Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
883 Cylindric Set Algebras
Cylindric Set Algebras and Related Structures By L. Henkin, J. D. Monk, and A. Tarski On Cylindric-Relativized Set Algebras By H. Andreka and I. Nemeti
Springer-Verlag Berlin Heidelberg New York 1981
Authors
Leon Henkin Department of Mathematics, University of California Berkeley, CA 94720, USA J. Donald Monk Department of Mathematics, University of Colorado Boulder, CO 80309, USA Alfred Tarski 462 Michigan Ave. Berkeley, CA 94707, USA Hajnalka Andreka Istvan Nemeti Mathematical Institute, Hungarian Academy of Sciences Realtanoda u. 13-15, 1053 Budapest, Hungary
AMS Subject Classifications (1980): 03 C55, 03 G 15
ISBN 3-540-10881-5 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10881-5 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of iltustTations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. 9 by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Introduction
This volume theoretical
is devoted to a comprehensive
treatment of certain set-
structures which consist of fields of sets enhanced by addi-
tional fundamental
operations and distinguished
elements.
The treatment
dimension
~
is largely self-contained. Each of these structures has an associated infinite ordinal; Let 3
R
R
their basic form is well illustrated
be an arbitrary
set, and let
of all triples of elements of
of subsets of relative to tions
3R
~
R .
Thus
~
~ = 3 .
is a non-empty collection
intersection,
We shall assume that
CO, Cl, C 2
in the case
be a field of subsets of the set
closed under union,
3R .
, a finite or
~
of cylindriflcation,
and complementatlen
is closed under the three opera-
where
C 0 , for example,
is the
operation given by:
CoX = {<x,y,z> : (u,y,z> ~ X
C0X is the cylinder formed by moving and
C2X
are similarly
t h a t the diagonal planes
for some
X
parallel
D02
(resp.,
and third (resp.,
D01, D02, D12
DI2 )
a r e in
3).
coincide.
is called a cylindric
3R
whose first
A collection field of sets
fields of sets and certain closely related
structures are the objects of study in this volume. as collections
We also assume
.
second and third) coordinates
Cylindric
CIX
~ ; h e r e , for example,
consists of all triples of
satisfying all of these conditions (of dimension
to the first axis.
related to the second and third axes.
D01={<x,x,y>: x , y ~ R } Similarly
u, with xER};
of sets, hut as algebraic
Considered not merely
objects endowed with fundamental
iV
Operations and distinguished elements, cylindric fields of sets are called cylindric set algebras. of dimension
~ , and
Cs ICs
is the class of all cylindric set algebras is the class of algebras isomorphic to them.
In much of the work, general algebraic notions are studied in their application to cylindric set algebras.
We consider subalgebras, homomorphisms,
products, and ultraproducts of them, paying special attention, for example, to the closure of
ICs
and related classes under these operations.
In
addition, there are natural operations upon these structures which are specific to their form as certain Boolean algebras with operators, such as relativization to subsets of 3S
with
3R
and isomorphism to algebras of subsets of
S ~ R , and there are relationships between set algebras of
different dimensions. Although, as mentioned, the volume is largely self-contained, we shall often refer to the book Cylindric Algebras~ Part I, by Henkin, Monk, and Tarski.
Many notions touched on briefly in the present volume are treated
in detail in that one, and motivation for considering certain questions can be found there.
Indeed, the present work had its genesis in the decision
by Henkin, Monk, and Tarski to publish a series of papers which would form the bulk of Part II of their earlier work.
Their contribution to the present
volume is, in fact, the first of this proposed series.
As their writing l
proceeded, they learned of the closely related results obtained by Andreka and Nimeti, and invited the latter to publish jointly with themselves. Thus, the present volume consists of two parts.
The first, by Henkln,
Monk, and Tarskl, contains the basic defintions and results on various kinds of cylindric set algebras. parallel to the first.
The second, by Andrlka and Nemeti, is organized In it, certain aspects of the theory are investigated
more thoroughly; in particular, many results which are merely formulated
V
in Part I, are provided with proofs in Part II.
In both parts, many
open problems concerning the structures considered are presented.
The authors Berkeley Boulder Budapest
Table
First
Part: L.
Cylindric set algebras Henkin, J.D~ Monk, and set
and related structures, by A. T a r s k i ..................
2.
Relativization
....................................
12
3.
Change
....................................
33
4.
Subalgebras
.......................................
56
5.
Homomorphisms
.....................................
66
6.
Products
7.
Ultraproducts
8.
Reducts
9.
Problems
base
..............................
i
Various
of
algebras
contents
1.
..........................................
References
Second
of
.....................................
4
73 86
...........................................
122
..........................................
126
............................................
128
Part: On cylindric-relativized set algebras, by H Andr~ka a n d I. N ~ m e t i .............................. concepts
O
Basic
i
Regular
2
Relativization
.....................................
153
3
Change
.....................................
155
4
Subalgebras
........................................
185
5
Homomorphisms
......................................
209
6
Products
7
Ultraproducts
8
Reducts
9
Problems
References
Index
of
symbols
Index
of
defined
and
cylindric
of
base
notations
131
set
algebras
.......................
132
.....................
145
........................................... ......................................
222 229
............................................
261
...........................................
310
............................................
314
...............................................
317
terms
.........................................
322
Cylindric
set algebras and related structures
by L. Henkln,
The abstract the book [HMT] mentioned,
J.D. Monk, and A. Tarskl
theory of cylindric algebras
by the authors.
primarily
is extensively
Several kinds of special
for motivational
purpose of this article
I)
purposes,
developed
set algebras were
in that book.
It is the
to begin the examination of these set algebras
more detail.
The simplest and most important kind of set algebras are
the cylindric
set algebras
introduced
we shall refer to items from
in 1.1.5.
(Throughout
Recall that the unit element of any
set algebra
(Cs)
is the Cartesian power
and the other elements of DKk
of
~
is
the set
A
and for each
of cyllndrlflcatlon
are subsets of
ix E ~U: x K = x k }
fundamental Boolean operations mentatlon;
K < ~
of
~
~U
of a set
~U .
K,h < ~
(the base),
"
are union, intersection,
the fundamental operation
their definition is similar to that of a cylindric (cf. 2.2.11),
U
cylindric
The diagonal element
for each
by translation parallel to the
set algebras
this article
~ - dimensional
CK
the
and compleconsists
Kth axis of the space.
Several other kinds of set algebras were briefly considered
cylindric
in [HMT], and
set algebra:
generalized cylindric
weak
set algebras
(of. 1.1.13), and what we shall now call generalized weak cylindric algebras
(cf. 2.2.11).
unit elements
in
[HMT] by number without explicitly mentioning
that book).
~
in
The algebras
set
of each of these kinds have for their
subsets of a special kind of some Cartesian
space
~U , while
I) This article is the first in a series intended to form a large portion of the second volume of the work Cylindric Algebras, of which Part I has appeared ([HMT] in the bibliography). The research and writing were supported in part by NSF grants MPS 75-03583j MCS 77-22913.
the fundamental a
Cs
operations
of any such algebra are obtained
, with unit element
to the unit element
~U
of the algebra discussed.
these several classes
of cylindric
general class of set algebras algebras, space.
, by relativization
set algebras
in any detail,
relativization w h i c h are
directly relevant
first-order
discourse
ourselves to the aspects
to our d i s c u s s i o n
of
of those set algebras
A
forms an
for
~
free
regularity:
if (Here .)
~
Furthermore,
x E B ~gx
Regular
,
f E x
,
, the dimension
of
of [HMT]).
and any
A
field of sets, (for the
Thus the above
set algebra i. Ii.i
K < ~ ~
~
cylindric
of variables
in the sense of
for the finitely many
in
structure
~ - dimensional
of a cylindric
is locally finite dimensional
stems from the
, the collection
see the Preliminaries
is the universe
except possibly
set algebras
Given any relational
is the length of the sequence
collection
1.6.1
set algebras.
the class of all cylindric-relativized
of cylindric
language
~ a formula of A]
notation used here,
see
that are
CA's
following construction.
g E x
of
set
of those algebras
from full cylindric
restricting
Much of the importance
occurs
2.2)
in w h i c h the unit elements may be arbitrary subsets of a Cartesian
We shall not discuss here, however,
~
To unify our treatment
that of the cylindric-relativized
These algebras are simply subalgebras
where
(in the sense of
set algebras we use here as the most
obtained by arbitrary relativizations
[~:
from those of
~
.
This algebra
, since
such that the
CK~ ~ Kth
variable
has an additional property of
g E
C
set of
, and x
, is
Ax~f = ~xlg
, then
~K E ~ : cKx f x}
set algebras will be discussed e x t e n s i v e l y
The article has nine sections.
=
later.
In section 1 we give formal defini-
tions of the classes of set algebras w h i c h are studied in this article and we state the simplest relationships found in later sections of the paper. ships are established,
b e t w e e n them; the proofs are In section 2 some deeper relation-
using the n o t i o n of relativization.
is concerned with change of base,
treating
Section 3
the question of conditions
;
of
under which a set algebra with base different base L~wenheim,
W
U
; the main results are algebraic versions
Skolem, Tarski theorems
are also found in section 7). subalgebra is investigated cular attention
is isomorphic to one with a of the
(some results on change of base
In section 4 the algebraic notion of
for our various
set algebras,
paying parti-
to the problem about the minimum number of generators
for a set algebra.
Homomorphisms
of set algebras are discussed
tion 5, and products, along with the related indecomposability are studied in their application 7, devoted to ultraproducts results in the paper.
of set algebras,
In particular,
less trivial of the relationships described
to set algebras
in section 6.
gives perhaps
in secnotions, Section
the deepest
it is in this section that the
between the classes of set algebras
in section 1 are established.
set algebras are discussed in section 8.
Reducts and neat embeddings of Finally,
in section 9 we list
the most important problems concerning set algebras which are open at this time, and we also take this opportunity of the problems
stated in
For reference
[HMT]
to describe the status
.
in later articles, we refer to theorems, definitions,
etc., by three figures, e.g. 1.2.2 for the second item in section 2 of paper number I, which is just the present paper
(see the initial footnote).
The very most basic results on set algebras were first described in the paper Henkin, Tarski [HT]. in Henkin, Monk
[HM]
.
Other major results were obtained
In preparing
the present comprehensive
sion of set algebras many natural questions arose.
discus-
Some of these ques-
tions were solved by the authors, and their solutions are found here. A large number of the questions were solved by H. Andr~ka and I.N~meti. Where their solutions were short we have usually included the results here, with their permission, theirs.
and we have indicated
that the results are
Many of their longer solutions will be found in the paper
[AN3]
4
Various
following
set algebras
this one, which is organized
I.i.i
parallel to our paper; a few
of their related results are found in
JAN2],
course of our article we shall have occasion of their related results.
JAN4]
, or [N]
.
In the
to mention explicitly most
We are indebted to A n d r ~ k a and N~meti for
their considerable help in preparing this paper for publication, The following set-theoretical If
f E ~U
that
,
~ < ~ , and
(fK)>~ = fk u
if
u E U , then
k # < , while
reasons we sometimes write
1.
D e f i n i t i o n I.I.I. V ~ ~U
.
For all
notation not in [HMT] fK u
is
the
(fK)K = u . u
f(K/u)
in place of
will be useful.
member of
~U
such
For typographical fKu
y~arious set al$ebr~as
(i)
K,~ < ~
Let
U
be a set,
~
an ordinal,
and
we set
D[V] = [y E V : YK = Yk}
and we let
every
C IV]
be the mapping
SbV
into
SbV
such that,
V
u E U]
is implicitly understood we shall write
simply
A
= [yEV
K
for some
(ii)
: Yu E X
- dimensional
is an
iff there is a set
and a set
U
D
cylindric-relativized
V ~ ~U
such that
A
V
and
K < ~ ), and containing as elements
C IV] K
(for each
closed under all the operations
(for all
0
; if we
set algebras, .
The inclusions
1.1.3
Various set algebras
7
C i s ~ .
crsreg
Gws ic
Gwsreg
i
~
~
sa,~
~ ~ G!reg
C s ~ .
Ws ~reg = W s
C reg 0~
Figure 1.1.3
ICrs I ~ C ~
eg
= I s
IGws
iCrs reg = IGws reg
ICs
r
iCs reg IWs
= IWs reg
Figure 1.1.4
In case
~ < w , the classes
and
; furthermore, under this assumption each member of any of
Gs
these classes is regular. to subdirect products of Gws's
and
Ws's
phic to a regular
. Gs
Ws
and
Cs
coincide, and so do
In the general case, Cs's
Every
Ws
Gs's
Gws
are isomorphic
and conversely; similarly for is regular.
Every
Gs
is isomor-
and to a subdireet product of regular
Cs 's.
Proofs of these facts and the relationships in the diagrams will be found at the appropriate places in this paper. We begin our discussion by describing some degenerate cases of the
8
Various
notions which
in
I.i.I
follow
, and giving
easily
omit proofs which
Corollary element
V
1.1.5.
Let
N
V = 0
iff
be a
(iii)
If
of > 0
and
U ~ 0 , or
Corollar~ ofU (p) = ~U
.
Cs
the paper we
w i t h base
U
and unit
Hence
Crs I
for
1.1.7.
V = 0 .
frequently
~ < ~
,
U
make
such a s s u m p t i o n s
is any set, and
~ < 0~ we have } , where
If
~
Gs
= Gws
p E ~d
, if
~of is the unique
as
, then
0 < ~ < w
Cs~
with universe
, where
the base
2
Gws
w i t h every
subbase
I ;
~I
and
Corollarx. 1.1.9.
Crs 0
(i)
(ii)
For
of ~ I ,
Crs
(iii)
For
of ~ 2 ,
Crs
N2
are the unique Furthermore,
is
0 .
For any
,
_c CA of
, where
Crs 0 = Gws 0 = Gs 0 = [~i,~2]
respectively.
of any
having
Gs
Crs I = Gws I = Gs I = Cs I = Ws I U [~}
with universe
and
is a
is a discrete
then
1.1.8.
Cs 0 = Ws 0 = [~2} I
= 2 .
Cs 0 = Ws 0 .
only one element,
Corollary
IAI
.
U = 0 , then
If
= Ws~ U [
Corollary
then
V ~ 0 .
1.1.6.
I , and finally
full, and
the classes
Throughout
V = [0}
V ~ ~0}
of this theorem we w i l l
universes
Crs
IAI = I ; if
of = 0 , then
we have
between
seem trivial.
If
of > 0 ,
inclusions
from the definitions.
(ii)
is the
those
1.1.5
.
(i)
Because
set algebras
~ CA
of of
Cs
U Gs
Crs0's
every
of
U Ws
Crs 0
~
U Gws
and with is
c CA
I.l.lO
Various Proof.
Both
we construct
~ E Crs
IuI > 1 , taking w i t h unit
cerning tion,
l9
Crs
trivial. by c h o o s i n g
, and letting We have
0 .
Thus
~
role
(iii),
U
with
be the full
D IV] = 0
Crs
so that
V =
axiom
(C6)
results
con-
'
to satisfy
,
is complete
w h y we
is introduced
shall not give many
Just
as indicated
to unify
in the introduc-
some definitions
and
results,
in our discussion.
If
I.I.I0.
any set
~
fails
in these papers;
an a u x i l i a r y
To e s t a b l i s h
~ ~ 2 , then
Cs
~ Gs
and
R~Cs
~ Crs
c~
We shall Gws
ffz2
9
O1
explains
this class
are
(l.l.l(iii)).
and the proof
Corollary SR%Cs
for
Ivl I0 ) =
the class
and plays
(ii)
~CA
V
'
~ ~ CA R
Corollary
and
v = ~u ~ -01 n [~U]
element
Ivl whence
(i)
set algebras
be able
c RgCs ~
although Rs [HR]
for
P~Cs O
c Crs
~ a 2 =
for
and Prop.2.3
Corollar~f C Gs
to strengthen
these
below by showing
(see 1 92 . 12 - 1.2,13).
Crs O
mnd
~ > 1
RgCs I
but we
=
Cs I
that
It is known
,
that
we have
shall not give an example
here"
see
and
Cs
of [AN3] (p.155),
I.I.II.
If
~ ~ w
, then
Ws
~ Gws
C Crs
~ Gws
Theorem
I 1.12.
Let
~
be a
9
where
results
Then
for all for any X
with
unit element
~
~. (pi) N ~ U (pj) = 0 ui j
~U! pi) E A l
Gws
E A
the
Also
assume
following
~u(pi)
~iEl
for all distinct
i E I .
~
i,j E I .
Assume
-i
that
~ ~ i . conditions
are equivalent:
'
10
Various (i)
X = ~u~i(pi)
(ii) and
&X
=
X 0
i E I ; (under
=
) such that
X ~ 0
.
(i) =
(ii)
~ U ~ pi) = 0 .
y E Y
for some
is a minimal element of
Proof. and
1.1.13
set algebras
and let
Clearly for any
Now suppose
x 6 ~, (pi) ui
,
such that
.
(~F)?y
that
.
we have
0 ~ Y ~ ~.(pi) ui
be arbitrary.
= (~F)~x
i E I
~us
There
and
0 ~ ~U~ pi) Ay = 0 .
is then a finite
x E c(F)Y = Y, and hence
Fix F ~
Y = ~u!Pi~ l
(ii) = # 0 . by
(i)
Assume
Since &~U! pi) = 0
(ii)
ui
established,
,
Gs
such that
X n ~U~ pi)
by 1.6.6
while by the implication
(i) =
Hence (ii)
already
,
~,u i(pi) = X N ~.u i(pi)
Now we discuss regular
iEl
(pi) &(X ~ U i )= 0
we have
X = X n ~ (pi) ,
Ws
(ii), and choose
Thus
set algebras.
(i)
holds.
In the case of the classes
the definition assumes a simpler form mentioned
Cs
in the
introduction.
Corollary I.I.13. X E A
.
~
be a
Cs
with base
U , and let
Then the following conditions are equivalent:
(i)
X is regular;
(ii) gEX
Let
for all
f E X
and all
(i) .
Trivial.
g E ~U , if
~XI f = AXJ g
then
.
Pro0f.
(ii) =
the hypotheses is obvious.
of
Suppose
(ii)
.
If
therefore
(&X ~, U I )~ fOgO = ( ~ X U I ) I g
The next two Corollaries
.
(i) =
(ii) .
Assume
(i)
0 E AX , the desired conclusion 0 ~X.
Hence
g E X
Then by
f0 gO E X
since
0 ~X
(i) .
are proved in the same way as
1.1.13.
and g E X and
1.1.14
Various
Corollary and
let
X E A
(i)
X
(ii) g E X
1.1.14. .
Let
Then
~
be a
the following
11
Ws
with
conditions
f ~ ~U ~p~
unit element are equivalent:
is regular; for all
f E X
and all
g ! ~U (p)
, if
AXI f = &X1g
then
.
Corollary where
Yi N y j
conditions
1.1.15.
= 0
Let
for
~I
i # j
be a
, and
Gs~
let
w i t h unit element
X E A .
Then the
~ i E l Y'l '
following
are equivalent:
(i)
X
(ii)
is regular; for all
i E I , all
&XI f = ~X1g
then
No analogous
simplification
known.
cylindric
Weak
Corollary
Proof.
Suppose X E A
f,g E ~U (p)
, there
Hence
[~N(F~X)]If=
I.I.14
,
X
' and all
g E CYYi , if
of the notion
set algebras
of regularity
are always
for arbitrary
Gws
regular:
Ws reg = Ws
that
~
is a
Ws
, f 6 X , g E ~U (p) is a finite
F ~ ~
[~(F~AX)]Ig,
w i t h unit element , and with so that
~X1f
= AX1g
(~NF)If
.
~U (p) Since
= (O~NF)Ig
g E c(FN~x)X
9
= X .
Thus by
is regular.
Corollary = Gs reg
f E X f3 ~Yi
g E X .
1.1.16.
A l s o assume
Gs
set algebras
1.1.17.
, and
Gws
If
~ < w
, then
Cs
= Cs reg
, Ws
= Ws reg
,
= Gws reg
reg Corollary
1.1.18.
If
~ ~ w
,
then
Ws ~reg c Gws
~
~ Crs reg
and
's is
~2
Relativization
1.1.19
Cs~ eg c Gs~ eg c Gws~ eg .
Proof.
To produce a member of
Gws reg N Ws reg , let
two disjoint sets with at least two elements, and let be arbitrary.
Let
~
be the full
Gws
U
p E ~U , q E
with unit element
Then it is easily checked, along the lines of the proof of E Gws reg ~ Ws
It is also clear that
the two-element Gws
with unit element
Crs
(provided that
Finally,
if
of the full
U
and
p
V
are as above and
Corollary 1.1.19. and
~V
If we let
~
that be
~ E Crs reg
For
~ m w
~
is the minimal subalgebra
~U U ~V
we have
, then
~ E Gs reg N Cs
Cs reg = Cs
, Gs reg c Gs
Gws reg c Gws
Proof. regular. Then
be
~U (p) U ~V (q) 1.1.16,
, we see that
V
has more than one element in its range).
with unit element
Gs
~ ~ Gs {p}
and
By
Let
~i
~2(P) E A
in a regular
Cs
I.I.Ii
it suffices to exhibit a
be the full and
Cs
A~2 (p) = 0 .
the only elements
with base
2 .
