
an arbitrary element of maxV . Let V′ =V \{v } and let V′ be an arbitrary
n n n n ω
monotone sequential cascade of rank 2 such that maxV = {v }. Now put
ω n<ω n
T =(n)" V and V =(n)" V′. S
n<ω n n≤ω n
Example 3.2. (T ⇛ V 6⇒ T ⊃ V) Let (V ) be a sequence of pairwise
n n∈ω
disjoint monotone sequentiaRl cascadRes of rank 1. For each n < ω choose v
n
- an arbitrary element of maxV . Let (B ) be a partition of {v }
n n n<ω n<ω n
into infinite sets. Let V′ be a monotone sequential cascade of rank 1,Ssuch that
n
maxV′ =(maxV \{v })∪B . Put T =(n)"V′ and V =(n)"V .
n n n n n n
Let V and W be monotone sequential cascades, let f : V → W be a finite-
to-one, ⊑ order preserving surjection such that F =id and
V |maxV∪∅V |maxV∪∅V
v ∈ f−1(v) for each V ∈ V. Then, it is easy to see, that V ⇛ f(V) and
f(V)⇛V, we call this property locally finite partition property (LFPP)
Let u, p be filters on ω, then u∨p we define as {X ∈ω : there exist U ∈u
and P ∈p such that U ∩P ⊂X}.
Let u be an ultrafilter and let V, W be monotone sequential cascades such
that V ⊂ u and W ⊂ u. Then we say that rank α in cascade V agree
with rRank β in cascaRde W with respect to the ultrafilter u if for any choice of
V˜ ∈ V and W˜ ∈ W there is: (V˜ ∩W˜ ) ∈ u; this
p,s p p,s s (p,s)∈Lα,V×Lβ,W p,s p,s
relationRis denoted by αRE β . S
V u W
Proposition 3.3. Let u be an ultrafilter and let V, W be monotone sequential
cascades suchthat V ⊂uand W ⊂u. Then 1 E 1 andr(V) E r(W) .
V u W V u W
Proof. First suppRose that r(VR) = 1 or r(W) = 1, say r(V) = 1. Clearly
card(L ) = 1. For each (p,s) ∈ L × L take any V˜ ∈ V and
1,V 1,V 1,W p,s p
any W˜ ∈ W . Notice that since V is a co-finite filter on maRxV thus
p,s s
for each (p,sR) ∈ L ×L the set (RW˜ ∩maxV)\(W˜ ∩V˜ ) is finite.
1,V 1,W p,s p,s p,s
ThereforethereexistWˆ ⊂W˜ ,Wˆ ∈ W suchthatWˆ ∩V˜ =Wˆ ∩
p,s p,s p,s p,s p,s p,s p,s
maxV. Thus (W˜ ∩V˜ R) ⊃ (Wˆ ∩V˜ ) =
(p,s)∈L1,V×L1,W p,s p,s (p,s)∈L1,V×L1,W p,s p,s
S (Wˆ )∩maxV. Since S (Wˆ )∈ W ⊂u and
(p,s)∈L1,V×L1,W p,s (p,s)∈L1,V×L1,W p,s
SmaxV ∈ V ⊂u thus S(W˜ ∩V˜ )∈u. R
(p,s)∈L1,V×L1,W p,s p,s
BeforeRwedealwithcSaser(V)≥2andr(W)≥2westatethefollowingclaim:
inassumptionofthisProposition,ifasetU issuchthatforeach(p,s)∈L ×
1,V
4
L the intersection U ∩maxV ∩maxW is finite, then U 6∈ u. Let f : ω →
1,W p s
L ,h:ω →L bebijections,andletG(i,j)=U∩maxV ∩maxW for
1,V 1,W f(i) h(j)
i,j ∈ω. Let ∆≥ ={(i,j)⊂ω×ω :i≤j}, ∆≤ ={(i,j)⊂ω×ω :i≥j}. Since
uis anultrafilter thus either G(∆≥)∈uor G(∆≤)∈u. ButG(∆≥) is finite on
each maxV for p ∈ L and so (G(∆≥))c ∈ V therefore G(∆≥) 6∈ u. Also
p 1,V
G(∆≤)isfinite oneachmaxW fors∈L anRdso(G(∆≤))c ∈ W therefore
s 1,W
G(∆≤)6∈u. R
Now let r(V) ≥ 2 and r(W) ≥ 2 and suppose on the contrary that K =
(W˜ ∩V˜ ) 6∈ u for some choice of V˜ ∈ V and W˜ ∈
(p,s)∈L1,V×L1,W p,s p,s p,s p p,s
SW . Thus Kc ∈ u. Put R# = {(p,s)∈L ×L ; RV # W } and
s 1,V 1,W p s
RR6# = {(p,s)∈L ×L ; V 6# W }. Define K = R R((maxV ∩
1,V 1,W p s 1 (p,s)∈R# p
maxW )\ (V˜ ∩ W˜ )) andR K =R (maxV ∩SmaxW ) and notice
s p,s p,s 2 (p,s)∈R6# p s
that Kc ⊂ K ∪K . Thus either KS∈ u, or K ∈ u. Since traces of W
1 2 1 2 s
and of V on maxV ∩ maxW are co-finite filters (on maxV ∩ maxRW )
p p s p s
thus KR∩maxV ∩W = (maxV ∩maxW )\(V˜ ∩W˜ ) is finite on each
1 p s p s p,s p,s
(p,s) ∈ R# and empty on each (p,s) ∈ R6#. By similar reasoning K is finite
2
on each (p,s) ∈ L ×L . Therefore by claim above K 6∈ u and K 6∈ u -
1,V 1,W 1 2
contradiction. Second statement of Proposition 3.3 is clear.
In the above Proposition 3.3 the inverse of the implication does not hold.
Example 3.4. Let (A ) be a partition of ω into infinite sets. Let V
α α≤2ω α
be a monotone sequential cascade of rank 1 such that maxV = A . Let W =
α α
(α)" V , V =(α)" V , and let u beany freeultrafiltercontaining
α≤ω α ω≤α<2ω α
A . Clearly V 6⊂u, W 6⊂u but 1 E 1 .
ω V u W
R R
Although the following Theorem 3.5 is stated using the ”⇛” relation, it is
worthtolookatthe proofofitasonthedescriptionofpossiblerelationsofcas-
cadeswhose contoursarecontainedinthe same ultrafilter,andas a description
of operation which leads from such cascades to others whose contours are also
contained in the same ultrafilter.
Theorem 3.5. Let ubean ultrafilterand let V, W bemonotone sequentialcas-
cades of finite ranks such that V ⊂ u and W ⊂ u. Then n E m implies
V u W
the existence of a moRnotone sequRential cascade T of rank
max{r(V),r(W)} ≤ r(T) ≤ r(V)+r(W) and such that T ⊂ u, T ⇛ V,
T ⇛W, V ⊂ T and W ⊂ T. R
R R R R
Proof. Before we start the proof, let us make the following remarks: in this
theorem we claim (in place of T ⊂ u) that (under the same assumption and
notation) T ⊂( V∨ W),anRdthisformulationalsowillbeusedintheproof;
cascadeTRbuildinRthisRproofhasranksnotlessthenmax{n,m}+max{r(V)−
n,r(W)−m}andnotgreaterthenr(V)+r(W)−1andthis isinductivelyused
in the proof. Without loss of generality,we may assume that each branch in V
haslengthr(V)andeachbranchinW haslengthr(W),andthatr(V)≤r(W).
We proceed by induction by r(W), and for each r(W) by sub-induction by
r(V). First step of induction and of sub-inductions is r(V) = 1 and then we
5
take T = W↓maxV which clearly fulfills the claim. Assume that the claim is
proved for all cascades V˘, W˘ which behave like in assumptions and such that
r(V˘)≤r(W˘)andeitherr(W˘)<r(W)orelse(r(W˘)=r(W)andr(V˘)<r(V)).