Hence
~2 (p)
X
such that
Cs
which is not
Let
p = o~ .
q = (I : ~ < ~ )
ments.
we have
be the full
V = ~U (p)
B C [XEC
is a
More specifically,
.
Say
Gws~'s with
R6Cs
and unit set
V E B ; in fact,
and
17
Let
widely-distributed
be a
(Pi) UiEl W i
is
We call
~
UO .
~ if
Usually a
Gws
J
Hence
9/ ~ ~L ~D . V
in that the subbase Gws
is normal,
, ~ ~ o~ , and suppose ~ (Pi) (P.) Wi n ~W i j = 0
W i = W. J
W i ~] W. = 0
A .
is unusual,
~ , with
normal if
,
or
whenever
W i [~ W. = 0 J
U1
in the
the unit
whenever
i,j E l
for all
i ~ J ; compressed ,
i,j if
;
and
18
Relativization
W i = W. J Cs
for all
i,j
Thus every
blished
in section
Gws
7
is a normal
~
is a direct
is a
P~Cs
It follows
of this p a p e r
widely-distributed
to a
Gs
Gws
factor of a
with
that
show below
in 1.2.9
It
is also clear
Cs
w h i c h has
Gws
at least
Gs
Gws
Gws
c P~Cs
Henkin,
lemmas
~ ~ w
, since
~
; so
by 2.2.10,
can always be
the same base as 3 ~ ~ < w
and
~ ~
~ ~ w
.
We
is a we have
the last part of 1.2.5 holds
then
Gs
~ ~ ~
that in case
of course,
, since we have
that
that every compressed
with
in case
Gws
~ Gws
= Gs
.)
We also have
1.2.9
is due
to
jointly by H e n k i n and Monk.
two lemmas about a r b i t r a r y recall
is i s o m o r p h i c
the same base as
having
1.2.11 has b e e n obtained
We n e e d these
(Thus,
~ < ~
in case
while
Cs
~
, we show in 1.2.11
= P~Cs
's with
from a
that this is true
On the other h a ~
Gws
same base.
o b t a i n e d by r e l a t i v i z a t i o n
for
, and every
from results w h i c h will be esta-
that e v e ~
We do not k n o w w h e t h e r a normal
Gs
Gws
~
is a c o m p r e s s e d
Gws
.
1.2.7
CA
~[b ~
's.
In c o n n e c t i o n w i t h
is a
CA
for e v e r y
kJ
~ CA R all
and every
K ,k
a ~
To say
) = 0 .
the first p o s s i b i l i t y
yields .
X
Gws
:
f s CI(X N C 0 ( C I X N
the form of
b = I .
does not hold
in every
the following
fol'l 0 = /m
fol E X , and
:
9 -c0(clY 9 -d01 ) = 0
CI(XNC0(CIXN~XNV))
let
33
b 6 U
and
b = 0 ,
f0 a = f E D01 6 V
.
to a
Thus
Gws
3
G i v e n a set algebra 9/ isomorphic IUI = IWI
to a set algebra
, where
Theorem
the a n s w e r
1.3.1.
(i) f
element
V
.
Suppose
For any
X E A
let
phism
from
~
9/
onto a
w i t h base
U , and given
w i t h base
W?
is obviously
Let
9/
is a one-one
FX = [y s ~W : Crs
be a
~
We
first consider
w i t h base
function
w i t h base
set
W
, is
the case
yes.
Crs
f-1 o y
some
~ X} W
from .
Then
U
U
and unit
onto a set F
W
is an isomor-
and unit element
FV
.
.
'
34
Change of base
(ii)
If in
(i)
~
is a
Cs
(iii)
If in
(i)
~
is a
Ws
is a
Ws
If in
S. N S. = 0 l j
for
U i E i f f f * S i , where (v) where
~
is a
i ~ j , then
Gs ~
(i)
is a
UiEl~(f*si )(f~
~
is a
Cs
with unit element
~U (p)
with unit element
~iEl Si ' where v
is a
f S i N f Sj = 0
~S (pi) N ~S (pj) = 0 i i
element for
If in
(i)
, then
for
for
Gws
Gs
with unit element
i r j . with unit element
i ~ j , then
, where
~
is a
~(f*Si)(f~
II ~(pi)'" ~iEl ~i Gws
with unit
N ~(f*Sj) (f~
the function
F
true, e.g., if
1.3.1
to the special case
~
is a full
is rather extensive.
Remark
U = W , in some cases
is an automorphism of the algebra
general kind of Galois theory.
1.3.2.
Cs
~
.
This is always
In this way one can develop a We shall not go into this theory, which
See, e.g., Daigneault
[D ] .
The less trivial question concerning change of
base arises when the two bases have different cardinalities. our discussion of this case we shall show that for each
base.
= 0
i ~ j .
If we apply
is a
then
Uw(f~
with unit element
(iv)
1.3.2
Cs
with an infinite base not isomorphic to a
First suppose that
equation
~
3 ~ a < w 9
holds identically in every
fails in some finite
Cs
Cs
To begin
~ ~ 3
there
with a finite
Then, we claim, the following Cs
with a finite base, but
with an infinite base
(this equation is due
to Andr~ka and N~meti, and replaces a longer one originally found for this purpose):
1.3.2
35
Change of base
c(~)[-c(~l)X
+ c(a N 2 ) x . d 0 1
0 1
+ c(~2
~ 9 SlS2C(~ N 2
I 9 -s2c(~N2)x]
To show this,
let
be a
hold identically.
Cs
w i t h base
U
)x
= 1 .
in w h i c h
does not
Then it is easy to see that there is an
XEA
such that the following conditions hold:
(I)
C ( ~ ~ z)X = ~U
(2)
C(~N2)X
(3)
C(c~2)X
N o w let
;
, 0 1 N SlS2C(c~.2)X
c
--DoI
R = [21u : u E X}
I -c s2C(c~.2)X
.
Then
R
.
is a binary relation on
U
satis-
fying the following conditions:
(4)
for all
u E U
there is a
(5)
for all
u E U , not
(6)
R
v E U
(uRu)
with
uRv
;
;
is transitive.
It follows
that
U
is infinite.
Thus
E
holds in every
Cs
with a
finite base. Now we construct a finite fails to hold in tional numbers and discourse Then
',-~ .
If
for some
subbase
Let
~
.
Let
In fact,
some
P E ~W
onto
W
(I)
also holds
of
IWI
D~
~W (p)
, and any
in
1.3.4.
~
with
of
9/ .
Let f
to a
has base Cs I
with
For each n o n - z e r o
U' = [u a : a E A}
Then
.
In a later article
Gs 2 (resp.
is an i s o m o r p h i s m
Cs2)
, and
of
9/
in this series
is isomorphic
to a
base.
and
.
9
be
Gws's W
and assume of
~
that
, then
IWI = IW'I
Then
~ 0
is included q E ~W (p) of the
9/ , and
in the unit element
such
this gives
Namely,
that
KI q
left side of
The i m p l i c a t i o n
9/ > ~
Crs 0
.
C~ ~ < ~ D
in the other direction. Cs
U'
Any
is isomorphic
.
By
w i t h a finite base.
~ < 2 .
E a
base.
9/ .
w i l l be a m e m b e r
Remark
a
~
Cs
for some subbase
< =
N~ 2
of
Gs
71 ,
IWI < ~ N 0~ ,
c~ such that objects,
the base of let
U
and
~ W
has c a r d i n a l i t y be disjoint
~ Iwl
sets w i t h
To construct IUI >
IWl = I .
these Let
1.3.5
Change of base
be the full
Gs
with unit element
with unit element fX = X n ~U
37
~U .
~ U U ~W , and
The mapping
is a homomorphism by
f
from
2.2.12.
A
~
onto
the full B
Cs
such that
Clearly the above properties
hold.
We can strengthen a part of
1.3.3
by use of the following
notion of base-isomorphism.
Definition 1.3.5. by
~
the function (ii)
elements
Let V
~
and
F
With
f
defined in
and Y
(1)
9
be
as in Theorem 1.3.1, we denote
1.3.1.
Crs's
respectively.
with bases
We say that
isomorphic if there is a one-one function is an isomorphism from
~
onto
9
f
U ~
and and
mapping
W
~ U
and unit are base-
onto
W
such
.
The following result is an algebraic version of ~he logical result according to which any two elementarily equivalent finite structures are isomorphic;
it is due to Monk.
Theorem 1.3.6. regular
Cs's
has power
Proof. bases
U
and
Let
~
and
, and assume that < ~ n w , then
~
Claim.
be locally finite-dlmensional
~ ~ ~ and
.
~
If the base of either
W
respectively, and that ~
onto
~
9
~
and
IUI = ~ < ~ N ~ . .
Note that by
9 Let
have G
1.3.3,
Now we need the following
For each regular
Cs
c~
~
or
are base-isomorphic,
Assume the hypotheses, and suppose that
be the given isomorphism from IWI = IUI .
9
having a base
T
of power
< ~ N
38
Change
there is a function s
of
T
C
into
(a) foT
S
C
of base
w h i c h assigns
1.3.6
to each
T E
~
a mapping
such that:
for any
x E C
and any
f s ~T
we have
f E ~s~x iff T
~ x ; (b)
if
F
is an isomorphism
w i t h base of power E~
< ~ n ~ , then
from
~
Fs~x=s T
onto any regular Fx
for all
x E C
and
.
This c l a i m is of course related to our considerations I.II.
For
In fact, let
~ ~ w
x s C
, and let
let
k0 =
s
.
k~_ I ...
f E s~y
k _1
f~ I
K 0 < . . . < Ks_ I , ordinals
be the first
K0
... s
TK~_ I sk 0
iff
, with
I~XI < k
, where for all
with unit element
with unit element
X E A .
~U (p) , then
9/
is
~W (p)
O~
(iii) phic to a
Cs
(iv) then
9/
9/
is a
with base If
~I
If W .
9/
Cs
and
is a
K = K I~I , then
9/
is ext-isomor-
W .
is a regular
is e x t - i s o m o r p h i c
(v) with base
If
Gs
(resp.
to a regular
Gws
then
~
Gs
Cs ) and (resp.
(i) (c)
Cs )
is ext-isomorphic
with
to a
holds, base
Gws
W .
48
Change of base
(vi) base
W
If
~
is ext-isomorphic
(ii):
Crs
We assume given w e l l - o r d e r i n g s
Note that
I~I ~ IAI
.
of
U
and
V .
There is a subset
TO
S U Rgp c T O , and
X N ~T 0 @ 0
N o w suppose that
0 < ~ <
: K< ~} 9
to-one function from a subset of
Theorem 1.3.1,
"
and
U0
~/' A
U I , and it can
onto
UI .
onto the fuli
By Ws
~"
are finite sets such that
I q = (~A)
( ~ N (F U &)) I (f' o p) = (~,~ (F U A)) I q 9
f is a one-
Thus
I q~ , clearly ~"(f~l op) = ~U q) , as
desired.
Remark 1.3.21. imply
(i)
It is easy to see that in 1.3.20,
in general.
(ii)
The condition of base-isomorphism
cannot be replaced by isomorphism. theorem of Andr~ka and N~meti.
in
does not (i)
This follows from the following
56
Subalgebras
Theorem 1.3.22. with base
U
~
is a locally finite-dimensional
q E ~U , then
and
unit element
If
1.3.22
~
is isomorphic
with
Ws
~U (q)
Proof.
Let
~/ have unit element
fX = [u E ~U (q) : there is a
Using the regularity of isomorphism
to a
Ws
from
~
~
K < ~
v E X
(1.1.16),
into
~g~
~U (q) .
Now let
and
so that
~CKXI u = A CKXI v .
with
X E A
If
Ws
f
is an
with unit element
u E fCKX , choose
w 6 ~U
let
~Xl u = ~XI v]
the full
X E A .
= ~u~
For any
it is easy to see tlmt
, ~
Define
~U (p) .
v E CKX
be setting for any
k
Iog2(8 + ~_-~i ) ,
generalizing 1.4.6 and 1.4.7. Note that these results on the function discussion in 1.3 of change of base. if a
Cs
q
are relevant to our
For example 1.4.6 implies that
cannot be generated by a single element, then it is not even
isomorphic to a
Cs
with base of power
~ ~ .
R0 J. Larson has shown that for phism types of one-generated [EFL]
Cs 's.
show that there is a countable
generated
2 ~ ~ < w
ICs
isomor-
Cs 2
not embeddable in any finitely
Cs 2 .
and
IWs
under directed unions. ICs re~=
2
P. Erdos, V. Faber and J. Larson
In 1.7.10 and 1.7.11 we show that ),
there are
(for
IGs
~ < w ) , and
and
IGws
ICs reg N Lf
Andr~ka and N4meti have shown that
are not closed under directed unions for
~ ~ w .
(for arbitrary are closed ICs
and
It remains
68
Homomorphisms
open whether
IWs
1.5.1
is closed under directed unions for
~ m w ; the
proof of 1.7.11 may be relevant to this problem. We also should mention that Problem 2.3 of
H. Andrlka and I. N4meti have solved
[KMT] by showing that for each
finitely generated
Cs
~ > 0
there is a simple
not generated by a single element; see
[AN2].
HomomorDhisms
5.
The following result about
CA's
in general will be useful in
what follows.
Theorem 1.5.1. = {x 9
d :x E I
Proof.
and
Let
~
be a
CA
and
I s Lb~ .
Then
Sg(~)l
d E Sg~[0}}
This is clear since
~)I/I
is a minimal
CA
Turning now to set algebras, we begin with a result concerning simple algebras.
Theorem 1.5.2.
(i) Any regular locally flnite-dimensional
Cs
with non-empty base is simple. (ii) (iii)
Proof.
Any locally finite-dimensional For
a < ~
any
Cs
Trivial, using
Corollary
1.5.3.
Let
is simple.
with non-empty base is simple.
2.3.14.
~ 6 Cs
Then the minimal subalgebra of
is IuI if Iul 0 , then
0 .
Let
Ws
is an error
.
p = K x [0]
Ws
subalgebras
, where
that
fact:
(resp.
or
U [I}
[De]
Let
Cs
~K,X 2
to show that
finite
, a contradiction.
HWs a ~ ICs a
struction based
5 = Hk
In contrast
element
IZd~
, so we only need
y ~ c(r)(U6Eg(U[c6y
k < ~
with
I g ( ~ ) [ c K x . - x :K < a}
x/l E Z d ~
Suppose
(*)
I
~ E H~
1.5.6
K
I = IX : X E A, that
to the c o n t r a r y 1.5.3
(resp.
,
respectively.
~
and
~I/l that ~
unit
IXl < ~]
.
is not iso~i/I have
is isomor-
The two formulas
I .5.6
Homomorphisms
~(K • K) ~ 0 ,
hold in
~'
Noting that
d((~ + l)
and hence in ~/I
k,~ < ~ 9
Set
X
of
A
such that
X
K .
to
(*)
0 ~ X/I ~ d (~/I) k~
for
k < 5}
.
the above construction
to
HCs reg ~ ICs Andr~ka and N~meti have shown that
(8)
It is clear that
~ E HCs
clearly
has power
satisfies the above conditions.
(7)
of any
~
for exactly one
Andr4ka and N4meti have modified
show that
Gs
~ HCs
ICs reg ~ HWs
, since the minimal subalgebra
is simple or of power
~ E Gs
(9)
I
by 1.5.3, while there are
without this property.
The inclusion
It implies that
IWs
In fact, clearly
Ws
J
Hence the base of
X = Ix E ~K :xk ~ 0
It is easily checked that (6)
.
• (K + 1)) = 0
is atomless, we can obtain a contradiction
by exhibiting an element all
~J
7~
lWs HCs
c ICs reg
will be established
in
1.7.13.
, and this inclusion is easy to establish.
~ SHCs
since the full
Ws
with unit element
~
~U [p) lWs
is a homomorphlc ~ SHCs
~ HSCs
Cs
with base
It remains open whether
(II)
Andr~ka and N~meti have shown that
"Cs reg''
ICs
= HCs reg
the inclusion holds trivially if or
(12)
Hence
or
H(Cs
"Cs "
HCs
= HCs r e g .
~ Lf ) ~ I C s
is replaced by
'Us " From the definition of characteristic we know that if
has characteristic K .
U .
= HCs
(I0)
By 1.5.1,
istic
image of the full
K ,
~ > ~
, and
IBI > 1 , then
The meaning of characteristic
~
has character-
for set algebras,
described
in 1.5,.3 and 1.5.4, is further elucidated by a result of Andr~ka and N~meti according to which for each cardinal
K > 2
there is a
Cs reg ~/
72
Homomorphisms
with base of power Gws
~
in
Recall
an
element
~
l~xl < W , then
(14) a
Cs reg g
simple. base
from
has base of power
1.3.9
N
The c o n s t r u c t i o n
Now if
f E Ho(~,~)
tN N Cs
Finally,
by
(15)
for every
c Ies reg
H~ n Cs
is simple:
let
F ~ ~
and
for any
~
2.3.14
there
is
and
full
N
is not
Cs
with
Note that for any
and every
K 6 ~ N F
, then each
we have
y 6 [f[x] : x 6 ~ ]
it
By
(4)
is clear
above,
that
N
it
feIlows
is not
K
and some
Contrasting to
Figure 1.5.7
that
IN ~ I C s r e g .
K ~ 2 with
(4)
above,
there is an 1Zd~ I > 2 .
(15) , Andr4ka and N~meti have shown that
there is an IZd~l
9 6 }~/
~ 6 Cs r e g
simple.
Generalizing the construction given in
not simple, but (17)
~ cs reg
b e the
: x s Re]
~ E Cs
~ Cs r e g .
K ~ 2
, if
~ , with
, a n d so by Theorem ~.3 o f [AN3] we b.ave
with base
(16)
~ E Cs r e g
, A n d r 4 k a and N4meti h a v e c o n s t r u c t e d
Andr4ka and Nimeti have shown that for any E Cs reg
> K 9
we have
= 0 .
(*)
(13)
~ = ~)[[x]
for every finite
satisfies
such that every
~ Cs reg
to
such that
~ , and let
that
~
x , not in the minimal algebra of
In contrast
c r( )
Thus
~
F~/ N Cs
Y 6 [[x] : x E ~ }
(*)
< , having a homomorphic image
isomorphic to
(13)
1.5.6
~ 2
~ 6 Cs reg for all
with base
K
such that
~
is
~ 6 }~ 9
simplifies considerably if we restrict our-
selves to set algebras with bases and subbases infinite and to
~ ~ w 9
Let us denote by
Then we
Cs , G s ~
, etc. the corresponding classes.
1.5.8
Products
73
obtain Figure 1.5.8; see (3), (4), (9) .
Here
= ?
means that we do not
know whether equality holds in the two indicated cases,
I Cs
= HSPaoGwsof
tt Cs e g
IooCsreg
HooWs
(~ > ~) Figure 1.5.8
6.
In terms of p ~ d u e t s , Gs's of
and
Cs's ~
Products
we can express a simple relationship between
, and between
Gws
~
this relationship more generally for
's
and
Crs's
Ws's of .
.
We first express
For this purpose it
is convenient to introduce the following special notation.
Definition 1.6.1. let
W c_ V .
Then
any
a E A
rtw~a = W n a .
,
Theorem 1.6.2. ~iEiVi for all
, where i 6 1 .
r~
Let
Let
9~ be a
Crs
of
with unit element
is the function with domain
~
V i [~ Vj = 0 Assume that
be a full for
Crs
i,j 6 1
9 E CA
of
of and
A
V
and
such that for
with unit element i ~ j , and
For each
i E I
~V i = 0
let
~/. i
be
74
Products
the full
Crs~
1.6.3
with unit element
there is a unique
Vi .
f 6 Is~,Pis
)
Then
~ ~ PiEI ~'i
such that
In fact,
r2.v. = PJi o f
for each
i
iEI
.
Proof.
Clearly there is a unique
f
mapping
By 2.3.26,
each
by 0.3.6 (li).
f s Hom(~,PiE~li)
into
PiEiAi
rLvi E Ho(~,~ i)
and satisfying the final condition. i E I , so
B
Clearly
for
f
is
one-to-one and onto.
The assumption
~ E CA
Corollar~/ 1.6.3.
is not actually needed in 1.6.2.
For
~ a 2
we have
IGs
SPCs
and
IGs; eg
= Si~s reg.
Proof.
First suppose that
U i 6 1 ~U i ' where
U i n Uj = 0
for all
Let
i 6 I .