We consider 3 cases
1) n<r(V) and m<r(W);
2 (n=r(V) and m<r(W)) or (n<r(V) and m=r(W));
3) n=r(V) and m=r(W).
Case 1) Let R# ={(l,s)∈L ×L : V # W }. Notice that exactly
n,V m,W l s
one of the following 3 subcases holds: R R
1.1) There is K ⊂ R# such that card(K (l)) < ω and
F−F F−F
card(K−1 (s)) < ω for each l ∈ domK , s ∈ rngK and
F−F F−F F−F
(V˜ ∩W˜ )∈uforeachchoiceofV˜ ∈ V andofW˜ ∈ W ;
(l,s)∈KF−F l,s l,s l,s l,s l,s l,s
S 1.2)∼1.1andthereisK ⊂R# suchthatcarRd(K−1 (s))<ω foRreach
∞−F ∞−F
s ∈ rngK and (V˜ ∩W˜ ) ∈ u for each choice of V˜ ∈ V
∞−n (l,s)∈K∞−F l,s l,s l,s l,s
and of W˜ ∈ W S; R
l,s l,s
1.3)∼1.1aRndthere is K ⊂R# suchthatcard(K (l))<ω for each
F−∞ F−∞
l∈domK and (V˜ ∩W˜ )∈u for each choice of V˜ ∈ V
F−∞ (l,s)∈KF−∞ l,s l,s l,s l,s
and of W˜ ∈ W S. R
l,s l,s
Let ∆≤ =R{(i,j) : i ≥ j;i,j ∈ ω} and ∆≥ = {(i,j) : i ≤ j;i,j ∈ ω}.
Let p : ω → L and q : ω → L be bijections. For X ⊂ ω ×ω define
n,V m,W
G(X)= (maxV ∩maxW )andput(p,q)(x ,x )=(p(x ),q(x )).
(i,j)∈X p(i) q(j) 1 2 1 2
SinceG(∆S≥)∪G(∆≤)∈uthuseitherG(∆≥)∈u(case1.1or1.2)orG≥(∆)∈u
(case1.1 or1.3). ForA⊂R we define also H(A)= (maxV ∩maxW ).
(l,s)∈A l s
Since u is an ultrafilter thus without loss of genSerality (by LFPP, for case
1.1usedtwice,its propertyofincreasingorderandtransitivityof”⇛”relation)
exactly one of the following subcases holds.
1.1′)ThereisK ⊂R#suchthatcard(K (l))=1andcard(K−1 (s))=
1−1 1−1 1−1
1 for each (l,s) ∈ K and (V˜ ∩ W˜ ) ∈ u for each choice of
1−1 (l,s)∈K1−1 l,s l,s
V˜ ∈ V and of W˜ ∈ WS;
l,s l,s l,s l,s
1.2R′) ∼ 1.1 and there iRs K ⊂ R# such that card(K−1 (s)) = 1 and
∞−1 ∞−1
card(K (l))=ω for each(l,s)∈K and (V˜ ∩W˜ )∈u for
∞−1 ∞−1 (l,s)∈K∞−1 l,s l,s
each choice of V˜ ∈ V and of W˜ ∈ W ;S
l,s l,s l,s l,s
1.3′) ∼ 1.1 and tRhere is K ⊂ R#R such that card(K (l)) = 1 and
1−∞ 1−∞
card(K−1 (s))=ω foreach(l,s)∈K and (V˜ ∩W˜ )∈ufor
1−∞ 1−∞ (l,s)∈K1−∞ l,s l,s
each choice of V˜ ∈ V and of W˜ ∈ W .S
l,s l,s l,s l,s
Subcase 1.1′) WiRthout loss of generaRlity, we may assume that maxV =
maxW = H(K ). Define a series of sets: R = L × L ,
1−1 n,V m,W
R(A) = {(l,s) ∈ R : card(maxV ∩ maxW ∩ A) = ω} for A ⊂ ω, uˆ =
l s
{R(U) : U ∈ u}, Vˆ = {v ∈ V : r(v) > n}∪R(maxV), Wˆ = {w ∈ W : r(w) >
m}∪R(maxW). OnVˆ wedefineorder⊑ by: ifv ,v ∈V ∩Vˆ,(l,s)∈Rthen:
Vˆ 1 2
v ⊑ v iff v ⊑ v ; v ⊑ (l,s) iff card(maxV ∩maxW ∩maxv↑V) = ω.