~ E CA
product
of
Clearly Cs ' s
We may assume that
Clearly
CI f
CI f
i 'j 6 1 ' and
for each
i 6 I , and let
~V i = O
for all
i E I
is an ~ omorphism of
~ ~d Pis
U i ~ U 3. = 0
i 6 1 , and again let -I
has unit element U.l ~ 0 ~, ~ , f since
~ ~ 2
~ onto a subdlrect
, as desired.
Second, suppose
for each
~
1
be as in Theorem 1.6.2; clearly and
; say
for distinct
V. = ~U. 1
~ s Gs
' each
1.6.3
a
Cs
with base
i,j 6 1 .
~, ~
be as in Theorem 1.6.2.
and ~
f
onto a
is handled by
In an entirely analogous way we obtain
Gs
.
1.1.15.
Let
U. ~ 0 .
for distinct
is an isomorphism of
The second part of
~.
V i = ~U i
1.6.4
Products
75
Corollary
1.6.4.
For
~ ~ 2
Corollary
1.6.5.
Let
~ E IGs
following
conditions
(i)
~
Proof.
,
By 1.6.3 and 1.5.2
PGws
Remarks
1.6.6. =
1.1.4,
are
For
IGws
None
1.6.8.
of set algebras
l~s
IAI > i ,
a < w
9
Then
the
;
R e m a r k 1.6.7.
Figures
= SPWs
is simple.
Corollary = IGs reg
,
IGws
are equivalent:
~ E ICs
(ii)
we have
,
~ ~ 2 and
PGws reg
1.6.4,
1.6.6
in Figure
1.6.9
= HSPGs
= SPCs
I
1.6.9
(c~ m 2)
,
extend
to
for
\
Figure
= IGs
H, S, P
W e n o w discuss
= IGs
separately).
PGs re~~
= IGws reg
under
and 1.5.8.
~ < 2
PGs
properties
summarized
= HSP~s
(treating
we have
of 1.6.3,
Closure
1.5.7,
(iii)
~ a 2
this
~ ~ I .
of our classes ; see also
figure.
= S~s
76
(I)
Products
For
d ~ 1
1.6.9
the diagram is different;
then the classes are just
five in number, increasing under inclusion:
(2)
[~ E CA
IWs d
(b)
ICs~ = [~ E CAd :~ is simple or IAI = I] ,"
(c)
PCs d
(d)
HPCs
(e)
HSPCs
=
[~ E CAd :~ is a product of simple CAd
's] ;
;
d d
The example
general, for
:~ is simple]
~
(a)
= CA
9/
=
d
SPWs
d
in 1.5.6 (5) also shows that
d ~ w 9
In fact,
(*)
HWs
~ PCs
d
continues to hold for all
in 9/ E PCs d
(3)
Andr~ka and N~meti have shown that
for
d ~ w 9
(4)
To show that
with bases
K,X
PWs
~ HCs
d
c~
for
Cs reg ~ PWs
d > w , let
respectively, where
~
and
and
I < K < k < w 9
since each non-trivial homomorphic image of a
Cs
~
Then
Cs
be
~ PCs reg
Ws's 01
~ X ~ ~ HCs
has a well-deflned d
characteristic, while
~ • ~ does not (cf. 2.4.61 for the definition of
characteristic). (5)
For any
d
we have
SPCs
d
~ HPCs
d
(for
d ~ w
this was shown by
01 '
1.6.9
Products
A n d r ~ k a and N~meti). while
if
For
9/ 6 H P C s
~ = 0
and
For
let
be the subalgebra of
elements.
Suppose
trivial
Cs
must have
0-
w i t h base
where let
~
.
Then
~
with base ~
~i
elements,
For each
But
~ E 9 N 2 ,
Let
a non-
IZ~I = m
1.5.2 PiE~j
we
(iii), it has only
a contradiction. let
and let
For, assume
i E I .
m , and
~.~
since
is simple by j c I .
,
zero-dimensional
result of Koppelberg,
~ HPCs
for all
Cs
= BA
by S. K o p p e l b e r g
PiEI ~ i > ~) , each
~ = P56~,~ 297~
~/
~
be the minimal sub-
that
~D = PiEl~i
be the minimal
h 6 Ho(PiEl~i,~ ) ,
.
For each
5 s ~ N 2
be the term
e(~)d(5• Thus
; say
(SPCs
IAI m = IAI
generated by
for some
~ > ~ .
~ , let
~i 6 Cs ~
~
dimensional
N o w suppose
then
be the full
Since each
~ = PiEj~j
finitely many
algebra of
~
By the above
follows that
Cs
let
~ E HPCs~
III < ~ .
this is w e l l - k n o w n
IAI > 0u
[Ko]). 9
0 < ~ < 0~
77
@
= 1
It follows there is an f 6 Pi61Ci
holds
that
i 6 1
in a
Cs
~ 0
~
~ E w N
is even, and
~i
fi = 1 1 ,
~
~(~9~)--" h f = 0
so
= 0 , so by 2.1.17
has base of c a r d i n a l i t y
has base of power if the base of (~i)
~(~) 9 f = 0
and
.
~ E ~ ~ 2 , so for all
(~i)
fi = 0
~(~)-- ~ hf A(~)hf
iff
for all
such that
b y defining
ments for some E w'~ 2
~3) &
~) .- c(5+l)d((5 + I) • (5 + I))
otherwise.
if
Now define
has
Thus
But
hf E S g ( ~ ) [ ~ ( ~ )
2~ ~
~ E w N 2 is odd.
in these two cases. (Ill),
~ E ~ ,~ 2
~ . ~i
~ .
(~)
ele~ f
Hence
~(~D)f -- 0 , :~6~2]
w h l e h is c l e a r l y impossible. (6)
From
(4)
it follows,
of course,
that
Pes
~ ICs
if
even for
,
78
Products
~ ~ bases
(7)
.
B u t we show in
is isomorphic
to a
Cs
that a product for
~ k w
A n d r ~ l ~ and N d m e t i h a v e n o t e d t h a t
for all
~ > 0 .
and
~
let
Then by
(*)
of
Cor.
1.4
similar: (8)
of
x
Dc
(*)
a n y full
.
Then
(**)
for all
x , if
holds
in every
Lf
infinite
Cs
and
Ws
w i t h base
by the atoms
N SPDc K ~ 2 ,
of
~
.
But the statement
k < ~
in every
(7)
we
then
x = 0
~ E SPDc
~ ~ SPDc
Since
The case of
Dc's
should m e n t i o n
not
In fact, write
IFI,II
full
9 E Cs r e g .
for all
with
every a t o m Ws's
is
has an atom.
with and
Cs's
Cs r e g ~ SPDc
generated
, we have
Ws
Lf's
~ k ~
,
~
of
.
be the
of
cBx = 0
falsifies
fact about
N
, and hence
In c o n n e c t i o n
for
let
JAN3]
, if
in every ~
In fact,
be the subalgebra
for all
holds x
1.7.21
1.6.9
found
= F U g
in
the following
[HMT] :
with
SPDc
F ~ & = 0
general
~ SPLf and
the s t a t e m e n t
c~x = 0
, hence
for all
in every
k E F
SPLf
then
, but
x = 0
fails
in some
Dc
c~ (9)
F r o m 1.6.13
(I0)
Among
important (II)
and subbases use
the q u e s t i o n s
seems
If we
it follows
about
to be w h e t h e r
restrict
1.6.9
of 1.5.7
PWs
Figure
~ IWs 1.6.9 w h i c h are open the m o s t
ICs reg ~ HPWs
ourselves
infinite,
the n o t a t i o n
that
to
~ > ~
simplifies
(17)
.
and
to set algebras
as in F i g u r e
1.6.10,
w i t h bases w h e r e we
1.6.11
Products
79
l=oCs~ = HSP Gws =? reg HP C s _ _
H Cs
re
Ws
tiP
"~ csreg
] ~
I
W
s
(~
> w)
Figure 1.6.10 Again
= ?
means that equality of the classes in question is not known.
Some of the theorems needed to check this figure are in
[AN316,2.
Now we discuss direct indecomposability, subdirect indecomposability, and weak subdirect indecomposability.
We give some simple results
about these notions and then we discuss some examples and problems.
Theorem 1.6.11.
Proof.
Let
0 ~ y E A , choose (~
~
Every full
be a full f E y 9
F) I f = (~ N F)I p .
Ws
Ws
is subdirectly indecomposable.
, with unit element
Then there is a finite
Thus
[p} ~ C(F)y .
So
~U (p) .
F = ~ ~
Given
such that
is subdirectly
indecomposable by 2.4.44.
Corollary 1.6.12. phic to a
Ws
Any subdirectly indecomposab~e
Cs
is isomor-
80
Products
Proof.
By I.I.II and 1.6.4.
Corollary 1.6.13.
Proof.
1.6.13
Every
Ws
is weakly subdirectly indecomposable.
By 0.3.58 (ii), 2.4.47 (i), and 1.6.11.
Corollary 1.6.14.
Let
Then the following two condi-
~ E IGws
tions are equivalent: (i) (ii)
~ E lWs ~ = ~
Proof.
; for some subdlrectly indeeomposable
(i) implies (ii) by 1.6.11.
Corollary 1.6.15.
Any regular
Cs
9 6 IGws
(ii) implies (i) by 1.6.2.
with non-empty base is directly
indecomposable.
Proof. By 1.4.3 and 2.4.14.
Remarks 1.6.16. (1)
Throughout these remarks let
Examples (I) and (II) in 2.4.50 are
~ ~ ~ .
Ws's
which are res-
peetively subdirectly indecomposable but not simple, and weakly subdirectly indecomposable but not subdirectly indecomposable. (2)
To supplement our discussion of homomorphisms we shall now
show that for any
K ~ 2
there is a
morphie image not isomorphic to a
Ws
Ws
with base
K
having a homo-
The first such example was
due to Monk; the present simpler example is due to Andr~ka and N4meti. Let
p =
~K (p) fK # 0
Let
and let
~
x = [f 6 V : ~ ] f ~ p
is even}
Let
be the full
Ws
or the greatest
with unit element K <w
such that
1.6.13
Products
i = ig -- [(c+q)i ]~c(K,s/F)i = [ (c+ q)i ]~si E a i
Thus, indeed,
qK
s f(a/F)
Hence
q 6 c[T]f(a/~)
9
Second, suppose
s/F that
q 6 c[T]f(a/F)
9
Thus
q E T
and
(~ ~ [K}~I q = (~ N [K] J p
for
+ some P E f(a/F) . Let M = [i E I : (c P)i E a i] ; thus M E F . Also + since q 6 T the set Z = [i E I : (c q)i E Vii is in F . Now let + + + i E M N Z . Then (c q)i 6 V i and (~'~ [K])I (c q)i - (c P)i E a i , proving that
(c+ q)i E C K[V i] a i .
We have now verified that
Thus f
q 6 f(cKa/~) , since
is a homomorphism.
M N Z E F .
For the second
part of the conclusion of the lemma, assume its additional hypotheses. Let
q = <wK/F :K < ~>
(hence
and
fl = Rep(c') . We show that
f(a/F) # 0 , as desired).
In fact, for any
i E Z
q 6 f(a/F) we have
+
= = s i s a i 9
So
q E f(a/~) , as desired .
Next, we derive some specialized versions of Lemma 1.7.2
which
are more easily applicahle~ they are due to Andr4ka and N~meti.
Lemmm 1.7.3. that
F
is
(i)
Assume the hypotheses of Lemma 1.7.2.
I~I + - complete (which holds, e.g., if for any
(F,U,~) -choice function
Rep(c I) ; (ii) (iii)
Rep(c) For any
is an isomorphism; a 6 PiEIAi
we have
c'
~ < w).
we have
Also suppose Then Rep(c) =
1.7.4
Ultraproducts
89
(Rep(c))(a/F) = [q 6 ~(PiEiUi/~) : there is w 6 ~(PiEIUi ) such that
qK = wK/F for all
K < ~
Proof. It suffices to prove arbitrary then implies So, let
a 6 Pi6iAi
and
q E
s ai] E F] .
(iii), since the fact that
(i), and thus
and
[i E I :PJ i ~
(ii)
c
is
follows from Lemma 1.7.2.
(Pi61Ui/F) .
We want to show that the
following two conditions are equivalent:
(I)
+ [i E I : (c q)i 6 a i] E F
(2)
there is a
and
[i E I :PJi ~
If
(I)
w E ~(PiEiUi )
(2)
w = ; then
holds, and let wK
and
K < ~ , both
c(K,qK)
set
Z K = [i E I : (wK) i = c(K,qK)i] is also in
F .
(2)
H = [i E I :PJi ~
any
Y = H N ~K~. ZK
for all
qK = wK/~
are members of is in
For any
F . i E Y
is clear.
E a i] -
(I)
Therefore the set we have
Assume the hypotheses of Lemma 1.7.2.
Nee (c)*(Pi61~i/F) . ~/ E IK
latter case, (ii) 6 K
for
~
Then for
6 ai ,
holds.
Lemma 1.7.4.
(i)
Now
qK , and hence the
+ (c q)i = c(K,u)i + = [(c q)i]c(~,u)i E Si,ji 9 Thus
q~ E Wj , as desired. Now
that
(I), (2), (3)
Wj = W k
if
immediately yield that
Let
~ E IGs
Since
(ii) . Gs
notation, where in addition for each Yij = Yik ~i
(4)
or
is a normal
If
Yij n Yik = 0 Gws ).
j,k E PiEiJi
In fact, assume
So, assume that
= Gws i E I
F
is
I~I + -
, we can assume the above and
j,k 6 J. l
we have
(that is, in the terminology of 1.2.6,
We claim
and
j/~ ~ k/F , then
y E PiEiYi,ji 9
H ~ ~i E I : ji = ki] ~ F , so holds.
, upon noting
j/F = k/F .
Now we turn to the proof of complete.
~ E Gws
Let
H ~ F
Qj ~ Qk = 0 .
H = [i E I :Y i E Yi,ki ] and hence
Y ~ Qk "
So
Clearly (4)
92
Ultraproducts
By
(5)
(I), (2), (3)
For any
To prove
j E PiEiJi
K < ~ .
(4)
it now suffices to show
we have
q E ~ Qj .
(5) , let
for all
and
~Qj _c f(V/F)
i E I
we have
Hence by Lemma 1.7.3 we conclude that It remains to check E lws
Since
each
Ji
Pi,ti
'
Pi61Ji
Ws
~ Gws
a singleton W
.
and
Q
Thus by
(ii)
[ti}
for (i)
for
Wj
and
Qj
(6)
For any
y E ~(PiEIYi ) Let
q E W
iff
, as desired.)
so that
So, we suppose that
Y.l
for
where
j
ZK E F
(3) ,
for
W = U q 6 w ~ Q (q) Now we claim:
~s
(6) , let
for all
K < ~ 9
For each
K < a
iff
Pi
is the unique member of
r 6 ~X .
To prove
qK = yK/F
Thus
Thus
Y.1,ti '
[K < a : qK # rK]
G = [K < ~ : qK # rK] .
[i 6 1 : YKi = pi K} 9
.
K = Ws
," we write
r = ~/~ :K < ~> 9
W = ~Q(r)
qK = yK/~
, we still have the above situation, with
Let
(Hence
and
PJi ~ y 6 ~Yi,ji ~ Vi "
q E f(V/F)
f(V/~) = W , and by
q 6 ~X ,
.
y E ~(PiEiYi ,ji)
Say
Then for all
1.7.4
K ~ G .
finite.
q 6 ~X .
let
Choose
ZK =
Hence the set
Z = N [ Z K : K s a'~ G] [~ N I l N Z K : K E G]
is in
F
(by
I~I + -completeness).
[K :YKi = pi K] = ~ ~ G . M 6 F
i E Z
we have
M = [i 6 1 :c(K,qK) i = YK i] 9
Thus
also.
Now if in
Let
Now for all
F .
For
c(K,qK) i = yK i is finite.
q s W , then the set i s M n N n Z
N = [i E I : (c+q)i E ~y~pl)]
we have, for all
and hence (by the definition of
K < ~ , N),
is
(c+q)i K =
G = [K :YK i ~ PiK}
1.7.5
Ultraproducts
On the other hand, any
K E ~ -- G
if
we have
G
93
is finite then for any
(c+q)i K = c(K,qK)i = YKi = pi K
+ ~y(pi), [i E I : (c q)i E i J 6 F
M n Z E F ,
i E M N Z
and
and
and so, since
q E W .
This completes
the proof.
Lemma 1.7.5. or
Gs
Assume the hypotheses
Then for every non-zero
choice function
Proof.
c'
If
such that
F
is
of Lemma 1.7.2.
a 6 PiEIAi/F
Rep(c')a # 0
I~I + - c o m p l e t e ,
Let
K = Ws
there is an
and
(F,U,~) -
Rep(cl)*(Pi61~i/F)
the desired conclusion
from Lemma 1.7.3 and Lemma 1.7.4 (ii). So we assume henceforth is not
I~I + - complete.
In particular,
~ ~ w .
as in the last part of the proof of Lemma 1.7.2 Assume now
~ 6 ~s
Let
easily seen that there is an
We let .
Let
s
(F,U,~)- choice function
follows that
and
w
X = PiEIUi/F
N = [i E I : IUil > I] .
E K .
F be .
Then it is
c'
satisfying
the following two conditions:
(I)
ct(K,wK/F) = wK
(2)
c'(K,y) i # wKi
Again let <wK/F:K q E~X
for all
K < ~ ;
whenever
K < ~ ,
f = Rep(c t) . < ~>
.
Y E X ,
By Lenmm 1.7.2,
We shall show that
y # wK/F
fa # 0 .
, and
Now let
f(V/F) = ~X (q) .
i E N .
q =
Note that
. If
N ~ F , then
~x (q) , so
(ct+P)iK = siK
as desired.
and hence
0 # f(V/F) ~ ~X = [q] =
f(v/~) = ~x (q)
Assume that
p E ~X (q) 9
IXl = I
iff
Since
N 6 F .
Let
y/~ = wK/F N E F
P E r
,
, and hence
it follows that
i E N , and
K < a 9
(c'+P)i E V i p 6 f(V/F)
iff
Then
iff p E ~X (q)
94
Ultraproducts
Next, assume is an
h E I
of
V.
l
Since
~ E IGs
such that
l
choice function
cp
is not
I~I + - complete, there
(see, e.g., Chang-Keisler
i E I
there is a subbase
Now there clearly is an
(F,U,~) -
l
satisfying the following three conditions:
(3)
c1(hi,y/~) i E Yi
(4)
et(K,Y/~) E PiEIYi
(5)
c~(K,wK/~) = wK
Let
W = PiEIYi/F (U)
for all
and all
K < ~
and all
f = Rep(e')
f(V/~) = ~W . that
Y E X ; Y E PiEIYi ;
~ < ~ .
Again let
o
such
i E I
for all
for all
Now we shall show that q E ~
F
, for each
s. E ~Y. 9
l
Now l e t
Since
[I/(h Ih-l)] N F = 0
such that
[CK], p. 180). Y.
~ E IGs~
1.7.6
qK ~ W
H = [i E I : c(K,qK) i ~ Yi] E F .
If
By
for
(4)
some
By Lemma 1 . 7 , 2 ~
.
we have K < c~ .
i E H , then by
fa ~ 0
~W c_ f(V/~) . Thus
the
set
(3)
we have
+
c(hi,qhi)i E Yi Since
while
H E F , it
C(K'qK)i ~ Yi ;
follows
that
hence
q ~ f(V/F)
.
(c q)i ~ V.I " This
completes
the
proof.
Our final version of 1.7.2 concerns regularity.
Lemma 1.7.6. Rep(e) . in
~i
Also suppose
and
assume that
a E PiEIAi
~a i = ~f(a/F) .
Proof.
show that
Assume the hypotheses of Lemma 1.7.2.
Then
and for each f(a/F)
q E f(a/~) .
q E f(V/~) , and
Let
and
is regular
F = I U Af(a/F) , and
FI P = FI q .
Let
+ H = [i E I : (c P)i E a i
ai
f =
is regular.
We assume all the hypotheses. P E f(a/~) ,
i E I ,
Let
+ (c q)i E Vi} -
We want to
.
1.7.7
Ultraproducts
Thus F I
H E F .
(c+P)i
implies that
+ e , for all
By the definition of
+
= F I (c q)i
Thus for any
+ (~ q)i E a i
Remarks 1.7.7.
Since
95
i E I
we have
i 6 H , the regularity of
H E F , it follows that
ai
q E f(a/F)
.
The above lemmas are algebraic forms of the ~o~
lemma for ultraproducts.
The exact relationships between them and
~o~'s lemmm will be discussed in a later article. Andr~ka and N~meti have shown that various hypotheses in 1.7.2-1.7.6 are essential, and that these lemmas do not generalize to arbitrary reduced products.
Now we shall use the above lemmas to prove various closure properties of our classes of set algebras.
Theorem 1.7.8.
Proof. UpK
UpK = IK
for
K E [Gws
By Lemmas 1.7.2, 1.7.4 (i), and 1.7.5, each member of
is a subdirect product of memebers of
IK = SPK ; if
,Gs ] .
~ = I
K .
If
~ ~ I , then
, then by 1.7.3 and 1,7.4 we have
UpK = IK ; the
proof is complete.
Theorem 1.7.9.
Proof.
UpCs
directed by
{I(Ws
UL E K
~ , for
N Lf ) :~
Proof.
for
~ < ~ 9
By Lemmas 1.7.3 and 1.7.4 (ii) .
Theorem 1.7.10. K
= ICs
an
whenever
K E [l@ws
ordinal]
L
is a non-empty subset of
,IGs ] U [ I C s
: ~ < ~] U
.
This is immediate from 1.7.8 and 1.7.9, since
for all choices of
K
except the last.
Let
K = Ws
n Lf
SK = K , , and let
96
Ultraproducts
L
be as indicated.
UL
Then
1.7.11
UL E Gws
i s simple by 1 . 5 . 2 ( i i ) ,
by what was already proved, while
2 . 4 . 4 3 , and 0 . 3 . 5 1 .
Hence
/JL E IWs
by
1.6.4.
The following generalization of a part of Theorem 1.7.10 is due to Andr~ka and N~meti
[ANI]:
Theorem 1.7.11o , then
let
~
be an isomorphism onto a regular
such that
P~6.L~/~
~
~
= [~ E L :~ G ~] .
E F
onto
b E P~EL B 9
for each
P~EL ~ / ~
Let
c
Cs
be an
with base
9 E L .
such that
Let
~ ~ w 9 Cs
F
~
For each
with base
be an ultrafilter on
There is an isomorphism
g(b/F)=
( ~ b ~ :~ E L)/~
(F,U,~) - choice function, and let
By Lenmms 1.7.2 and 1.7.4 (i), onto a
is a set directed by
By Theorem 1.7.10 we may assume that
U~ ; further, let L
0 ~ L ~ ICs reg n Lf
UL E ICs reg n Lf
Proof. E L
If
f
is a homomorphism from
P~EL U~/F .
Now for each
g
of
for all f = Rep(c) .
P~EL ~ / ~
b E ~EL B
let
~) h'b = (b : b E B, ~ E L) U (0
and let P~6L~/F
hb = hlb/~ .
Then
h
f o g oh
is simple, by 2.3.16 (ii).
2.1.13,
~ ELf
b E U~EL B 9
Now let
is an isomorphism, and
First note that each member of DL
To show that
Then for each
E L),
is an isomorphism of
(see the proof of 0.3.71).
We claim that
:b ~ B , ~
Thus ~
9 E L ,
L
UL
into
~ = (f o g oh)* UL ~ E Cs reg n Lf
is simple, by 1.5.2 (i); hence
f o g oh
is an isomorphism.
is regular we apply 1.7.6. ~(hJb)~
is regular in
By Let
~
, and
1.7.12
Ultraproducts
~(h'b)~
since
07
= ~ ( h ' b ) ~ G ~ b G ~(fghb)
f ~ g oh
is an isomorphism.
Therefore by 1.7.6,
fghb
is
regular, as desired.
The following
lenmm is due to A n d r ~ k a and N~meti.
Lemma 1.7.12. V , and let
F
Let
~
be a
be an u l t r a f i l t e r
(F,
we
infer
9
.
Note
that
=
Iv E L : v > z] E F PzELBz/~
by
.
for
setting,
a 9 -c(~)v
it s u f f i c e s = 0
; since
On the o t h e r
if
hand,
a . kz # 0 ; thus
by
h E Hom(9/,PzEL~z/~)
this
for all
.
z E L
E Hom(9/,PzEL~z )
[v E L : a 9 k v = O]
Thus
Let
into
E Ho(9/,~ z) for e a c h
that
h 9/ ~ 9//L ; to show , then
kz = -c(~)z
,
hence
We w a n t
let
ha = / ~
a E L
direct
for a
~ IGs
~ < 2
is z e r o - d i m e n s i o n a l
O.3.61
present
to the a u t h o r s
.
Case
Now
known
and N ~ m e t i .
1.7.15.
for a n y
The
result,
Theorem
9/ E Gws
kz
or
I. 7.15
9
to show v > a
and
2.3.26,
0.3.6(ii).
Iv E L : v > a ] E
a E A N L
by
Hence
N o w we c l a i m that , so F
z E L
so
h-l~o
by
that
= L
.
If
Iv 6 L : v > a] =
, it f o l l o w s then
that
a ~ c(~)z
ha = 0 .
and
ha # 0 , as d e s i r e d .
h 9/ ~--9//L , so
9//L E S U p G s
By
1.7.8
,
9//L E IGsc~
1.7.15
Ultraproducts
Case 2. a E A " L ha # 0
~ ~ ~
and
and
.
Using
1.6.3,
find a h o m o m o r p h i s m
h L = [0]
101
it is enough
h
of
let
F
& ~ F} 6 F
be an u l t r a f i l t e r for all
onto a
Cs
such that
Let
I = L• [ r ~ - ~ : l r and
~
to take any
I
we have
IGs
= HSPGs~ = H S P G w s
=
= HSPCs reg
HSPGws
= HGws
= IGs
c IGws
using
parts of the theorem are easily established by using
The following result is due to Monk.
1.7.15; 1.7.13.
the other
,
1.7.17
Ultraproducts
Theorem
1.7.17.
A n y direct
Proof.
We may assume
103
factor
of a
Cs
is isomorphic
to
Cs
a
pressed
Gws
in
1.2.6.
that
interesting
(I)
= [~ :~ is isomorphic
compressed
The
Gws
theorem
zero-dimensional element Say
result
element
the unit element
of
the notion
from
~
on
is
V
~
such
that
(2)
for all
Let
f = R e p ( F , ( U : ~ < ~),~, ~
In fact, and
applying
Gws
U
proof
of
Since
~
By
1.7.13
such that
from
9/
and b y
we
9/ we ~
find
~ > ~ ~
1.6.2
, as desired
to a
(3)
for some Cs
that
we get a in
.
w i t h unit e l e m e n t
9/ > ~
is isomorphic
~
~
looking
the base of
Gws
; let
;
Ws
~ g
V
wi~ at
~
the
is
X .
w i t h unit element be a h o m o m o r p h i s m
. 9/
~ X (f ~
is isomorphic
9.I
is
h = < g a U f6a : a E A}
It is e a s i l y
that
W
Gws
see that we may assume
0.3.6(ii),
function
to the full
is zero-dimensional,
onto
By
to
1.7.13,
W ~ ~X
w i t h unit element
2.3.26(ii)
then restricting
base
W
for some
1.7.18
checked
that
to a subalgebra
isomorphic
to a
Cs
is an i s o m o r p h i s m
ha n'~v
= ~a
for
of
9/i
; in
from
all
~ x (f ~
~1
fact,
into
a E A , so
9/ ,
the ~/'
(1)
holds.
We and
shall e s t a b l i s h
relationships
between
after
discussing
about
the results
Remarks
again
the obvious ways; (I) 2 ~ IUI < w such
that
If
~/~
does n o t extend and choose
and
~ ICs to
9/
is a
is a set This was
IUI > ~
an u l t r a f i l t e r
several
, by F
over
in this
above
specific
in
1.7.28-1.7.30,
some
cannot
U
by M o n k
In fact, that
in
to this effect.
such that
and an ultrafilter
such
remarks
be improved
arguments
w i t h base
1.7.22.
properties
section.
first noticed
I
closure
First we make
Cs I
about
of set algebras
of the results
we now make
, then there
of base.
established
Many
~ ~ ~
results
our classes
change
already
1.7.18.
a few more
let
I IA/~I
F
over
[M2] I = 22~Z = 2221~
I It
1.7.19
Ultraproducts
(see, e.g., Chang, Keisler
[C~
IUl , and hence so does
~/F
has base of cardinality
IUI
Thus
I~/~ ~ I C s (2)
~ ~ w
ultrafilter
~
Now
But any
~
Cs
has characteristic
of characteristic
, and hence has at most
, any n o n - d i s c r e t e
F
x =
a C A , and
=Gws
, say with unit element
K = UiEllUil Now let
we have
U i is infinite for all
.
we denote by
having all subbases infinite.
9/ be a
(Pi) Ui
unit element
For
First we formally give a defini-
~/ ~-- I c Pa6A~a
.
By
1.6.2
we have
1.7.22
Ultraproducts
Remarks 1.7.22. closed class for
By
1.7.16
~ m w .
and
showing that any
CA's
.
Gws
1.7.21,
I Cs
Thus for many purposes,
with infinite bases are the simplest represent abstract
107
CA's
is an algebraically
cylindric
set algebras
with which to isomorphically
Andr~ka and N~meti have improved
is sub-isomorphic
to a
1.7.21 by
Cs
Before turning to results concerning change of base by increasing the cardinality of the base, algebraic analogs of the upward L~wenheimSkolem-Tarski The example,
theorem, we give an example supplementing I
K < m , then there is a then the base of
~
k < K
base
K o
9/ ~
E Cs
~
Now
K
~
I < ~ < ~
such that if
(whether or not
9/ ~
and 6 Cs
K < ~ ); because of
in most of our theorems that the base is infinite. a k = [ K .
C(C ~
let
Thus
Cs
has power
this example we assume For each
l
due to Andreka and Nemeti,
those of section 1.3.
Then
fa0,...,faK_ I
has at least
k I
for some
(c0fa k," fa k) n
,~l)fa k = 0 , contradiction.
The following general lemma and theorem about increasing a base !
I
are due to Andreka and Nemeti;
their parts dealing with
Ws
and
Cs reg
are due to Henkin and Monk.
Le-r~ 1.7.23. V .
Let
F
there is an
be a
Let
~
be a
Gws
with base
I~I- regular ultrafilter
U
on some set
(F,,~)- choice function
c
and unit e ~ m e n t I .
Then
such that, letting
108
Ultraproducts
f = Rep(F,,~,,o) in
1.7.12,
6
and letting
and
~
be as
we have:
(i)
f o6
(ii)
~ 9/
(iii)
1.7.23
is an isomorphism is sub-isomorphic
from
9/
(f ~ 6) 9/ ;
onto
to
(f o 6)
is
IU/F
;
~ = r6~ o f o 6 ; aV
(iv)
~ V = ~(E*U)
(v)
the base of
n f6V ; (f ~ 6) 9/
;
w (vi)
9/ E K
implies
K 6 [Ws
Proof.
(f o 6) 9/ 6 K
,Cs ,Gs ,Gws
Assume
for distinct
X = IU/F
o
R = {<j,k>
relation
on
Let J .
reg ,Gws~ ] .
the hypotheses.
~Y!PJ)3 n ~y~pk)
Let
K
for all
V = ~Hj E J - -~Y(PJ) j
Say
j,k E J .
Let
E 2j :yj = yk ] .
be a subset of
in co~Inon with each equivalence
J
Y = Thus
R
class under
R .
Let
in common with each equivalence
~ ( < K :is
<j : i 6 I> 6 L
functions
w s X(Iu)
and
for each
v E X(Ij)
.
and
one element
L ~ IK
have
class under
j E K .
Let
O =
is an equivalence
having exactly
exactly one element , with
' where
y E X .
Now we define Choose
k
= Y
k 6 IK some
such that u 6 U .
y N PiEIYki @ 0 , and
Let
y N Pi61Y(vy)i
vy
if
y = Eu .
Now
wy s y
for all
(2)
weu =
w
and
y 6 X ; for all
of
if
L N k/F
wy s y ~ Pi61Y(vy)i
properties:
(I)
constant
be the unique element
~ 0 , so we can pick
wy =
k
u E U ;
v
y = Eu .
Thus
with
have the following
for
1.7.23
Ultraproducts
(3)
for all
(4)
if
Since
y s X
k,q s Rgv
F
is
and
i E I
and
we have
(wy)i 6 Y ( v y ) i
[i E I :Yki = Yqi ] s F , then
1 5 1 - regular,
[i E I : K 6 hi] 6 F
109
choose
for all
N o w let
K < 5 ,
P(vy)i K
k = q .
h E I[F ~ 5 : Irl < w]
K < 5 .
choice function such that for all
;
if
c be an
(F,,~) -
y 6 X , and
K ~ hi
and
such that
i 6 1 ,
y ~ e U ,
c(~ 'Y)i L(wy)i
Note that
c(K,y) 6 PiEiY(vy)i
otherwise.
for all
be as in the statement of the lemma. hold. For
(iv), use
be the base of obvious
that
in fact say
Z = X .
q 6 5X , and f o r any (2)
By 1.7.12,
(2) and the definition of c.
(f o 6) ~
u E Yj
K < 5
; we are to show that Suppose
with
y E ~ U ; say
j E J 9
i E I
Now let
c .
y E X .
(i),
Let
f,6,~
(il) and (iii)
To prove
(v), let Z
Z = X , and it is y = s
with
u E U ;
q = <EpjK : K < 5>~
9
Thus
(c+q)i = (pj)~ E 5YIPJ) ~ V using
we have
and the d e f i n i t i o n of
and
It follows
that
q E f6V , and hence
. y 6 Z . i61
N o w let
y E X ~ e U .
Let
q =
9
Then for any
,
(c+q)i =
= ( 5 - hi)IP(vy)i U E ~Y(vy)i (p(vy)i) c_ V . Hence a g a i n
q E fSV
and
y E Z .
N o w we turn to the parts of K = Cs
5
(f o 6)*~
are
taken care of by 1.7.4.
b y 1.7.6,
since
f ~6
So
(v)
(vi) . If
~
holds. The cases
K = Gws 5
is regular,
is an isomorphism.
and
then so is
N e x t suppose
110
Ultraproducts
K = Gs
Thus for all
For each
r E Rgv
(5)
if
For,
suppose
Then
j,k E J
we set
r,r j E Rgv
Qr
and
1.7.23
we have PicIYri/~U
r @ r I , then
y 6 PiEiYri
,
(6)
fSV ~ ~r ER gv Qr
"
For,
let
Thus the set
We claim that
q E ~Qvqo
(c+ q)i E ~y ji " and also tion of
.
~
c(K,qK) i E Y(vqK)i
[i E I :Yi = zi} E F .
r = rl
i E M
i E M
Yji = Y(vqK)i
, and
+ M = [i 6 I : (c q)i E V}
and
(c+q)i K = c(K,qK) i E Y(vqK)i c , so
Yj ~ Y k = 0 .
Qr N Qr I = 0 .
(4)
For each
Now for any
or
Now
z E Pi61Yr~i
[i E I :Yri = Yrli ] E F , so by
q E fSV
Yj = Yk
"
choose
K < ~
"
ji E J
we have
Thus
i E M
and
qK E QvqO
F .
so that
(c+q)i K 6 Yji
by the note following
So for any
= Yji = Y(vqO)i
is in
the defini-
K < ~
for any
we have K < ~
as desired.
(7)
If
r E Rgv
For, choose c
,
K < ~ , and
z E Y ~ PiEiYri
we also have
c(K,y)
{i E I : z i = c(K,Y)i] we infer from
(8)
For all
In fact, c(K,qK)
let
(4)
q E ~Qr
E PiEiYri
-
' then
is in
F o
.
By
Hence 9
c_
~Q
(7)
r
Thus
E PiEiYri
9
c_ f6V
the set
M =
follows.
.
for all
+ = i 6 I , (c q)i
Hence
of
M ~ {i E I :Yri = Y(vy)i } '
(7)
we have,
for all V .
9
Since
vy = r , so
we have
c(K,y)
By the remark after the definition
6 y ~ PiEiY(vy)i
that
r E Rgv
9
Y E Qr
q E f6 V
.
K < ~ ,
,
1.7.24
Ultraproducts
By (5),
(6),
(8)
In the case #
we have
K = Ws
(f o6) ~ s Gs
we redefine
K < ~
let
K < ~ .
We may assume c(K,~u)
Choose
that
IUI > 1 .
= IYjl
, and
9
j,k E J , where W.j ~ Yj
with
Furthermore,
.
By 1.6.2 we have
9/ ~ I - c Pjs
with unit element is given by
that
U
Kj ~ IYjl .
Proof.
h
with base
be a cardinal-number-valued
for all
v=%nl
Gws
Kj ~ (IAI n 2 I~IUIYjl) U ~
Then U j E j ~'(PJ) w j
K
IYjl ~ ~
9
~j
, where
~j
is a
Ws
~Y(PJ). for each j E J ; in fact, the isomorphism JI (ha)j = a n ~Y[.PJ)j for all a E A and j 6 J 9 Note
and
IBjl IA! n 2!IulYjl
114
Ultraproducts
for each
j E J
for which
isomorphic to a
Ws
yj = Wj
and hence
We may assume that
IYjl ~ Kj .
Hence by
with unit element
Wj n W k = Yj n Y k For each
f. E Is(~j,~j) . J
j E J
H
g
0~(pj)
~JEJ"j
~
and
~j = ~j
for all
j E J 9
~W~ pj) n ~W~ pk) = 0
aEA
;
onto a
(as is easily checked)
is sub-
fj = if R6
=,
IUI
q6
u,
and
R = qlq-I
, then there is
R l = q'lq '-I .
is clear.
,and
q 6 ~U
, then there is at most one
1.7.28
q
t
Ultraproducts
( ~ E ~U "q"
(4)
such that
For every
For, say
R E e
Since
i-1
= R .
there is an
R = qlq -I , where
I~/RI < ~] Choose
q'lq
K < w
such that
R' 6 @o~
q E ~K . we have
~ E ~'~ F , and let
~17
Let
R--- R' .
F = U [ B / R : 8 < ~ and
I~/R 1 < •
and hence
IF[ < ~ .
q' = (~ ~ F)lq U ;
q'
is
as desired. Now by -I pilp i = R i
(I) - (4) for all
Now let each of power is a
~ < k
i E I , and
Set
such that
q 6 ~YB
with
qlq -I = Ri] .
(5)
if
i,j E I
and
(6)
if
i E I , then
i 6 1
.
and
W , and the
Next, for each
whenever
such that pilPi I = pjlpi I
be a system of pairwise disjoint sets,
W = U~ ~ .
1.7.19
Ws
such a
be one-to-one and onto. Ws
a 6 KC .
For every
~/ with base
U
Namely,
such that
if
IAI ~
either has only one element or else has base
To construct
the full
are necessary.
with unit element k < K
let
Let
Ws
, let
Let
p = , and let
V = ~U (p) . kk
IU[ = K .
~
be
We now construct
be a one-to-one
function mapping
120
K
Ultraproducts
into
{v E U : X < wv}
Then for each
a x = ~q ~ V : ( k m e x { w q ~ Now i f
0 < ~ < ~
wq~ ~; m x
so
(1)
~
qo
~ if
"
,
:0
system of pairwise disjoint non-empty subsets of
that is a
[q 6 W : qlq -I = R] .
Hence
l [ q E V : q / q -1 - R} ~ I[q E W : q l q -1 = R]I --k'l[q E ~
and
: qlq "1 = R] I 9
122
(f)
Reducts
The assumption
upon considering (g)
III ~ K
1.8.1
in
1.7.29
cannot be improved, by
R = ff X ~ .
Andr~ka ar~ N~meti have proved the following algebraic version
of the various logical
theorems to the effect that elementarily equiva-
lent structures have isomorphic elementary extensions: and
(e)
~ ~
.
Then
respectively
~
and
such that
~
Ot
are sub-isomorphic
and
O'
8.
We restrict ourselves
to
Let Cs's
Ot
E Cs and
~'
are base-isomorphic.
Reducts
in this section to the most basic results t
about reduets.
~
t
A more detailed study is found in Andreka, Nemeti
[AN3]
to which we also refer for the statement of various open questions.
Lemma 1.8.1. V .
Let
~
For each
Y
+
y+~Su. Then
f
Proof.
be a
Crs~
f
Y E ~U
For all
Y~A
W = fV . x~
preserves
and unit element
be one-to-one.
Fix
p-l)
let m--~y~=U:y+~Y~ ~
Clearly
(p)91 into a
f
preserves
6 fX , i.e.,
fX ~ 0 .
dKk
U
set
is a homomorphism of
Let
with base
p E ~
= ((6 N Rgp)Ix) U (Y ~
( x o p ) + = x , we have check that
~
be an ordinal and let
x E X E A .
thus
Let
for
K,k < ~ 9
Crs
+
, and
and
fX ~ 0 .
Sir= e
It is routine to
Now suppose that
Y 6 A ,
1.8.2
123
Reducts
K < ~ , and
y EW 9
fc
zz , let
~ ~K
Y+ 6 c[V]ypK "
Thus
Y Efc
y+ s V
easily checked that and so
For brevity set
)Y . Thus
and
The other inclusion
Theorem 1.8.2,
If
~
and
~
,
for some
,
u E U .