1 V 2 1 Vˆ 2 1 Vˆ l s 1
On Wˆ we intrtoduce order in the analogicalway. Notice that uˆ is an ultrafilter
6
on R and that Vˆ and Wˆ are monotone sequential cascades (on R) and that
r(Vˆ)=r(V)−n, r(Wˆ)=r(W)−m and that Vˆ ⊂uˆ and Wˆ ⊂uˆ.
By inductive assumption there is Tˆ monoRtone sequentiRal cascade (on R)
of rank max{r(V) − n,r(W) − m} ≤ r(Tˆ) ≤ r(V) + r(W) − n − m − 1
and such that Tˆ ⊂ uˆ and Tˆ ⇛ Vˆ, and Tˆ ⇛ Wˆ, also by inductive as-
sumption, for eRach (l,s) ∈ R# there is a monotone sequential cascade T
l,s
of rank max{n,m} ≤ r(T ) ≤ n+m−1 such that T ⇛ V , T ⇛ W ,
l,s l,s l l,s s
T ⊂ V ∨ W . Define T¯ = Tˆ " T . Take
l,s l s {(l,s);maxVl∩maxWs∈maxTˆ} l,s
Rany A ∈R T¯, tRhus there exist Aˆ ∈ Tˆ and A ∈ T such that A =
l,s l,s
AR . Since A ⊃ V˜ ∩ W˜R for some V˜ ∈ R V and W˜ ∈ W
(l,s)∈Aˆ (l,s) (l,s) l s l l s s
Sso A ⊃ (V˜ ∩W˜ ) = (V˜ ∩W˜ R)∩ (maxRV ∩
(l,s)∈Aˆ (l,s) (l,s) (l,s)∈R (l,s) (l,s) (l,s)∈Aˆ l
maxW )S. Since (maxV ∩SmaxW ) ∈ u thus S(V˜ ∩W˜ )
s (l,s)∈Aˆ l s (l,s)∈Aˆ (l,s) (l,s)
= (V˜ S∩ W˜ ) ∩ (maxV ∩ maxWS) ∈ u and so A ∈ u
(l,s)∈R (l,s) (l,s) (l,s)∈Aˆ l s
andSso T¯ ⊂ u. Consider seSts U = maxT . By in-
i {(l,s)∈R#:r(T(l,s))=i} (l,s)
ductiveRassumption - upper limitationSof ranks, only finite number of these
sets are nonempty, and since U ∈ u thus U ∈ u for some i . Let
i<ω i i0 0
T = T¯↓Ui0. Clearly T ∈ u. CSalculation of the rank of T follows easily. Take
anyP ∈maxT,withRoutlossofgenerality,wemayassumethatot(f (P))=ωb
V
for some b ≤ r(V). Split R into following sets R = {(l,s) ∈ R : ωa−1 ≤
a
ot(f (P ∩maxV ∩maxW ))<ωa} for a∈{1,...,b+1}. For some a, say a ,
V l s 0
wehaveot(f ( maxV ∩maxW ∩P)=ωb,thusot(f (R )≥ωb−a.
V (l,s)∈Ra0 l s V˜ a0
Therefore ot(fS(R )) ≥ ωb−a and so ot(f (P)) ≥ ωb, and so T ⇛ V. Proof
Tˆ a0 T
that T ⇛W is analogical.