K + 6 Y , so (yu) =
It is
YuK E fY
is established similarly.
are ordinals with
Rd ~P)GwsB ~ I G w s
is one-to-one, then
To prove that
y E fC
. +.pK iY )u 6 Y
(y~)+ = [Y . +,pK )u ; hence
y 6 c~W]fY .
p E~
~ = ~(P)9/ .
, and
~ m 2
Rd (p)
and
GWS~
I Gws
Proof.
First we take any
9/ E Ws~
and show that
~
(p)
9/ E I G w s
To this end, by 2.4.39 and 1.6.4 it suffices to take any non-zero
X E A
and find a homomorphism
fX ~ 0 .
f
of
~(P)~
into some
Ws
such that
Say 9/ has unit element ~U (p) , and x E X . For each y 6 ~U define + y as in 1.8.1; then define f as there also. Applying 1.8.1, we see that
f
is a homomorphism of
~ (p)9/
Now it is easily checked that Fy = [K 6 Rgp :yp-IK ~ pK} Ip -I* Fy I < ~ , and ~U (p=p) , so
~
into a
Crs
f(BU (p)) = [y 6 ~U
for all
y s ~U .
Ws
fX ~ 0 .
:Iryl < |
, where
Clearly
p-l*Fy = [K < ~ :yK ~ ppK] .
is a
~ , and
Thus
IFyl < ~
iff
f(~U (p)) =
as desired.
,
The theorem itself now follows easily from 0.5.13(iv) and 1.6.4, the firml statement being clear from the above.
Remarks 1.8.3. we also have and
Rd (p)~ C s
Under the hypothesis of 1.8.2 and using 1o8.2
~ IGs ~ R d(P)Gs8 ~ = l~Cs~if
have shown that if
Rgp ? ~
and
~ ~ ~ then
g e n e r a l i z i n g examples of Monk.
Rd(P)Gsreg ~ 8 -~ l~ u s reg by 1.7.21 .
by
1.7.14,
l
But Andr~ka and Nemeti
Rd~ Cs~ ~ ICs~
and Rd~O)Ws~ ~ I W s
124
Reducts
Theorem 1.8.4. one-to-one and onto. Cs,Ws]
Let
~
Then
y
is a
The arguments
1.8.1
= IK
for
K = Crs
-I
The hypotheses of
fX ~ 0
~(P)9~
K E [Crs,Gws,Gs, Gws reg,
K = Gws r e g .
and unit element
into a
Crs
~
f
y+ =
defined there
with unit element f
fV ,
is one-to-one.
Thus
It is easy to check that
Y E A , y E fY ,
For any
c~)Y
K < ~
we have
c~)fY
~(~)fY = p-l*~(9/)y .
p'lo E ~(~)fY , and so
= fY
iff
First suppose that
Clearly
by the assumed regularity, so
a(~)y
.
Now
= Y
iff
pK
+ 0
(Y)yO
' so
c_ IK 8
Second,
(using
+0 (z)zO
E V , and
E Y 9
Hence
in our two representative cases.
For the other inclusion, it suffices to note that Rd~ p- I ) K
z+ E V , so
E V
+0 (z)yO
Then
Hence
as desired.
Clearly also
+0 +0 (A (9/)Y U 1)I ( y ) y O = (~(9/)y U 1)I ( z ) y O + z ~ Y and z E f Y , as desired. IRd(P)K~ _c I K
so
and
~ = ~&(P)9/ .
c(9/)Y = Y .
O s ~(9/)y 9
y+ E Y
z E fY
Y+ E Y ,
+0 (Y)yO E Y .
~/ E Gws~) , and hence
We have shown
Now let
yp-lo = zp-lo , i.e., y+O = z+O .
(gCg/)Y LJ I)IY+ = (~(91)y U l)Iz + 9
O ~
((~(B)fY) U l)ly
z E W , and z E fY 9
one and onto,
Now the function
x ; we have simply
X E A , i.e.,
9/ E Gws r e g .
Assume that
assume that
First suppose
V .
= ((~(~)fY) U l)Iz ; we want to show that
+ z E Y
be
, as desired.
Now suppose that
Thus
~ 6 ~B
1.8.1 hold, so the function
for every non-zero
~(P)9/ E ICrs
E Gws
U
and
does not depend on any element
is a homomorphism of and
be ordinals and let
being very easy, we restrict ourselves to
Crs B , say with base
in
y ~p
B
IRd (p) K
two representative cases,
+
and
.
Proof.
9/
1.8.4
p
-i
E
B~
is one-to-
by what was already shown, and clearly
1.8.5 Rd(P)Rd(p-l)K
= K
N o w we turn to neat technical
.
embeddings,
for w h i c h we a l s o
require
a
lenmm.
Lemma V
125
Reducts
1.8.5.
Assume
that
Let
~
~ ~ 8
be a
and
Crs
W ~ ~U
with base
.
U
and unit e l e m e n t
We also assume
the following
conditions: (i)
V = {x : x = ~ l y for some y E W]
(ii) K x u EW
for all
Then
XEA
there
preserves
~x
6 V
x E W N
Let +
Crs~ ~
u E U , if
and
~
.
c~)fX
be t ~ f u l l
N o w let
x E W
Hence
f
preserves
(i)
there
is a
y E W
shows
x E f(V N X)
that
f
~Ix K u E V
then
; we w a n t x 6 W u 6 U
xKu 6 W
.
Finally, x 6 W
and
.
(i)).
to show that and
CrsB
w i t h unit element
and
Hence suppose
X E A
-
If
such that
.
suppose
9 ,
x uK 6 fX
for some
f
.
preserves X E A
fx .
,
.
Clearly
iff x s X .
Y E fX dKk
K < ~
.
for ,
and
By the definition
~Ix E V
K 6 B N ~
such
we have
Thus
and
our a s s u m p t i o n
u 6 U
.
, choose
x 6 C~ W] fX , as desired. ,
W
0 # X E A x ~ y
Hence
(i)
W
K 6 8 " ~
~Ix ~ X
that
x ~ c~W~ K
X E A
By
and
iff
Clearly
(~Ix)Ku 6 V
that
X s A
~Ix E V N X
~ I x E C~ V ] X
Since
a n d unit element
for all
iff
Now
U
.
= fX
is one-to-one.
(again u s i n g
for some
w i t h base
9
we have
Thus
is a
fX
x E C ~V]x
that
, and
fX = Ix E W : ~ I x E X]
Hence
K,k < ~
f
K < ~
o
T h e n by This
let
f 6 Ism(9/,~)
Proof. f
,
. For any
that
x E W
;
(~IX)Ku E X
(ii)
The converse , and
Hence
yields is similar.
_[W] fX x 6 ~K
x uK E W
of
and
9
126
Problems
~Ix K E X .
Since
1.8.6
~Ix E X , and hence
K ~ ~ , this means that
U
x E fX , as desired.
From this l e ~
it is easy to prove
Theorem 1.8.6. Then
K
~ ISNr ~
Assume that
~ ~ ~
and
K E [Ws,Cs,Gws,Gs}
9
.
Corollary 1.8.7.
If
2 ~ ~ ~ B
then
IGws
= SNr IGws B =
SNr IGs B = I G s
Proof.
By
1.7.14, 1.8.2, and 1.8.6.
Remark 1.8.8. ordinal
~
we have
It follows from 2.6.48 and 1.8.6 that for any Cs
U Ws~ U Gs~ U Gws~ c SNr Dc0t~
result of the representation paper, is that if
~ ~ 2
theory of
then
CA's
9.
A major
, to appear in a later
= IGs
SNr Dcct~
.
=IOws
.
Problems
We begin by indicating the status of the problems listed in [ HMT ] as of January 1981.
In Problem 0.6 one should assume that
than the first uncountable measurable cardinal Under this corrected
formulation,
relative to the consistency shown by Magidor
of
[Ma] and Laver
as less
(see Chang, Keisler [CK] ).
the consistency of a positive answer
ZFC
plus certain other axioms has been
[L ]
!
affirmatively by B. Sobocinski
~
Problem 1.2 has been solved l
[S ]
Andr~ka and Nemeti solved
Problem 1
Problems
Problem 2.3 affirmatively;
see
127
[AN~
solved affirmatively by J. Ketonen hence for discrete D. Myers
[M~
CA's.
and
[N ].
[K ]
Problem 2.4 has been
for Boolean algebras,
and
Problem 2.8 was solved affirmatively by
and Problem 2.9 negatively by W. Hanf
[ H ].
2.11 was
J
solved negatively and
[ N ].
(except for
~ < 2) by Andr~ka and Nemeti;
see
Problem 2.12 was solved negatively by R. Maddux
[Md].
Now we shall list some problems left open concerning
Problem i. is there a
Let
~ ~ w 9
Given a normal
Cs~ 9/ with same base
U
Gws
~
[AN2]
set algebras.
with base
U ,
such that
Io2.6-Io2.13).
Problem 2.
Let
q
be the function defined in 1.4.8.
For every
+ 6 w N 2
let
q ~
be the largest
8 s w
such that
q(~,B) = I . +
Give a simple arithmetic
Problem 3. (Cf 9
Is
description of
IW~
q , or at least of
q
.
closed under directed unions for
1.4.8 and 1.7.11.)
Problem 4.
Is
Problem 5.
Does
Problem 6.
ICs
= HCs reg
I Cs
= H~ W s
Is
Icsreg=
HPWs
Problem 7.
Is
H=Ws
Problem 8.
Is
HP=Ws
HP=Ws
or
~.
or
._ reg HCs ~ = n~ss ~9
(Cf. 1.5.6.)
(Cf. 1.5 6(17) and 1.5.8.)
ICs
?
ImCs~ ?
For these two questions cf. 1.6.8 and 1.6.10.
= HPWs
?
(Cf.
1.6.8.)
128
References
Problem 9.
Is every weakly subdirectly indecomposable
morphic to a regular
Problem I0. (or
Problem 9
Cs
Cs
iso-
?
Is every weakly subdirectly indecomposable
Cs reg) isomorphic to a
Ws
Gws
?
For these two questions cf. 1.6.16. Problem Ii.
Is the condition
IUI ~ IAI
in
11.7.27 needed?
(Cf. here also 1.7.30.)
REFERENCES
[ANI] Andreka, ' H. and Nemetl, ' " I., A simple~ purely algebraic proof of the completeness of some first order lo$ics, Alg. Univ. 5(1975), 8-15. !
[AN2] Andr~ka, H. and Nemeti, I., On problems in cylindric algebra theory, Abstracts Amer. Math. Soc. 1(1980), 588. l
l
[AN3] Andreka, H. and Nemeti, I., On cylindric-relativized this vol.,
set algebras,
[AN4] Andr~ka, H. and N~meti, I., Finite cylindric algebras generated by a single element, Finite algebra and multivalved logic (Proc. Coll. Szeged), l
!
eds. B. Csakany, I. Rosenberg, Colloq. Math. Soc. J. Bolyai vol. 28, North-Holland, to appear. [CK] Chang, C.C. and Keisler, H.J., Model theory (second edition), North-Holland 1978, xil + 554 pp. [D] Daigneault, A., On automorphisms of polyadic algebras, Trans. Amer. Math. Soc. 112(1964), 84-130. [De] Demaree, D., Studies in algebraic logic, Doctoral Dissertation, Univ. of Calif., Berkeley 1970, 96pp. [EFL] Erd~s, P., Faber, V. and Larson, J., Sets of natural numbers of positive density and cylindric set algebras of dimension 2, to appear,Alg. Univ.
References
129
[H] Hanf, W., The Boolean algebra of logic, Bull. Amer. Math. Soc. 8(1975), 587-589. [HM] Henkin, L. and Monks J.D., Cylindric set algebras and related structures, Proc. of the Tarski Symposium, Proc. Symp. Pure Math. 25(1974), Amer. Math. Soc., 105-121. [HMT] Henkin, L., Monk, J.D., and Tarski, A., Cylindric Algebras, Part I, North-Holland (1971), 508pp. [HR] Henkin, L. and Resek, D., Relativization of cylindric algebras, Fund. Math. 82(1975), 363-383. [HT] Henkin, L. and Tarski, A., Cylindric algebras, Lattice theory, Proc. Symp. pure math. 2(1961), Amer. Math~ Soc., 83-113. [K] Ketonen, J., The structure of countable Boolean algebras, Ann. Math. 108(1978), 41-89. [Ko] Koppelberg, S., Homomorphic images of Proco Amer. Math. Soc. 51(1975), 171-175. [L]
~ - complete Boolean algebras,
Laver, R., Saturated ideals and nonregu~r ultrafilters,
to appear.
[aM] Magidor, M., On the existence of nonregular ultrafilters and the cardinality of ultrapowers, Trans. Amer. Math. Soc.249 (1979),97-111 . [MI] Monk, J.D., Singularu cylindric and polyadic equality algebras, Trans. Amer. Math. Soco 112(1964), 185-205o [M2] Monk, J.D., Model-theoretic methods and results in the theory of cylindric algebras, The Theory of Models, Proc. 1963 Symp., North- Holland, 238-250~ [Md] Maddux, Ro, Relatio~ algebras and neat embeddings of cylindric algebras, Notices Amero Math~ Soc. 24(1977), A-2980 [My] Myers, Do, Cylindric algebras of first-order languages, Trans. Amer. Math. Soe. 216(1976), 189-202. I
IN] Nemeti, I., Connections between cylindric algebras and initial algebra semantics of CF languages, Mathematical logic in computer science, eds. B DSmSlki, T. ~ergely, Colloq. Math. Soc. J~ Bolyai, vol. 26 North-Holland (1981), 561-606 . IS] Soboclnskl, B., Solution to the problem concerning the Boolean bases for cylindric algebras, Notre Dame J. Formal Logic 13(1972), 529-545~ [TV] Tarski, A. and Vaught, R.L., Arithmetical extensions of relational systems, Compos. Math. 13(1957), 81-102.
On cylindric-relativized
by
This theory
is b a s e d
of cylindric
[HMT3. are
work
Most
H. A n d r 6 k a
on the book
algebras
cylindric-relativized
voted
to t h e
study
classes
of Crs-s
Gs reg.
The
played the
fundamental
the
introduction
ting
classical
in a s e n s e ture
of
of
tion
t h i s way.
tion
to
by
first
proved Gs r e g
much
order
in
[G]
Following
these
use
the n o t a t i o n s
is a c o n t i n u a t i o n
[HMTI].
refer
to
We
individual
[HMTI3. first
[HMTI32.2
The
in t h i s paper
this paper,
i t e m of
[HMTI3
sections;
items
figure
the present of
For
is
is role
example,
Gs reg,
the c l a s s
given
see
connec-
GsregnLf,
to t h e m e t a - s t r u c -
interpretations
structure
we
class
is e x a c t l y
all
considerable
introduced
paper
The
and
motivations
The present
as
that
was
is d e -
c a n be r e p r e s e n t e d
insight
shall
be-
and
give
simplifica-
special
atten-
G s reg.
shall
of t h e
attention
this
work
to the
theory.
to C A - t h e o r y
that
achieving
is s i m i l a r
IN3
of CA-s
distinguished
and CA-theory
theories
theory
a distinguished
in
The abstract in t h e b o o k
The present
algebra
proved
theory
developed
to c e r t a i n
Such
theory
It w a s
Recently
all
(Crs-s).
in B o o l e a n
[HMTI3.
abstract
in C A - t h e o r y
model
model
for t h e
precisely
Gs reg
[HMTI3.
It w a s
isomorphically
by
finitary
is e x t e n s i v e l y
[HMTI].
algebras
at least.
them.
We
set
in
I. N 6 m e t i
a n d the p a p e r
examples
more
link between
consisting
tween
introduced
by Boolean
[HMTJ
set algebras
of C r s - s ,
role played
and
(CA-s)
of the motivating
set a l g e b r a s
in
paper by
moreover
i t e m 0.5.
recalling
a n d is o r g a n i z e d
we have
the
means
this
EHMTI3
without
discussion
therefore
item of
1.2.2
figures,
item
In g e n e r a l ,
practically
are n u m b e r e d
I and
strings
[HMT3
f o r an i n t r o d u c t o r y
[HMTI]
is a l w a y s
of
in
the
of
by three
we omit
figures
O.5.1
when
read
O, from
like
the
refer
refers
in s e c t i o n
to
contents
section-titles
e.g.
We
parallel
of t h e
figures
it,
[HMTI3.
e.g.
is f o u n d
same
them.
1.2.2.
reference
to i t e m s
of
to i t e m 0 . 5 . 1
a n d it is a s u b left
to r i g h t
132
correspond We
to the
shall
proved
subdivisions
be g l a d
(or not
to send
proved
Acknowledgement. guiding logic
us
full
work
of
grateful
statements
as in o u r
work,
research
to is found.
claimed
but
whenever
to P r o f e s s o r
J.D.
not
requested.
Monk
concerning
for
algebraic
in g e n e r a l .
O. B a s i c c o n c e ~ t _ s _ a n d
We use
the n o t a t i o n s
recalling Ws,
item referred
in the p r e s e n t
are m o s t as w e l l
the
proofs
in detail) We
in this
in w h i c h
Gs
them. , Gws
-relativized normal
set
we use
, C s ~ eg,
algebras
[HMTI3
EHMTIJl. I
G s ~ eg,
were
of
and
where
the
Gws reg~ , Crs rega
introduced.
E HMT3
without
classes
Cs
,
of c y l i n d r i c -
All
these
algebras
to
CA -s.
are
Bo -s.
Notations:
Let
%%
exists
since
Mn(~)
~ Sg(~){l
Let unit
and d e f i n i t i o n s
Especially , Crs
notations
V
i
is a c o n s t a n t ~
be a
V.
This
be an a l g e b r a
}
Crs
and
notation
symbol
~(~)
-unit.
similar of
~
Then
CA -s.
~(4A){I~
~V
is a m b i g u o u s
i
We d e f i n e
}.
denotes
if
Then
V=O
the
full
but we h o p e
Crs
with
context
will
help. Let
xCVC~U.
Then
AEVJx
~
{ie~
: c~V3x#x}
and
A(U)x
~ AE~U]x.
1
of
Let
H
be any
H
and
Gc
H
set.
Then
denotes
Sb H that
denotes GESb
--W
As
the
set of
all
finite
subsets
H. W
a generalization
of the
notation
f~
introduced
in
EHMTI],
U
the
following
and
let The
H
be a set.
notations
f : A >~ one-one
notation
B into,
mean
will
be v e r y
Then
fEH/k]
f : A ~ B, that
one-one
A1f onto
f
useful. ~
(Dof
: A ~-- B,
is a f u n c t i o n respectively)
~
f
Let
f,k
H)If
U H1k.
: A >-- B,
mapping B.
be two
A
functions
and into
In a c c o r d a n c e
(onto, with
EHMT3,
0.1
133
fEIs(~,~)
means
We
use
shall
[HMT~
to
Crs
not be a
CA
A(~)x
{ {iE~
~ By of
-s
as w e l l ,
E.g. : c~x 1
2.2.3
1.1.1
in t h e i r
true
>
-s,
for
Crs
the
-s,
of
apply
1.2.1-1.2.12
1.6.2, Let
1.6.5-1.6.7 ~,~ECrs
Zd A ~ Z d ~ will
Boolean
holds
V
we
Subu(V)
the
of
we
U
let
At
be a n y
an a t o m
of
set of a l l
too.
the B o o l e a n subunits
of
~ u{Rgp
subbases
. Then
base(~)
~ base(l~),
subunit
of
, 1~ .
and
Y = base(W)
is s a i d Y
is
1.6.5-1.6.7
( C o ) - ( C 3)
to
we
sets.
shall
-s.
By
let
applicable of
and
we
Crs
Therefore
to
Cr
The
Particularly
for of
V~U
-s
above
for a n y
By a s u b u n i t
field
Zd Sb V
V
I.e.
: pCV}. some
We
of
sets.
Subu(V)
=
say t h a t
Y
W ~ Subu(V).
V Subu(~)
to b e a s u b u n i t
said
1.6.2,
~CCrs
let
set o f all
W
(as it
B ~ Ate.
the
= Subb(l~).
(Co)-(C3)
every
fields
set a n d
base(V)
only,
in the p r o o f s
notions
set a l g e b r a s
CA
use only
~=/~
iff
Subb(~)
of
implies
(C 7)
-s
EHMT]
for
V
denotes
a subbase
~
We define
1~
~CBA
need
d Ax =
1.2.1-1.2.12
for
1.6.5-1.6.7
In g e n e r a l ,
Let
understand
~J~ECrs
that
A=B
for Boolean
Zd Sb V
Let
1.6.2
Crs
(C 5) a n d
EHMT]
are
the a b o v e ,
to c y l i n d r i c - r e l a t i v i z e d
denotes
Subb(V)
of
proofs
a
C A -s in
2.6.18.
stated used
for
and
(C0)-(C3),
EHMT3) . A l s o
and
~ECrs
axioms
their
Because
Then
O.1.
is a s u b b a s e
of
and
etc.
Then
: O}
Therefore are
that
xEA.
: A(~)x
axioms .
fact
let
1.6.1
they
only
we have
set a l g e b r a
Definition
= At
for
be applied
argument
of
.
Crs
since
1.2.2-1.2.12 EHMT~
in
on p.177
EHMT].
[HMT]
for Hom
introduced
the
and
{xeA
the
etc.
of
,
Zd~
that
similarly
Zd,
despite
although
proofs
noted
A,
cf.
we have
Crs
and
~ECrs
are v a l i d
for
is e x p l i c i t l y
let ~ x},
EHMT]
because
are
notations
+,',-,O,1
true
A1fCIs(~,~)
the
( Zd~,
[HMT3
are
that
of
to be a s u b b a s e
= Subu(l~)
and
~
is a
of
iff ~
W iff
Y
is
134
0.2
The that
above
a subbase
Notation:
Let
subbase
be e m p t y
KCCrs
and
iff
~
~ {~eK
: (VUeSubb(~))IUlk~}
that
notation
agrees
the one-element
with
Crs
agrees e:O
with
EHMTI3
i.i
(vii).
Note
.
be a c a r d i n a l .
: (VUESubb(~))IUI=x}
above
has
might
of
K ~ {~EK K The
definition
Then
and
EHMTI3
is in
5.6(17)
and
Crs N Crs
EHMTI3
for a l l
7.20.
Note
x since
it
no s u b b a s e s . In t h i s
I•
That
=
{~eCAa
is
Lemma
0.2.
recall
is a v a r i e t y
consisting
EHMT]
we
of
2.6.54
Let
and
EHMTI]
$J[eCrs
Then
is t h e d i s j o i n t
ii)
Let
WeSubu(~). some
~EGws
Moreover,
=
O . ~ At
Zd~J[,
lemma.
Let Then and
zd (i)-(ii) (VxEA)
below zd(x)
hold. :
135
0.2.1.
= H{yEZd~ closure" (ii)
If
Proof.
as
Proof Zd0~
2~
and
i E
n+l
x E Is
x>2
and
and
Ws
and
is finite.
and
assume
EHMT3 O . 3 . 6 ( i i )
a
~
I = ~x
the p r o o f of the first
t ~ Is(~,6%)
Let
then
rI(_X)~.:(~ "
V~ = ?~
is not a
.
~.
and
Let
Observing
Let
~ K
for any set
LHMT3 0.3.6,
is a b a s e - i s o m o r p h i s m By
~d
d ~U(5)
n
E
either.
f = ( < znX
statement
~t(X)
in
x>2
~X
Gws norm ,
-units)
If
which
let
Then
but
Is(4.~/d~)
-unit.
k(X)=-X
V
and
E
h< 1,0 ) = ( 0 , i >
completes
for some {x n}
t
E K
Ws
-unit
Ws
By
x2 d a { 2 , O } ( 2 )
t(Xo)=Xiux 2 d -- ~
is a
and
define
of
-unit.
since
with
the s e c o n d
=d ~{1,2}([), is a
X
~X
~ d ( s : iE~ >
• ]r~. C l e a r l y
-unit
n
to the
proof
H~ and
DmH~IH={O},
QED (Lemma
~6CA
and
1.3.3.)
thus
XCA
Then
.
Let
B 6 Su~
and
B C Sg (~) (XNI)
is g e n e r a t e d
{b/I
by
: b9
and
is o n e - o n e . ) of L e m m a
be fixed. XNIH~Dm H
by
Then
by L e m m a
1.3.3. iH E
Ii~
1.3.2.
D m H = S g ( X n D m H)
~J~
Let
,
,
Dm H E Su~,
Hence by
be g e n e r a t e d
by Fact(e)
D m H 9 Su~)5
.
by
148
I .3.4.
Lemma
1.3.4.
Let
~>O
Statements
(i)
b.
x
is
{i}-regular
e.
x
is
H-regular
for
some
d
x
is
H-regular
for
all
9
and
x
Proof.
is
Proof
of
exists
bERgkQRgq f~P)
is
we
in
O
(i):
If
=
d.
GCHCa x
, holds
then
. Then by Ax#O
~>0
a.
G - d.
that
finite.
the
Then
is
obvious
We .
Let is
k~P)Ex--
by
observing
. Let prove
Since
FC
that H#O
there
P ~ F~(HUAx)
finite,
we
have
Then
.
since
kex
H -regular. H#O
F -regular, not
(ii)
was but
essential while
~i
0 -regular). of
the
-regularity are
following:
-regular).
is
assumption
by
Suppose
By (AxUHUF)Ik~P) i q~P)~x
is
it
the
c qEx
P
is
the
kEx)
R -regular3
fixed.
have
element
Wlu~{l} c.
if
hold
direction
-regular
in
and
-regularity.
this
~
is
-regular.
Since
x
that
a.
be
x
. G
HUF
is
HUF
of
of
~
is
-regularity
-regularity d.
x
.
R1 Z A d RI(Z){)~d Sg(~)
We shall omit the s u p e r s c i p t s
4)5
rlz: A
and
A
and
if there is no danger of
confusion.
The above d e f i n i t i o n of f r e q u e n t l y use the fact that Z C Zd Sb i~.
by
we have the following c o n n e c t i o n s with earlier
The n o t i o n of "i-finiteness"
~niteness"
Let
1.6.2.)
By P r o p . l . 6 . 2
JAN13
h d fko'O
lOAx~kChEx
We have seen that
QED(Proposition
of
since
~CGws ~
Now
To save space, we omit the proofs of
papers.
~ ~ Gws ~
0-regular.
f,kel~ , c~
( V f , k e l X ) [ f O : k O ~ f=k3.
r l z " agrees with A (~,~Z ~) rlzCHO
[HMTI36.1. for any
We shall
~]~6Crs
This fact follows by the proof of [HMT32.3.26,
and a
d e t a i l e d proof can be found in IN1]. P r o p . 2 . 2 below says that r e g u l a r i t y can be d e s t r o y e d by
rlz,
154
2.2.
unless both
A[V3z=o
and
needed by P r o p o s i t i o n
Proposition
2.2(ii)
Let
(ii)
For every
6~ECrs reg
~
Proof of
and
Let
pEy
and
by
y
Then
~ C C s reg
~Z ~
and
y
there are an ~Z~)~
~ E C r s reg
be arbitrary. by
is regular.
ZEA
such that
is regular
in
~,
in
Cs[ eg
and
Z E Zd#]~
by
y
is regular
By
~)~, and therefore
too. This proves
Let
that
~
iE~.
ciY~ = (c~iy)NZ =
(iUA(~)y)Ip~q. A~y = A~y.
Let
Z E A.
and therefore
be such that
is regular
yCA
We show that
by
~6
is not regular.
and let
Then
Z E Zd{%
(iUA(~)y)Ip~q
in ~ .
Then yEA
~
q E i~
and
qEy.
Let
~ E C r s reg
We have seen
is regular,
since
was chosen arbitrarily. Proof of
Let
~
(ii):
be the
{X,Z}_c Sm @5 X
Let
q E i~ = Z
Z!l ~ and
that
~2
A(~)y = A(~)y.
we have that
y
(i) :
yER
Thus
Z E Zd6%.
such that
ciY~ _< c~Z = Z
= ciY.~
are
(iii).
there are
Z ~ Zd Sb 1~
Then
Both of these conditions
~ ICs reg.
For every
{~Z~)~ .
and
and let
~w
~Z~]~c Cs
Proof.
hold.
2.2.
(i)
(iii)
ZCA
since
Let
Cs
Since
X
and 1.3.
Clearly
X#O
and
Let
(VFc_~) (~i~F)
regular by Theorem
= Z = ~2.
.
p ~ ~xl,
with base 3 and generated
since
XcZ.
~
XeR
X#I ~
Z
c~c(F)Z=O,
are regular by Let and by
~ ~ . ~X:O ~w
X ~ ~2 (p) by
{X,Z}.
Then
~ECs
Therefore since
QED(Proposition
2.2.)
{]L is 1~ =
(despite the fact that A~X = a). we have that
~
is not regular,
by EHMTI34.3. (iii) of P r o p o s i t i o n
Z ~ ~2.
and the same holds for
AX=AZ=~. Then
and
2.2 is a consequence
of Prop.4.11.
2.3
.
155
About
Proposition
Proposition Proof. ~d
2.3.
Let
> }.
A = {V,O,{(O,1 _
x d
[HMTI]2.10.
), < 1,2 >, < 2,3 > } show
> }, {< 1,2 > ,< 2,3 > ]}.
Let
> }EN
{< 1,2 ) } E R I v ~
~J~ECrs 2.
and
We
{O,
Let
q C Z r . N r. +.
eJt
n>O
Nr d u{~U i
Say
if
Then
is a f u n c t i o n
: n > ~ - U i.
rEmn
n>O.
(~icI) IUiI=n.
m>O.
Set
every
qEx.
qEN r
m#O
3.4.1.1.)
of r e s i d -
lemmas.
xCSg(~)G x
I. T h e n
: m]qE{kior
and
Since
and
Further,
Moreover, there
: (3i~I)qo=kiv].
For
suppose
~.
and
IG(x) Ii
which
~
and are
not
~>O. lower
is s u b - b a s e - i s o m o r p h i c is h e r e d i t a r i l y
nondiscrete
~>i).
"lower
Zd B
Then
h 6 IS(~a~,~h(a~
: a E At
of
in P r o p . 3 . 4 ( 2 ) , ( 3 ) .
there
There
by
h(a)
is a b a s e - i s o m o r p h i s m .
condition
isfying
(iii)
Let
an asymmetry
3.5.
Then
if
by
hence
Let
3.4.)
necessary
(ii)
Then
in a k i n d o f d u a l
Proposition (i)
2.2(i).
Is(~,~).
Zd A,
WdSubb(~),
is a c o n s e q u e n c e
3.4
exhibits
h e
is b a s e - m i n i m a l
base(~)
that
Let
a c At
some
since
R1 a A 1 h -c ~a.
seen
Then
for by
characteristic.
>~
base-minimal.
a ~ ~Y.
h(a):eW
IY I< @ @ ~
: base(~)
section
are
E csregnLf
that
We have
~
base-isomorphism"
cannot
be r e p l a c e d
with
"base-iso-
182
3.5.1.
-morphism"
in P r o p . 3 . 4 . ( 3 ) .
Then
are
are
there
"Lf" For
GsregnMn~
not base-isomorphic.
between (iv)
~,~c
GsregNLf
cannot every
-s
If
there
Let
~6w~5
such that
H /F
the
strong
Proof. (F, ( U and
Let
:
In fact,
f,8,g
for any
ga = {qC~(Iu/F) so it is c l e a r
that
so
{jEI
: (ViEAa)(
of
a,
=@W.
Hence
QED(Lemma We
b e as
the
IU/F
and
~]L+ { ud~f)~
e = c -1
Let
F
by
b e an
in
such that
is a s t r o n g s {
e
from Let
~+
c
We
claim
to
45 .
be an
c(i,eu):u
EHMTI]7.12.
a E A ).
u d -I c ~ c I s ( ~ + , ~ )
in t h e h y p o t h e s e s .
function
:
ud and
Then
induced
ga c u d ( a ) . 6 F.
For
Now
all
for a l l that
iEW.
qEV.
~
We have
and this means that by
implies
Assume
d q =
q~Ex
and
u~W
~
c0x=clx=1. : V.
q~ E xnv by
q~E\7.
that
It is enough
{<m-1,M >,<m+l,M > } ~ W. Let Then
rl V E
5 W.
_EV3.[xAV) : C ~V3(xAV) so UO
qCV
u = (m-1,M)
Z•
~W (P) E Subu(V).
p(0/<m,M ) ) (1/< m,M ) ).
E C-[V3(xnV) O
Gws -unit and assume that
(VWESubb(V))(3MEK)
to show that this implies _
be a
Therefore
Then
q E
for some
u.
Similarly,
q E
< m + 1 , M >EW.
QED (Claim 3.5.2.2.) NOW we prove Assume
that there is a
Then there is an that
3.5.2=3.5(d) (c). Gws
~eGws
rl V E Is(~,~)
rl(~V) e Is(~,~*~).
sub-base-isomorphic with unit
and there Let
is
V
to both
and with base
f : y >4 U
h ~ rl(~v)-ioforlv 9
~ y c U
and
such
such that Then
~
h E Is(~,~).
.
165
3.5.2.
By C l a i m that
3.5.2.2
(3 LEK)
_c W.
that
and
3.5.2.1.
Let
c_ Zx{L},
zeZ.
Let
therefore LEK
be
9 VnQ]
N ~ {uEY'
: (3qeV)q~
9 V~Q_}
the
since
definition
regular
and
of
~
Q
n
: w >-
EZ) l{iEw order For
and
even,
= "least
let
wE f*N
Q9
that such
that
Then
Let
y,
d Zx{M}.
Let
by the
and
and
AQ = I.
definition
in t y p e
mappings
t
and
~.
ITI=INI=m
Since
h(Q)
of
h
(VuCf:~N)q~E-h(Q)~.
Such
that
c Z•
Y' : Z•
t i : "least
such
be
show
f(z,M) ( 1 ) : f ( z + l , M ) (1) by Claim
f~(Z•
be two o n e - o n e
f~:N
to s h o w
eW (p) 9 Subu(V)
is r e g u l a r
: t 1.+z:n.}I-~
be
of
3.5.4. and
claims,
Proposition
3.5(ii):
We
shall
Let
~=[ylk~.
(VVC_I~]EIVI
:
Mne
there exists
ha
(( ha(Pa ) : aEL > F d < Eiha(X,a) =ha(a):f(a)
and
Gco.
be two
and f C_ F. Let
Hence G --w c B
LCZd~)~.
Since
:
~L=I ~
Let
: aeL}
proves
[~(~)Rgf. Let
~ d [~(~)~)L and
d L : zd
we have
and Let
aeL.
Then
n, hence by EHMT]2.5.25
e iS(Pa~L ~ a ~ , P a E L ~ f a ~ )
and hence
(~':) below.
(:.~) (VGC_ B) (3F e Ism( ~ 0 1 ) G , s
g~(0~)Dof ~
By EHMT~O.3.6(iii),
: xcQ > c Is(~,Z). L1f _c F
Then there
by EHMT]2.4.7
are both of characteristic
: p~paCL(RIa~)>
both of cha-
be arbitrary.
xER > C I s ( ~ , P a E L ~ f ( a ) ~ ) .
e is c Is(~,pac L ~{~a~)
: aeL >
~fa~CE
statement
to be the unique
We base the proof on the
(:'{) there exists an
Assume the hypotheses.
< <x,f(a) ~a~'
By
and hence
~(s
< < x,a
~(H)
see HausdorffEH].
BA-freely
FEIsm(~(~A)B,/[)
Proof.
Sb H.
(VzEC~{O}) IzI=y +.
: ~ey+}
3.5.5.
exists
we define
3.5.4.)
racteristic
d
H
Then
_
V~
Z ~(y+)BA QEm(Lemma
B=2 x .
~ C ~(y+)
IzI>y.
such that
below,
be a cardinal.
ICl=y +
(Vzs
d At
~ li~l 9
(VxCA~{O})IxI=~.
BA. Let
Clearly
(~) and
such that
Lemma
I~[
set algebra with universe
Let
with
with
For any set
well known result (e)
~
c_ F.
Let
aEL.
G1f _c F.
Then
and hence F(a) =
We have proved
170
3.5.
By ~
3
clearly,
Z#Y.
we have
For
EIs~
(by
and
and
rlQ ~ I s ~
Therefore
that
@i
is such that
seen that
W(f2)Nz=O
rlo E I s ~
and
~ ~5
~
rl(Wn)
n•
and
is
and
IKI=~. let
that
= ~v>~
Then
that
be f i x e d be
IHKI
T { {{qC~U
v~L CH
KcU
~ U'
:
every
be such
k e U~K
d x = {qE~U
For
Hey
Let
Let n
in the h y p o t h e s e s .
Let
~ ~ ~U.
K c U'
function
and
that
Let
exists
Then
U,~
IU'I>~.
Such
a
We d e f i n e
HIqcHu'}.
that
ve. (~iEHNI)y
Suppose
there
Hom(~,~)#O
Suppose
-< -d(~)Oi
h : {~-
since
is a subbase
W
(~ieH~l)X~ of
~5
O i
such that
IWI~.
(Then
Then there exists NOW,
U>e.
AX=IUH.
Next we show that for every
~
Let
I~HI~.
~ d= [~ (s
and
such that
q~c0Y
Therefore
a
since (~)
co
.
~
y~l.
qe~W~
W~O
since no subbase such that
(~w6W)[(3iEH~1)qi=w But we have
(~
Co)X=l~
is empty
qe(H~1)=W, and thus (by
if
since
~#O.) ~eGs
.
qwO~vc ~ - - d(~)] Oi "
IUI>~),
contradicting
3.9.
175
h~Hom(~,~). every
We have
subbase
It remains in section
of
seen
~
that
Hom(~,~)#0
and
~EGs
imply
that
is of p o w e r >~.
to show that
Reducts,
~
is regular.
in Prop.8.24,
6 ~ E C s reg
will be p r o v e d
because
it uses m e t h o d s
remains
true
of that
section. QED(Proposition
3.8.)
We c o n j e c t u r e "IAI~" by
is r e p l a c e d
~
elements
Conjecture = Sg(~)G Assume
Let
K E {csreg,ws
~ = ~
~u
to some
that
condition
(VxEA)~IAxI".
lwOGUSl
< • _ < _
K
}.
Let
Ibase(~) I.
the above
with
" 0g can be g e n e r a t e d
In p a r t i c u l a r :
~EK,
Let
Then we c o n j e c t u r e ~E
if the c o n d i t i o n
that
SCbase(~)
A =
i d = U{ iAxl +
: xEA}
6~
is strongly
ext-
SCbase(~)
conjecture
is i n t e r e s t i n g
only
in the case
~w. and
R d O{qi~.:Axi : i function.
Let
AxltoqCk. f(x)EB
d d [1{dij : q(i)=q(j)
and i,jCAx}.
Now
by
~eLf
qEd(~)Ax.
Thus
kCd(~)Af(x) : ieAx}. pE1 ~
Thus
qexEA,
IAxI<w
Clearly,
fCIs(~,~).
and
f]t and
(Existence of partial b a s e - i s o m o r p h i s m s )
@5,~csregALf
Proof.
to both
.
~hw.
(ii) :
Then there is some
with the same base,
Let
Proof of
e#{.
~ Ecs[egALf
~ .
have d i s j o i n t units and
for some e x t - i s o m o r p h i s m s
of
and
base(~)Abase~):O
Then there is
on
~
Assume
(iv)
Let
Then some
Then
and therefore
d(~)eA
f(d(0t)Nx) = d(~)Af(x)
t : Rg(Ax]q) Ax]pCtoq.
by [HMTI31.13
is regular by
is a term in the language
be arbitrary.
be such that
pEf(x)
d
and i,jEAx}.
~ C C s reg.
since
Define
# O
by
t d
>- base(~5) Then
and
is a one-one
Ax]pC_k,
Ax:Af(x),
by
kCf(x),
3. I0.1 .
177
2. Let
new
and assume
that
qOeXo,...,qn+leXn+l
.
permutation
such that
A
and
of
AXn+ 1
and
~w
and
~ ; see
have
~ are
Let
3.10.1
that
iff
O.
Let
be a finite
Such a
T
operation
By [ H M T ~ I . I I . I O , By
9
exists
and by
Recall
By
both
since
~,~EDc in
#]LECs NDc
EHMT]1.11.12(X)
i~n.
Xo,...,Xn+lEA,
D ~ T*AXn+ 1.
is a d e r i v e d
T
Let
we
we have
the n o t a t i o n
f[H/k]
from
We d e f i n e
d Yi = x i ' S T X n + l
d
h i = q i E D / q n + l O T 1].
and
Then
hiEY i
Then,
by our i n d u c t i o n
since
: R 1 >~ base(~) pEf(yi)].
~
with
Let
>- base(Q) iNn+l
Let
poT--IESTXn+I 9
A ( S T X n + 1 ) ~ T * A X n + 1 = D. section
s
~9
n.
Let
T*(AXn+l)nA=O.
by
[HMT31.11.9.
PEXn+ l
for
A ~ U { A x i : inn}.
finite
it follows
holds
is regular. hypothesis,
R I ~ O{Rg(AYil hi)
we have
the p r o p e r t y
that
a one-one
pEI ~
function
be such
function
and let
such that
that
: i~n}. tl :
(VpEl~)(ViSn)[AYilp~tlohi
R = U { R g ( A x i l q i ) : iSn+l}
be any o n e - o n e
and let
Let
t : R >~
(RnR1)Iti~t.
Axilp~toqi.
~
Let
We show that
p 9
9 f(xi). Suppose
first
Ayi~D~tlohi)
i~n.
by
pel ~
and
Thus
p'Ef(xi)
show
Axilp~p'.