Subcase 1.2′) Without loss of generality, we may assume that maxV =
maxW = H(K ). Consider cascade V′ - such a modification of cascade
∞−1
V that in the place of the cascade V , for each l ∈ domK we put a fol-
l ∞−1
lowing cascade: (s) " V↓maxWs∩Sk>q−1(s)maxVl⌢k. Notice that
{(s):(l,s)∈K∞−1} l
H(K∞) ∈ V and so H(K∞) ∈ u so without loss of generality we may as-
sume that KR∞ =K and so V′ =V′↓H(K∞). Notice that V′ is a monotone
∞−1
sequential cascade of rank r(V′) = r(V)+1 and that V′ ∈ u. Calculation
of the rank is straightforward, so take P ∈ V′ and foRr each l ∈ domK
∞−1
label elements of the set {s : (l,s) ∈ K }Rby natural numbers by preserv-
∞−1
ing lexicographic order bijection, s is a resulting name. If P ∈ V′ then
n
there exists Pˆ ∈ (V | ), also for each l : v ∈ Pˆ there exRists a co-
v∈V:r(v)≤n l
finite subset A ofRω that for each (l,s ), such that v ∈ Pˆ,n ∈ A , there is a
l n l l
set P ∈ V↓maxWs∩Sk>q−1(s)maxVl⌢k, such that P = P .
l⌢sn l l:vl∈Pˆ n∈Al l⌢sn
Since for eacRh pair (l,s ) there exist sets V˜ ∈ V andSW˜ ∈S W such
n l,sn l l,sn sn
that Pl⌢sn ⊃ V˜l,sn ∩ W˜l,sn thus P ⊃ l:vl∈Pˆ Rn∈Al(V˜l,sn ∩ W˜l,snR). Clearly
maxV 6∈ u so S SmaxV ∈ u. On the other
Shaln∈ddom(bKy∞a1s:svul6∈mPˆption ∼l 1.1) Sl∈domK∞1:vl∈Pˆ (V˜l ∩W˜ )6∈u, where
l∈domK∞−1 l⌢sn:n6∈Al l,sn l,sn
W˜ = maxW , V˜ = mSaxV and AS= ω for (l,s ) ∈ K such that
l,sn sn l,sn l l n ∞−1
v 6∈ Pˆ. Thus (V˜ ∩ W˜ ) ∈ u, therefore since
l l∈domK∞−1 l⌢sn:n∈Al l,sn l,sn
S S
7
P ⊃ maxV ∩ (V˜ ∩W˜ ) we have
P ∈uSal∈nddomsoK∞1V:v′l∈∈Pˆu. l Sl∈domK∞−1Sl⌢sn:n∈Al l,sn l,sn
WewillshoRwthatalsoV′ ⇛V holds. TakeanyA⊂maxV′ andnoticethat
itsuffices to proveot(f (A∩maxV ))≥indec(ot(f (A∩maxV )) for suchV
V′ l V l l
thatr (v )=n. SowefixsuchlandconsiderA∩maxV assuming,withoutloss
V l l
ofgenerality,thatot(f (A∩maxV )=ωc forsomec≤n. Considerafollowing
V l
oserqtuheenrceeiosfOset-si(nmfianxitVel⌢sukb∩seAt)okf<ωω,stuhcehretihsakt0otth(aft (oAt(∩fVV(A∩))Vl=′⌢k0ω)c)−=1 fωocr
V l⌢k
each k ∈O. Notice that each V is split, during the construction of V′, into
l⌢k
finitely many pieces by sets maxW ∩ maxV . So there is s such
s k>q−1(s) l⌢k 0
that ot(f (A∩maxV )) = ot(f (AS∩maxV ∩maxW )) = ot(f (A∩
V l⌢k V l⌢k s0 V′
maxV ∩maxW )). Therefore either ot(f (A∩maxV ))= ωc for some
l⌢k s0 V′ l⌢k
k < ω, or ot(f (A ∩ maxV )) ≥ ωc−1 for infinite number of k’s. Thus
V′ l⌢k
ot(f (A∩maxV))≥ωc and so V′ ⇛V.
V′ l
WenoticethatforcascadesV′ andW conditionsdescribedas1.1hold. Now
weproceedlikeinsubcasce1.1’. Defineaseriesofsets: R′ =L1,V′×L1,W,R′# =
{(l,s)∈R′ : V′# W R′(A)={(l,s)∈R:card(maxV′∩maxW ∩A)=ω}
l s l s
for A ⊂ ω, uˆR= {R(RU) : U ∈ u}, Vˆ′ = {v ∈ V′ : r(v) > 1}∪R(maxV′), Wˆ =
{w ∈ W : r(w) > 1}∪R(maxW). Observe that (l,s ) ∈ R ⇐⇒ (l⌢n,s) ∈ R′
n
and (l,s ) ∈ R# ⇐⇒ (l⌢n,s) ∈ R′#. On Vˆ′ we define order ⊑ by: if
n Vˆ′
v , v ∈ V′ ∩Vˆ′, (l,s) ∈ R′ then: v ⊑ v iff v ⊑ v ; v ⊑ (l,s) iff
1 2 1 V′ 2 1 Vˆ′ 2 1 Vˆ′
card(maxV′ ∩maxW ∩ maxv↑V′) = ω. On Wˆ we intrtoduce order in the
l s 1
analogical way. Notice that uˆ is an ultrafilter on R′ and that Vˆ′ and Wˆ are
monotone sequential cascades (on R′) and that r(Vˆ′) = r(V′) − 1 = r(V),
r(Wˆ)=r(W)−m and that Vˆ′ ⊂uˆ and Wˆ ⊂uˆ.
By inductive (or sub-induRctive) assumpRtion (for V′, W and u) there is Tˆ
monotone sequential cascade on R′ of rank max{r(V),r(W) −1} ≤ r(Tˆ) ≤
r(V)+r(W)−2andsuchthat Tˆ ⊂uˆandTˆ ⇛Vˆ′,andTˆ ⇛Wˆ,alsobyinduc-
tiveassumption,foreach(l,s)R∈R′# thereisamonotonesequentialcascadeT
l,s
ofrankr(T )=1suchthatT ⇛V′, T ⇛W , T ⊂ V′∨ W . Define
l,s l,s l l,s s l,s l s
T¯ =Tˆ " T . TakeanyAR∈ T¯,tRhusthRereexistAˆ∈
Tˆ and A{(l,s)∈;maxTVl′∩msauxcWhst∈hmaatxATˆ}=l,s A . SiRnce A ⊃ V˜′∩W˜ for
l,s l,s (l,s)∈Aˆ (l,s) (l,s) l s
RsomeV˜′ ∈ V′aRndW˜ ∈ W soA⊃S (V˜′ ∩W˜ )= (V˜′ ∩
l l s s (l,s)∈Aˆ (l,s) (l,s) (l,s)∈R (l,s)
W˜ )∩ R (maxV′∩RmaxW ). SiSnce (maxV′∩maSxW )∈uthus
(l,s) (l,s)∈Aˆ l s (l,s)∈Aˆ l s
(SV˜′ ∩ W˜ ) = S (V˜ ∩ W˜ )∩
(l,s)∈Aˆ (l,s) (l,s) (l,s)∈R′ (l,s) (l,s)
S (maxV′ ∩maxW ) ∈ u and so A ∈ uSand so T¯ ⊂ u. Consider sets
(l,s)∈Aˆ l s
SU = maxT . By inductive asRsumption - upper lim-
i {(l,s)∈R#:r(T(l,s))=i} (l,s)
itationSof ranks, only finite number of these sets are nonempty, and since
i<ωUi ∈u thus Ui0 ∈u for some i0. Let T =T¯↓Ui0. Clearly T ∈u. Calcu-
Slationof the rankof T followseasily. Take anyP ∈maxT,withRout loss of gen-
erality, we may assume that ot(f (P)) =ωb for some b ≤ r(V). Split R′ into
V′
following sets R′ ={(l,s)∈R′ :ωa−1 ≤ot(f (P ∩maxV′∩maxW ))<ωa}
a V′ l s
for a ∈ {1,...,b+1}. For some a, say a , we have ot(f ( maxV′∩
0 V′ (l,s)∈R′ l
S a0
8
maxW ∩P) = ωb, thus ot(f (R′ ) ≥ ωb−a. Therefore ot(f (R′ )) ≥ ωb−a
s V˜′ a0 Tˆ a0
and so ot(f (P)) ≥ ωb, and so T ⇛ V′ and T ⇛ V by transitivity of ⇛
T
relation. Proof that T ⇛W is analogical.