Now,
H1h i--qi c
Let
R g ( A Y i l h i ) ~ D o t 1.
Rgtl~base(~ ) .
H1tlohiCtoh i
since Let by by
p, d= p l a y i/tlohi].
Then
Suppose
AXn+llp~toqn+l Now
H { A Y i n A x i.
H A D C--A x i A D = O
i:n+l. and by
Let
p' 9 ~
since
by
: HIp
Now
,
feIs(~,~).
to show
We
H1p~p'. Therefore
Then
by the above
Hip' and by
Axilp~toqi. D1(tOqn+lO~
D = ~*AXn+ I.
H ~ AYoAD.
~eCs
h i : q i [ D / q n + l ~ -i].
(as desired)
We have
f(Xn+ i)
since
A Y i l P ' ~ t l o h i. by
It is e n o u g h
and
Dop'=~
Rg(Hlhi)~Rg(Axilqi)NRg(AYilhi)~RnRl.
p ' ~ f ( y o )= f ( X o ) ' S
D I p ' ~ p o ~ -I
p ' e f ( y i)
p' = p [ A Y i / t l o h i 3 ,
next
Therefore
f(yi ) = f ( x i ) - s T f ( X n + l )
= H 1 t l o h i = H 1 t o h i = H1toqi H : AYinAxi,
Now
Let
-i
) ~ pot
-1
by
p' { ( p o T - l ) [ A Y o / t l o h o ] .
just as in the case H 1 h o ~ q n + l O ~ -I
iSn.
We show
and t h e r e f o r e
:
178
3 . I0.
HIp'
= H 1 t i o h O = H 1 t o h 0 = H1toqn+iOT--i = HIpoT--1
~ R n R I. and
Then ~eCs
QED(Lemma
poT-les
reg.~
EcsregALf
I.
Let
§) = ud AF e Is({J%,~)~
U,W
.
Let
U + -- IU/F.
Let
Then
since
Gi ~ELf
Define
3.5.1.
~ "
since
By L e m m a
Let
such t h a t
(Vis
b
and
qex.
Then
it is e n o u g h definition
ud B
and
{ieI
Then c < bq(J)i
the d e f i n i t i o n
~
in
Lemma
3.5.1.
of
qjER i : j t..
Then
is
Then
1
: R. >~ U 1
Let
To p r o v e
beW(Iu)
Z d m(<x,Axlq
: weW > . Let
x~B
boqeud(fx)
) ).
Then
Let
iez
bq(j) i -- ti(q(j) ).
by
be
by the
(~ieZ)q ~':(Ax)CRi.
boqEud(fx)
Let
: k~G.} m "
b d E G i.
jegx.
tio(Axlq)
b o q E ~ ( U +)
to p r o v e
of
and
N e x t we s h o w that
e d= ~ - 1
and
is
ultra-
ud =
IK] E G i) E t i o q C p
Then
F
Let
be a r e g u l a r
#J~+eCsreg
Then
~J[,~E
feIs(~,~).
Let
and
G.! d {keK
is finite.
F
IEl = I I l
such that
and
f(Xn+l)EB
Let
e : U >-- U +, ~
: qexeB}.
exists
Let
Let
as in L e m m a
such that
E
be o n e - o n e .
i~I.
1--W
be
S u c h an
: K >~ E
R.C W
ECF
~_>m.
respectively.
K d {< x , A x l q >
is finite) .
ZeECF
Let
fit >- IBuwu~I.
be d e f i n e d
Let
~ELf
Fix
p'es
EHMT]I.11.10.
is a s t r o n g e x t - b a s e - i s o m o r p h i s m
3.5.1. by
by
to the p r o o f of 3 . 1 0 ( i i ) .
be any set s u c h t h a t
ud-lc_e
A(sTf(Xn+I))~D,
pef(Xn+l)
be of b a s e s
f i l t e r on
m
Then
by
R g ( H 1 h 0)
3.10.1.)
N o w we t u r n
I
f(Xn+i)
by
- U, Note that
i.e.
base(~)~Dof
and ~ u ( r ) ~ W # O . Let p ' E e u ( r ) n ~ V
for some
p''eV=ee (p)
q'=g.q''
~,~e
and since
and
and therefore
for some
I{iCa
and
to
q''EW:~ g
(q)
and
is one-one.
: p'(i)*q'(i)}i~w,
s
V~Zd Sb~H Then
~w.
Let
By
and this
respectively V ~ U{eH (p)
~ e G w s c~ with unit
and
,
: perU}
are
Ibase(~)l =
and
an~
ext-
base(~')
W ~ U{eH (p)
fo~ some
J5
Let
~',~'eCs
(~U)EIs( ~', ~ ) .
U~Y=O
~
rl(ey)eis(~+,~).
by [HMT]0.2.10(ii)
By
such that
and
such that
and such that
Similarly, W.
~
there are
D' ~I rl v ~Ho( , ~)
~'=rl m rlD(~u)orl
Omp
respectively.
be a cardinal
respectively
and therefore
we have
to
~
U,Y
Then by [HMTI]7.25(ii),
rlD(au)eIs( ~ ,f/)
we have
sub-isomorphic
, ~
Let
be of bases
rl(eU)eIs(~+,~)
IHI=~
= H.
Cs
~+,~+eCs
Then
-isomorphic
~Gws~
base(V)Mbase(W):O.
rls
we have
Let
to
Then
some
such
Ay:Ax:Af(x) .
lUAxIp~q.
f : ~ >- U
IRgqi:i~i~w
~>IAUBUUOYI.
H ~ UUyU~.
~U~V
xEA
Let
p', q'E~U (r)
= I b a s e ( ~ +) I=~.
V.
by
IRgq'i~w
sub-isomorphic
unit
There is
q ~ (0
~ : ~ U (r)
Also,
, U~Y=O
= base(~')
Let
p'=fop''
IRgq' I~w
~
and
~(~W)
eu(r)N~V#O
IRgp'l~,
the
~
definition
(F, 9149
filters
(D,(base(~)
Then
s149
3.13.
=
there
IHI~,
3.12.
Problem
.. UddD'~(~
by
parts
: ). 1]
d
g = < {qE~
fE
is regular
x E A >.
i,jE~.
(i)
~
a construction
be a
is a s u b - i s o m o r p h i s m
the h y p o t h e s e s .
: pEF/q]Ex}EF}
:
5 -
: ic8 >
f
:
Define
is a f u n c t i o n
and
Rgp~Dof
rd P ~ rb p* "
Lemma
4.7.1.2.
Then
(i)
and
Let (ii)
(i)
rd P E
(ii)
If
below
Is(~
V
p
: B >~ hold
p~V,
is a
~
and
be one-one.
Let
rdPV
is
CrsB-unit.
and
rb p
~rdPV)
Gws
-unit
with
a
V
: V >~
subunits
be
a
rdPV,
{~y(pi).
Crs
if
: iEI}
-unit.
8#0. then
1
rdPV
is a
Gws
-unit
with
subunits
B {B(yi•
Proof.
Let
-unit.
Then :
g
Then
P
Notation:
Proof and
: i~I,
of
y(q)
d
(i) : {gop
: B >-- ~ For Let
be
every qEV
: gEV(q)} "
f(q)EHo(~P~V(q), and
Rgf(q)
b(q)
defines
a base-isomorphism
geV be
f(q) by
Let function b(q),
g,
d V(q)
d < {gop
: gEx }
d
on
base(Y(q))
see
EHMTI]3.1.
d {gEV
since
< (u,q')
V
be
a
Crs
(~NH) Ig.
Define
EHMTI]8.1.,
b(q)
Let
H = Rgp.
denote
fixed.
Let
d
Let
we
~Y(q))
= SbY(q).
is a o n e - o n e
one-one.
g e ~ Y i(pi)} .
:
: g, _c q} xESbV(q)
( V g C V ( q ) ) (gop) + :
: uEbase(Y(q))> and
) .
therefore
9 b(q)
-
192
4.7.1.3.
Let
= "b(q)(Y(q)) d -~
W(q)
~}W(q))
and
: b(q){gop every
W(q)=W(g)
~>i.
Let
iff
if
~W(q))
q'=g'
and by
we obtain that and
Proof of
~ ~P[@v,
rb P
(ii):
rdPeHo(~,~),
is one-one Let
{~y!pi)
on
to
: q E H / P i ] E ~ y i(pi)}.
and therefore
Then
therefore Y(q)
for some
is one-one
: iEI} = Subu(V) V(q)
= {q[H/g]
: ieJ}.
Let
~y!pi)l : ~y~pk)
is a
Then
Let
. By
6~V(q),
because
qEV
W
Let
J ~
: (~iEJ)g E Hy!H1pi)}1 { By!piop)
for
iEJ.
l
q[H/g~
which implies
Gws -unit with subbases
~
"
1
i,kEJ.
g~EV(q)
and
"
Y(q) = o{BY! pi~
gEwiAw k
~
by
V.
1
Let
then
SbV(q)IrdPEHo(~P
1
d= {iEI
iff
RggNRgg~#O
since
rd p
Let
A(~)W(q)=O
iEH, g C V ( q ) , g ~ e V
(VqEV)
~W.
V(q)=V(g)
Therefore
[HMTI]6.2
for
~{
iff
g~eW(q)
Therefore we may apply
EHMT]0.3.6(iii)
rd p = rb P'"
then
=
: gEx} = rdPx
V(q)mV(g)#O
since if
h(q)EHo (~f6~V(q),
h(q)x = b(q)(f(q)x)
: qEV},
iff
i gaEW
A(~)V(q)=0
Now
: gEx} = {rbP(g)
W ~ o{W(q)
gEW(q),
g, = (gai ) ,
[HMTI~6.2,
since
base(W(q))Nbase(W(g))#O.
iE6,
Also,
d : ~b(q) of(q).
= SbV(q)Ird p
Then clearly
because
h(q)
: gEx} : {b(q)~gop
x ! V(q).
g,qEV.
by
h(q)
and
p k ) l
Wi=W k.
{By!piop)
and
This shows that
: iEJ].
This
immed-
1
iately yields Lemma 4.7.1.2(ii). QED(Lemma
Let ~U
4.7.1.2.)
V~U,
f,qEV
as follows.
t(f,q,H)(s)
{
and Let
H~.
sEV.
EHIs3
Define
the function
t(f,q,H)
: V
Then
if
( ~ H ) lq!s
if
(~--H)IfCs.
otherwise
Lemma
4.7.1.3.
Let
V
that the bases of the coincide. E Is(~P~v,
Let
H c
~P~v)
(unique) ~
.
be a
and
Gws -unit,
~.
subunits of p : IHI > ~
V H.
Let
f,qEV
containing Then
be such f
and
t(f,q,H)* 6
q
4.7
. 1.
193
Proof~
Let
= {~y!pi) 1
Let
V,f,q,H
: iEI}.
W ~ rdPV.
Then
b
: base(W) and
(ii)
and
by
Lemma
4.7.1.2(i)
IHI
f[H/p]EV
ge~y!} pi) l
: iEIHI> ) = q E H / g ]
to the p r o o f
of
f~y
since
Is(6~W,~W)
rd p E
~
~
=
g':(~H)Ig
: iEIHI> ) = g = t ( f , q , H ) g .
and
is e n t i r e l y
(i),(ii)
such
~ 6
b(w)
Subu(V)
4.7.1.3.)
unit
(I)
: ieI,
Let
we d e n o t e
: W
= rbP-1(b~
: iCIHI) ) = rbP-l( n ) q k : n } .
(VX,yEG)Ex#y
n{DmAx
by
satisfied
V.
small
{qE~w
is said
of h e r e d i t a r i l y
(3).
Crs
in
is i - s m a l l
eG) J A ( U G ) ~ A X I < ~ .
Let
(3).
V.
(iii)
y
immediate.
are
of
below.
y : {qEe~
i-small,
by
be a
(i)-(iii)
(i) ~
4.7.2
zEIg(~)S(HOO)
_~ z
conditions
Then
of L e m m a
the p r o o f
4.9.
QED(Proposition
4.10.
is not
Then
c(@){q}
conclusion
complete
Since pE c(@){q}
disjointness
is w s m a l l .
We h a v e
4.10.1.
element
c a n be o b t a i n e d
from
small
ones
200
4. I I.
by u s i n g
4.10.1.
x = {q~2 then
and
z
Propositio~ and 4.9.
About (15)
of this
: (3n~)[qh:qn+l:l
y
4.7,
An example
4.11,
For more
4.11-4.13
see s e c t i o n s
below
(V~)
construction
see
there.)
(V~2)
is g i v e n
Prop.4.11
by r e l a t i v i z a t i o n
element:
if
O~{y,z]CSb_ x
disjoint.
of P r o p o s i t i o n s 5 and
given
(~ ~
In [ H M T I 3 5 . 6
in [ H M T I ] 5 . 6 ( 4 )
cs~eg) (~ ~ H 6 % )
in P r o p . 4 . 1 l
below.
a zero-dimensional
can
[Zd~l>2.
(Actually,
a l s o says that r e g u l a r i t y
with
4.6,
6.
[HMTI75.6(15).
that the c o n s t r u c t i o n
to s h o w t h a t
~cs~egmDc destroyed
applications
wsmall
Indeed,
4.13 b e l o w are a p p l i c a t i o n s
it is a n n o u n c e d
This m o d i f i e d
following
, (~k>n+l)qk:O]}.
are not h e r e d i t a r i l y
Propositions
be m o d i f i e d
is the
can be
element;
see
Prop.2.2(iii).
Proposition ~& ~W
4.11.
cs~eg~mc ~
~
Let
le--Hl~w.
~kw
and a
( G w s rC ~ e
Proof.
Let
g
~Zd
,
~
saw
and
h
: w >~ H
Let
and [~
IZd~w4)[l>2.
~2.
Let
HC~
be o n e - o n e
and
~a2.
Let
Ri
{q6a~
: HI qEHx (pi)
~ ~ [~
Claim
4.1~.i.
Proof.
Ax : H,
Then
[Hl:w
be such that
and onto.
Such
and
H,h
Let
and
~{ and
( H ~ h ~ i ) l q C_p i } .
[ ~ ( & ) {x}. R
w
4AEcsregnDc
= {q6~
p
pEW+l(H~) e x i s t by
and be IHI=w
and t h e r e f o r e
QEZd~
.
Let
Note
x d= U ( R i
: iS~}.
that
(Vi<w)R.eSS 1 : HI qEH~(Pw) }.
.
is r e g u l a r by u s i n g P r o p . 4 . 7 . E H ~ ( p i ) }.
are an
Define
an d
U{R. : i<w} q Sm s 1
There
such that
and
( V i < k ~ w ) p i ~ H (pk)
Let
1~
be a r b i t r a r y .
a
such that
iSw.
~A2
~EDc Let
G ~ {x}.
by
l~--HIkw.
Q ~ {qE~ Then
~Gws
We s h o w that
: (3i}
k~H~G.
Yi~A
9
since
: jEj]
be such that
subbases,
Let
: k,j~H}.
I~NHI~,
naYi(a) (pi(a))
Hc~
is a set of disjoint
is nondiscrete.
{~y!pi)
.
Then
Proof of 4.17.:
[HMTI?7.16,
let
Let
such elements.
QE__~D(Proposition
E Gws
i~G
(21a~HI) +
: i~G}CA_
@i~ICs
there exists a
with unit
~ q l Cs
(Vi~G)y i ~ ~{dkj only
and
by a single element
ci(x-dik)-dik. {Yi
~
(i/< l,i>).
with
have to show
Then
Let
For every
: ~2, Cs reg
and
,
for all
IEII~
and
Ws
is the following.
~EWs
and
and for all
rlvEHO~
and let
Then there are
q6x
(3F ~
~)f6c(F){q}.
since
rlveHO~.
{ rlv*~ . and
fE1 Z Nx.
Let
x:VNy,
A contradiction.
one could say that some
By
yEB.
A(~)x=O
~EWs Then
and
and
d {~0%6CA
V
such that
Let
~Ws
x q{O,1~
i~ ~ i~
, }.
we have
fEVAc~F)y = C(F)X = x,
A n a l o g o u s l y to e x t - i s o m o r p h i s m s ,
csreg-s are " e x t - h o m o m o r p h i c "
d e c o m p o s a b l e CA-s, w h i l e
Dind
Suppose
By 4.11
V6Zd ~I ~
The latter statement can be seen as follows. ~
JEIl fS
cf. 4.13 and [HMTI]6.16(2).
~EcsregADc
for some
I Z d ( r l v e ~ ) I2
5.1.)
IZd(rlv*4)[) I>2
However,
Let
Then by [HMTI]6.11,
and clearly
But a d i f f e r e n c e between we have
~.
(i)
to d i r e c t l y
Ws -s are not.
: IZd4il_,
has b e e n defined.
By Thm
8A2.
Then
~@ ~8
where
1.3,
and Ord
~
8A2.
5.7.2.2.
(ii)
~6
~ I 8Cs~
Proof.
Proof
ordinal
p
: ~xPS/F
~ PB
6he.
for e v e r y
BeOrd~2.
(i) :
Let
be an
UdcFEIS(~6,Z~)
Let
= IP~/FI
where
By
-choice
F
exists.
function
: j
.
~ EbC s
~+Ecsreg8 ~
be an u l t r a f i l t e r
Such an
[ = --
since K E
and
let
~(h)~
~
be
to be
changes.
~L
the
~
IHI.
4.7.1.1
there
we
Recall
and did
{Ws,Cs,Gs,Gws,~ws
the
4.7.1.2
function apply that
rd (n)
to
the
not
assume
8
is
norm
_ comp _ wd _ creg ,~ws ,~ws ,~rs ,
define
d ~ A creg = mH CrSH
Related
be
=d < A , + ~
the
and
K H ~ {(rd(n))~(n)~ reg KH
~
~(h)~
CAT-S. with
and
.
Let
Crs}.
three
to
above
: H ~ ~
define
C h ( i ) 'd ~h ( i ) h ( j ) ) i , j E H "
present
(iv)
original
6.O.
{ ~(LIId)~ (iii)
the
restrictive.
~
Defin• (i)
of
Products
notions
: ~EKIHI}.
if
like
K H c_ G W s H.
BOH,
Nr H
See
etc.
1.6.1-1.6
are
9
defined
2.
analogously.
6 . "!.
223
Remark
Correctness
and
section
of
KH
n
: H >~
K
CA H
IHI
:
gives
Let
~ 6 K B}
of
all
UESubb(~).
that
if
a natural ,
then
rl(~U)*#J~,
UESubb(~L)
is v e r y
from being
property
different
[HMT]2.6.2 definition
enumeration
p : e >--
B
we h a v e
Reducts
: UESubb(~)>
one
the
be r e g u l a r .
the o t h e r
and
might
of
into
e Cs
be t e m p t e d
subdirect
below,
properties
-s
is a s u b d i r e c t
6.1 b e l o w
by the F a c t
Gs
decomposition
natural Thm
of
rl(~U)*~
subdirect
s o m e of
Thus
The
of the
and c l e a r l y
the n a t u r a l
true.
from
(iii).
decomposition
EHMTI]I.15
will
from
~,B
< rl(au)
or at least
in
choice
subdirect
of
follows
~ Hbase(~).
EHMTI]6.2
In v i e w
csreg-s
be as
of the
I~
then
~)i by
{)lEGs reg
far
K
by s e c t i o n
Crs H ~,
~EGs
decomposition
definition
for all ~ r d i n a l s
for any
EHMTI]6.2
yields
Let
since
that
-s.
Namely:
is i n d e p e n d e n t
= { r d ( ~ ) * ]~(~)~
Cs
above
Reducts.
and
We n o t e
of the
Gws
to t h i n k of
factors
states
that
regularity -s
for
this
is a
introduced
in
[HMTI]I. Fact:
Let
~eKE{Gs
rl(~u):~
eK
Proof:
Obvious
Theorem
for all
6.1.
(i)-(iii)
,Cs
,Gws
a~O3
and
x~2.
q
(ii)
rl(~U)*O[
~ Dind
(iii)
rl(W) * ~ ~ C s ~ eg ,
moreover
empty
that
I
d
Sb
O3
c~.
For
XH :d {q~e(~•
QED
There
is an
e G s ~ eg
for w h i c h
hold.
rl(aU)~:~
Let
Then
UeSubb(~).
(i)
Proof.
,Gws wd ,G w s ~ O r m , G w s ~ O m p } "
by the d e f i n i t i o n s .
Let
below
,Ws
W~2 ~
a->w any
ICs reg
such
and HEI
(< n,H>)
for
all
U~Subb(~
for
all
UESubb(~).
~>2. we
rl(W)::~
~ Cs n D i n d
for any non-
rlwEHO~.
For
any
define
: ne~
).
and
H•
set
s
let
s d <s
: i.
224
6.1.1.
d
y = U{x H : Hel},
F i r s t we
Claim
V d U{~(Hx{H})
show that
~
6.1.1.