Subcase 1.3′) Proof is analogicalto 1.2’.
Case 2) In both subcases proof is an easier version of proof in case 1 (sub-
cases 1.2 and 1.3).
Case 3) a) Case r(V)=r(W)=1 was done at the beginning of the proof;
b) If min{r(V),r(W)} = 1 and max{r(V),r(W)} > 1 then 1 E 1 by
V u W
Proposition3.3,andbyalreadyprovedpart2oftheproof,the requiredcascade
T exists;
c)Ifmin{r(V),r(W)}>1then1 E 1 byProposition3.3,andbyalready
V u W
proved case 1, the required cascade T exists.
Inclusions of contours is straightforward by monotonicity of contour opera-
tion with respect to the confluence.
By Proposition 3.3 and Theorem 3.5 we have
Corollary 3.6. Let u be an ultrafilter and let V, W be monotone sequential
cascades of finite ranks. If V ⊂ u and W ⊂ u then there is a monotone
sequential cascade T of finitRe rank not lesRs then max{r(V),r(W)} and such
that T ⊂u and T ⇛V.
R
Proposition3.7. [18, Proposition 3.3 redefinedinvirtueofit’s proof] LetV be
amonotonesequential cascade of rankα. If uis suchan ultrafilterthat V ⊂u
then ot(f (U))≥ωα for all U ∈u. R
V
Proposition 3.8. [18, Proposition 3.6] Let α be a countable indecomposable
ordinal, let n < ω and let u be an ultrafilter. If there is a function f : ω → ω
1
such that ot(f(U)) ≥ ωα+n for each U ∈ u and for each g : ω → ω there is
1
U ∈ u such that ot(g(U )) < ωα+ω, then there exists a monotone sequential
g g
cascade V of rank n such that V ⊂u.
R
Theorem 3.9. (ZFC) The class of strict J -ultrafilters is empty.
ωω
Proof. Suppose that u is a strict J -ultrafilter, thus by definition of this class,
ωω
for each n < ω there exists a function f : ω → ω such that ot(f (U)) ≥ ωn
n 1 n
for each U ∈ u and there is no function f : ω → ω that ot(f (U)) ≥ ωω
∞ 1 ∞
for each U ∈ u. Let K be a set of all such n < ω that there is f : ω → ω
n 1
such that ot(f (U))≥ωn and that there is U ∈u such that ot(f (U ))=ωn
n n n n
1). By Proposition 3.7 there is a sequence W of monotone sequential cascades
n
such that r(W )>r(W ) and W ⊂u.
n+1 n n
We will build a sequence (T )Rof monotone sequential cascades such that
n
1) (r(T )) is an increasing sequence
n
2) T ∈u
n
R
01) InfactK=ω,butsincewedonotneedthisinthetheorem,weomitashortproofof
thisfact.
9
3) For each n < ω there exist sets A0 and A1 that Ai# T for i ∈ {0,1}
n n n n
and there is such in ∈{0,1} that Tn+1 ⇛Tn↓Ainn; R
4) maxT ∩maxT =∅, for T =(j)"T ;
i∈{1,...,n},j∈{1,...,i} i,j n+1 i i,j
5) S maxT =∅.
n<ω n
DefiTne T as the monotone sequential cascade of rank 1 with maxT = ω,
1 1
clearly T ∈u. SupposethatcascadesT arealreadydefinedforn≤m. Fora
1 n
cascadeRT = (k)"T consider sets Bi = maxT for i ∈{0,1}.
m m,k m k=2j+i m,k
Clearly B0 # T and B1 # T , and since BS1 ∪B0 =maxT one of these
m m m m m m m
sets belongs tRo u, call i theRi for which it happens. Let C = ∅ for m = 2
m m
and Cm = i∈{1,...,n},j∈{1,...,i}maxTi,j for m ≥ 3. Clearly Bmim \ Cm ∈ u.
So Tm↓Bmim\SCm ∈ u. By Corollary 3.6 applied to Tm↓Bmim, Wm+1 and to u,
therRe is a monotone sequential cascade T of finite rank not less then m+1
m+1
that Tm+1 ∈ u and Tm+1 ⇛ (Tm↓Bmim). We define Aimm = maxTm+1 and
Abm =Rω\Aimm for b∈{0,1}, b6=im. Clearly Tm+1 with Aimm and Abm fulfill the
claim for n = m+1. To see that maxT = ∅, it suffices to notice that
n<ω n
maxT ∩maxT =T∅.
k∈{1,...,m} 2,k m+1
S ↓(ω\Ain+1)
Define T = T n+1 ordered by:
n<ω n
1) If t ∈T Sand t ∈T (n6=m) then t ⊏ t ⇔n<m
1 n 2 m 1 T 2
2) If T ,T ∈T then t ⊑ t ⇔t ⊑ t , where ”⊑ ” is anorder onV .
1 2 n 1 T 2 1 Vn 2 Vn n
Let f : ω → ω be a preserving ⊑ order function. Take any U ∈ u and
∞ 1 T
n ∈ ω. Since T ∈ u thus U# T and so U# T for infinitely
n+1 n+1 n+1,k
many k, take kRfrom this set. R R
0
Since U# T thus ot(f (U ∩ maxT ) = r(T ) ≥ ωn.
n+1,k0 Tn+1 n+1,k0 n+1,k0
By conditionR4) U ∩ maxT ∩ maxT 6= ∅ only for a finite number of
n+1,k0 i
i > n + 1. So {maxT ∩ U ∩ (maxT \ maxT ) : i ≥ n + 1} is
n+1,k0 i j>i j
a finite partition of U ∩ maxT . Thus thSere is i ≥ n + 1 such that
n+1,k0 0
ot f maxT ∩U ∩(maxT \ maxT ) =
(cid:16) Tn+1(cid:16) n+1,k0 i0 j>i0 j (cid:17)(cid:17)
ot f (maxT ∩U) ≥ ωn, Sand since T ⇛ T thus
Tn+1 n+1,k0 i0 n+1
(cid:0) (cid:1)
ot f maxT ∩U ∩(maxT \ maxT ) ≥
(cid:16) Ti0 (cid:16) n+1,k0 i j>i0 j (cid:17)(cid:17)
S
indec ot f maxT ∩U ∩(maxT \ maxT ) ,
(cid:16) (cid:16) Tn+1(cid:16) n+1,k0 i0 j>i0 j (cid:17)(cid:17)(cid:17)
and since f | =f | S thus
∞ maxTi0\Sj>i0maxTj Ti0 maxTi0\Sj>i0maxTj
ot f maxT ∩U ∩(maxT \ maxT ) ≥ωn.
(cid:16) ∞(cid:16) n+1,k0 i0 j>i0 j (cid:17)(cid:17)
Therefore ot(f (U))≥ωω. S
∞
There is a straight correspondence between cascades and < - sequences.
∞
Let u be an ultrafilter, take sequence u = u > u > ... > u and
0 ∞ 1 ∞ ∞ n
functions f :ω →ω - witnesses that u > u .
m m−1 ∞ m
We will build a monotone sequential cascade V which correspond to the
sequence above with respect to some U ∈ u. In this aim we build a sequence
of cascades (W ) . Take any monotone sequential cascade W of rank 1 and
i i≤n 1
label elements of maxW by natural numbers by any bijections. Clearly W ∈
1
10