I~AzI<w
Let
zCJ--{O}.
Proof. be fixed. infinite exists
Since
a finite
that
AZD~.~L.
Claim
6.1.1.1.
Proof.
there of
ra m E Ism( ~ L Then
~,
z~B
d i P : g( n,K>"
Let
Y : (Hx{H})x{(@~L)lq},
Then
by 4.7.1.2(ii).
Let
=}.
~
be the
of
rbL(P)
QED(Claim
be fixed.
F
H~I
by
There
z E S g ( ~ L ~ J ~ ) {y}.
We show
rb p : V - R g r b p
be
and
Recall
and
T h e n by and Let
: kor b L ( q )
J[ .
with unit
We h a v e
Since
q e z N c ( F ) X H.
y+NLz
pEc(s
lZ
Then
: W >~
W
k(y+):y+ Then
qez
N1kCmd
w e have
rbL(P)
E rdL(Z )
y+NLY =
: (VjEH)hj: be such that and
Then
(W.~(YUZ)) IkC
~Is(~
Then
is
~(y+nLy):y+NLz
since
m:Sg (~[) {y+}.
rbL(q)ErdL(Z). Since
rb L
Then by
is o n e - t o -
implies
pEz.
Now
for some
nEH.
Let
K.
p E ~ ( H x { K } ) ((n,K>)
,~ )
because
6.1.i.i.)
p E z N C ( F ) X K.
pez.
Y,ZeSubb(K)
we h a v e
: {hELz
.
: j
nEH
kok=W11d
Then
Let this
Notation:
Let
: {hELy
z~0.
for some
qez.
and
and
as in 4.7.1.1.
d rdL~.:~ "
zEJ,
J { Ig ( @ ) { y } .
zNc (F) XK#Od @ : L IId
rb L = < < (fi, (~--L)If) then
is
such that
i6~NL
Let
zCc(F)y_
c(F ) .
Lc_~ Let
Let
defined
Then
{)[ d @~(@@V) {y}.
and
is regular.
for e v e r y
z#O
additivity
: He1}
6.2.
225
mE~.~{n}.
By
iEK~F
we have
i P<m,K) @ z
by
arbitrarily
we have proved
QED(Claim
z~c(F)y.
A = Sg{y}
and
U = •215
y
iEAz.
AZDa~L
Since
and hence
pEz
but
ie~L
was chosen
la~Azl
N ILl < w.
/A
for some
satisfies Suppose that
~
and
Then
is regular
~ { rlw'%~
]YI~2
IYI~2
@ Dind nCs
D Cs reg.
QED(Theorem
6.1.)
6.2(i)-(ii)
U@Subb(~). Then
: nC~} E Z d Z ~ { O , ~ U } .
since
Y 5 x•
and since
Let
~ ~ %Z[(~U)~ .
(i) is a c o n s e q u e n c e
Let
Proposition
(i)-(iii).
Let
(< n'H>)
rlwEHO~
rlwEHO~.
satisfies
H c ~. --w
(ii).
Y ~ ] ~ Subb(~).
and 1.3.6 we have that
since
is regular.
C(H)X H = U{~(x•
: tEy},
Hence
4.7.2.1
Next we show that
and
Then
_
6.1.1.)
By Claim 6.1.i,
and
PSm,K> q c(r)y._
and
of
E Cs~.
Then
W~I~ W=~Y
~ ~ Co-dol=l
zq{O,W),
H
= yn~u E R
We have seen that
(ii). Let
z { C~H) . ~(Why)
we have
x
Then
by Then
~[)z=O. -
,
Wr
for some
@i~ Co-do1 =I z=O{aY ([)
Thus
:
rlw*~
below was quoted in [HMTI]6.8(3), (ii) and in
[HMTI]6.10. Propositio ~ 6.2. (i)
CS nLf
(ii)
csregnDc
Let
•
and
~ P Dind
.
~ P Ws
(iii)
H(xcs~egNDca)
(iv)
M(xcsregnDc~ e) _~ P Csa
(v)
H(•
(vi)
H ~Ws
NDc
.
--~ P GwsC~
reg if
x<e.
) _~ p GwsCOmp~ reg _~ P~Cs reg.
(vii)
C S ~ eg
Proof.
Let
c ~
saw.
~ P WS a.
~_>2 and
be an atomless
~_>w. Let BA.
~ d ~[e•
(such a
o~
Proof of
exists.)
Let
(i) : ~
Let
d ~(~
~ c )B.
226
6.2.1.
Then
~ E Cs NLf
PDind
and
since
Zd~:B
~
by e.g.
is atomic
[HMT32.2.24(iii).
for every
Let
K,L~CA
be such that
~ q
~ ~PDind
In the rest of the proof we shall use the following Fact 6.2.1.
Now
fact.
Dind nK ~ LUloCS
. Then
K
PL.
Proof.
Let
and hence
~EDind ~L
Proof of
~loCS ~.
implies
(ii):
Let
Then
~ @PL. HC~
~
is directly
QED(Fact
indecomposable
6.2.1.)
be such that
IHIAI~HI~w.
Let
x
/J~ ~ ~ ( ~
){Xo,Xl].
--
{q~ax
: (ViEH) qi=n),
Excs~egADc
by 1.3 and
(Vi,jEH)XoUXl~dij
IAI>I
and
and
is a corollary respectively,
Theorem 6.3.
Let
(iii),(v)
Let
~PWs
a
and
: HnlfCo},_
XomXl=O
by Fact
6.2.1
since
Let
and
(vii)
of 5.7(iii),(iv) ; and follow from
(v) and
(iv)
(ii)
~Ae.
~.
Then some weakly
is not subdirectly
eZw.
Let
indecomposable.
(H n : new) E w(Sbe)
subsets
of
a
subdirectly
such
that
be a system of laNU{H
n
: nEw}l~w.
we let
G d {x n : new}
Q d ex(O)
(vi)
and
infinite
Xn d: (fe~x
Thus
are corollaries
csregNDc
xZ2
disjoint
x~2
new
of 4.7.2.
Now
since
theorem was quoted in [HMTI36.16(7).
For every
Let
~qlWs
Then
6.2.)
indecomposable
A(Xn)=H n.
IHIkw.
by choosing
The following
Let
Let
la,~Hl~w.
of 5.4(iv).
QED(Proposition
mutually
nE~.
cs~eg~Dinda.
By Fact 6.2.1,
Proof.
n
for all
where and
~J~EDc
{i,[d ~ ( ~ @ ~ ) G . by
We show that zEIg(~)L)G,
~ :d e•
For every
I~NU{H n : nEw}I_>w ~)t ,G
and
Q
z#O. Then there is
new
we
have
and by [HMT]2.1.7.
satisfy ne~l
the condit• such that
6.4.
227
z6Ig(~i){xi icI.
[HMTI]8.2
~>i_>~
and by
IGSF _9 SRd F I G s
on Let
and by IF1<w.
I such i d ~>_w. Then
by [HFfPI]8.5, 8.7. Now
7.2.
233
~
I ~ PieI ~ i / F
can be seen
similarly
to the proofs
of
[HMT30.3.71,
0.5.15. Then
~(~)ESUp{
SUp{~(~)
} :
_c I (u{ G s C_ SUp
SUp{ ~ ( I ) }
= I coC s
l_>w.
EHMT~2.4.64,
for every
imply
If
(iv)
Clearly,
Cs
:
c K.
I GS
from
by Lemma
is easy
V
0~ has c h a r a c t e r i s t i c
For every
~E~
let
a ~
a :I)
: Oi
together
see 3.15,
with
completes
7.1.
for
Lf~ASUpCs~
Proof:
(i) follows
follows
QEm(qorgllary
such that
KE{~Ws~,
(ii)
maw.
be a cardinal.
xCs~ eg,
f r o m 4.15. 7.2)
PK C SUpK
~(~)ESUpK. xCs
= Ud (I C s a n L f
from
Then
)
, ~Ws Nmf but
,
cs~egNLf
SUpCsa
(1) in the proof
for e v e r y
In p a r t i c u l a r
of
K C
PK ~ SUpK , ~Cs~nLf~}"
~ Udl Cs~.
[HMTI37.17
:
[HMTI]7.21,
7.1.)
Gws~ ~
(ii)
K ~ I
NOW
Since
Ax
C_
_c S U p ( l c s r e g n L f
~#i
u {Vx(xw.
then
= SUp{ ~ (~)} = S U p ( I C s regnLf~ )
then
I•
By [ H M T I ] 7 . 2 5 ( i i )
: }i_>~} _c SUp{ ~ ( l ) }
I ~Cs
~E~2
hence
7.i is proved. Let
by 7.1.2.
: ~>_~}) _c S U p { ~ ( x )
Cs
every
~(~) }
and f r o m
7.1.
234
7.3.
Let
Theorem
7.3.
(i)
SUpCs reg
(ii)
For O
any
lz.
~
Cn
y ~ ~(0)
~,x,y
x~y.
in
~
~/I.
A(y/I)=O
Then
and
H~Wsa
~
and
~, ~ ~ a ~ )
satisfy the conditions
by
proving
of 7.3.3.
: qn~O}=2m
~ ~V
and
Then Let
x/mSdij
and
Then
Let
~n"
of 7.3.3.
by EHMTI37.15 ~
Let
~
~
x~I,
Then
i,jca and
~n
mew}, I ~
-y~I.
Hence
~
: ~(i+1)lq~
Ix-dijl~2
(in
and
~/I).
Let
x/I, y/I
for every
x =d
){x,y}.
ciY--y ~ {qEV
i,jea.
for all
Thus
for some
4J~ ~ ~ ( ~
IEII~
iCw.
~<w.
~ el Gws
the hypotheses
: U{nEw
: qEz}I<W}.
O<x/I~y/Ii,
Let
There are ~ x~u
(iii)
Proof. ~
~ ~(Z
~,~e
f UpGwsCOmp r e g
)G
O = ~ A t ~
for all sets s
while
Then
~ ~ UfUpDind
respectively,
corollary
~ •
Thus
either
{~V,
occurrence ~+
~=
~ { 9(~x~J~)({
Ibase(~][) i ,
by the h y p o t h e s e s
Z d u{C~y!pi)l Z.
let
~ IA[ -> IJl
we h a v e
Ws
then
(VaEA~{O}) ( ] j E J ) a N ~ Y ! P J ) # o . 3
: iCJ][
: iEl}i.
Thus
: i~I]]
0[ ~
We may
rlz~ e-Is(~J[, /S) I _c p i e J ~ i
assume
for some
where
iBii
~- Subu(~
easy base-isomorphism
Assume
f~0[
Ws}.
the h y p o t h e s e s .
rl(~U)6Is(~
QED(Lemma
and
K E { G w s c~
easy base-isomorphism Let
for f i n i t e
h ~ ( x u ~ ( x N ~ Y ~ pO))
Let
: Subu(~
Assume
conditio~of
sub-isomorphism
: Po(i):r}I~w
F i r s t we p r o v e
sub-isomorphic
rl(~U)
h ~ < xU~x
Ws}.
or
strongly
is a s t r o n g
be a p a r t i t i o n
I {iE~ Let
(2):
7.14.1.
I~U~l
h
f~eGws
K e { G w s c~
Lemma
: i-- ~ any
is d e f i n a b l e
: ~6E}.
Let
: FCF',
K
be o n e - o n e .
$(~)
equation
Let
B.
be an a l g e b r a
8.4(ii)
definable
Then
of c l a s s e s
[ E YB
6($)
i.
be
from
of the p r o o f
: ~EOrd>.
Let
in
KB ~
x Sb
of classes
An outline
K~ = Uf R d P K B = H S P K ~
( K eL
of
Construction
follows
of c l a s s e s .
systems
is the
-
then
EN].
systems
of e q u a t i o n s -one
of e q u a t i o n s
that to
such
B
be o n e - o n e . I
such
I. Let
p"~r1~ c p CA - s .
that
I _9 ~ P ~ ,
For
~Hi
that
Hi ~
d I =
Let
(V< F , A ) C I )
AUp*F and
-2
every
iEI
~ i = ~i~.
by t h e proofs of
let
d~ i
Let EHMT]O.3.71,
O.5.15.
Claim (i)
(ii)
i Let
K
be a s y s t e m
If
~ ~K
~
is e l e m e n t a r i l y
of
classes
definable
then equivalent
to
~P~.
by a s c h e m e
of e q u a t i o n s .
8.4.
265
Proof.
(i) Let
occurring
be
K5 ~
in
e.
Let
a permutation
RdnK8
b
e
Rd K 5 k
be any
6
such
R d n K 6 ~ KB,
n(e)
by
n*H _c e,
i.e.
equation.
i = < F,A)EI
of
by
~i(e),
e
~s
b
be
Let
such
H
that
that
n D ~i.
hence
KS k
n(e)
hence
K~
n(e)
e
be the HCs
set of Let
indices
q : S >~B
Then
>
by F a c t ( * ) .
by F a c t ( * ) . = {i(e).
Then
~
Then
Thus b
~J[
e.
Now
i
{( F , & > e I
: H~s
Proof in the ~.
of
iff
~
p(~)
~P~
~
QED ( C l a i m
of
tion
Up
Let
K,L
~>_w
and
K~
Then
C_ HSP
: Uf
~
proof
let
set of
indices
HCF.
Then
s
Thus
~
~
k
9
occurring
~
in
iff
iff
o~ P
p(~)
: HCF}~F.
and
Rd0KB of
p : ~ >~
~
by C l a i m RdP(KBALB). 8.4.)
HSP
8.4(i). all
be a s y s t e m
and o n e - o n e
of
p : ~ >~
classes. 5
then
is d e f i n a b l e
~->w
p : ~ >~
5
be o n e - o n e .
and
let
Clearly, ~
theorem
Uf
RdPK~
constructed Up ~ P ~
(see
We h a v e
_C K NL
constructed
from
(7[
i and
I c_ ~ 0 ~ .
by
.
Lemma
to
show
Let
Thus
i and
~
Let
by the 2~
8.4(i)
E is p r o v e d .
of e q u a t i o n s . K AL
: HSP
fjleK~NL .
in C o n s t r u c t i o n
We
R d P K B ~ K~.
Thus
schemes
by
f)[ in C o n s t r u c -
by C l a i m
EHMT]O.3.79).
definable
RdP(K6NLs)
_C HSP from
EHMT]O.5.13(viii).
be o n e - o n e .
4][ ~
K
K
classes ~
~_>w
6~ e Uf
by
Let
Suppose
algebra
ultrapower
Clearly,
QEm(Lemma
formula
of e q u a t i o n s .
R d P K S.
the
be s y s t e m s
of
for
Let
EK E
R d P K B : Uf
the a l g e b r a CK~AL B
order
be the
P(~)"
{( F,A>~I
to the
Consider
(KsNLs).
H
P
by a s c h e m e
Keisler-Shelah E Uf
by
first
that
~i
RdPK 5 : HSPK~
show
i.
Let
be such
of e q u a t i o n s .
~][EK .
be any
CA -s.
~
is d e f i n a b l e
to
~
iff
turn
K~ : Uf
a scheme
the proof.
i)
N o w we
have
Let
i = < F,s
~i~i
K
(ii):
language
Let
If
finishes
Let Rd p
Consider
i. T h e n
E IsRdP(KsAL6)
~
c_
266
8.5.
Corollary Then
8.5 9
(i)-(iii)
Let below
(i)
K~ : Uf
(ii)
Mn ~Ke _c Uf
(iii)
~.
Proof.
Let
: Rd~UpKB
: Up
implies
(ii) .
(ii)
Rd K F :
Rd K B
8.5
and
~>_~) .
corresponding
[HMT]2.11
Theorem
and
be p r o v e d
Rd Gs B : R d ~ U p G s B
We h a v e
The
7.18(iii)
8.3
we h a v e
Rd Gs 8 : Uf
B>~ I
in the h y p o t h e s e s .
By the L ~ w e n h e i m - S k o l e m - T a r s k i
Rd Gs
Hence
be as
8.4(i),
By the K e i s l e r - S h e l a h
eUfUp
K
by 8.3(i),
89
O~
If
W=HS
Thus
8.6.2.
271
By 8.6.1.2(3)
we have
231(2)CINr 2 C S 8 .
Then
{f[= 0[(2)
completes
the
proof. QED(Claim Claim
8.6.1)
8.6.2.
~
Let
uE22.
Proof.
~ Nr 2 CA B. Xu
E(u,TRUE)
~(x)
denotes
O SlClX-
the term
1 9 SoCoX. Lemma
8.6.2.1.
(i)
i {bCB
(ii)
~
(iii)
CA B ~
(iv)
(V~eCAB)
Proof.
Proof
(XIo)
: ~(~
2s(O,1)
of
(i): Then
therefore
(ZOK).
Then
: reQ}
Z = {~{X u : uES} seen
Proof Co(Do1.
~
,
.
since
: b~Xol}
: b~Xlo}i
D2K
Z
since
and
: RIve5
IRI v ~
l{bCK
Recall
rl v
: SC_w 22}U{D01}
by
]rlv*(ZOK) l~e. By
and let by
B~D,
R1 v ~ By
z!m. and
: Sg
rlv*K
By 8.6.1.1
and t h e r e f o r e
Now
is an e n d o m o r p h i s m
Therefore
Irlv~K[~w.
that
: b~Xlo}l>w.
DESu~J[ ! S u ~
~ RI v ~
I~.
VeZ~D.
to show
we have
: rER~Q}.
Z =d {c O~ a, c ~ a : aEA]U{D~I]
Let
to show
it is enough
= {p(,r)
l{beB
D2B
{bEB
~[~s
K { PN{p(< O,l>,r)
clearly
Then
algebra
Thus
We have
Let
it is e n o u g h
the B o o l e a n
: b~X10}l>~.
e Ism(~417;2~,
: bSXoI}I~.
(ZUK).
V d= XO1.
T{bEB
2s(O,1)c2x~T(c2x).
i{beB
~(~(~)
and
= XO1.
)K.
we show
Let
: b~Xol}JEl~Jt•
t
i,jE~,
sEl ~
(SI)
:
that
Cyl
(str(fA))
Lemma
8.13.4. and onto.
: V W[ >-
and
consists
, ~ , e x t , E , C , D,+
~
: SEX]
xEA.
the
from
Let
eU ~
I
C(i,x)
Clearly,
Bo
the o t h e r s . for
,
C ~iX
str(~l)
cyl~
.
Cy~ I and
tr~
xEX, is
on
A
~,6~Tm(X,CA )
C(~i,t~)% , t6dij
(resp. terms) X.
in the d i s c o u r s e
be one-one. Tm(X,CA
from
) and is
We define the and
)
Let language "trans-
Fm(X,CA~)
~,%EFm(X,CA
respec).
D(6i, <j) , t~(T+6)
Then is
284
8 .i3.6.
is
t~T+t~6,
Atr~@,
t~-T
tr~(~)
is
is
Crs-structure
-t~T.
~tr~
and
such that
is meaningful.
Similarly Lemma
(t6T)
8.13.6.
one-one b
and onto.
tr~(~)
Proof.
b
Let
).
Cylc(~)
b
and
~
QED(Lemma
for every
e
since
Let
~
t~
are in the
(T[t,i>ie I
of
Therefore
b
tr6~.
~ : ~ >-- I
Let
cyl~ TT[ b
).
: A 9Tt
~
(t6~) ( < ~ , i ) i e i )
Let
~eTm(X,CA~).
~Fm(X,CA
cyl~
be a
be
~ ~ Cyl~(~7~). and
t~,
tr~Ek]
This implies
>~-- C
is one-one
that
cyl~
iff
~
~
tr~
~
Then
iff
and onto.
~,
let
S
be the class of all
Crs-structures
~T[ for
IIWII=I~I.
Ws}.
in
Hence
Let
9.
Now we turn to the proof of Theorem
Let
Tf[ .
tr~A
8.13.6.)
For every which
and
is
Then
by the definitions
(9]l)k = ~ ( ~ ) (cyl~ok).
CyI6(T/[)
tr~
the term function
eeFm(X,CA
k : X ~ AWt
~Ecyl~ok],
of
3xtr~.
~1~ be a Crs-structure.
it is easy to check, (~)
is
tr~(~A$)
Instead of this we shall write
iff
Let
t~:t2.
g~c 9 Let
hence
rs eHom(~[ , I ), is proved.
a-regularity
by
for
by
there
Gwsa-unit
~Hom(~7~ ~ ,
qe
Then
Then
(i) f o l l o w s
from
a~.
is a
and t h e r e ~
rs
)9
G w s [r eg is an
with unit
a-regular
Indeed,
let
V
such that
~ ECrs~
a d:
~
~
d=
rs a V
such that ~+~,
p
d =
ER.
THis is a
antireflexive.
Hence
Let
and and
iEw.
Thus
Case 2
Let
Then
: i,j~F}o{a
< d,e>ER.
IRgbI~
F --w C and
IHi<w
i,jeF.
and
If
i,jE2
ie2,
jeF~2
then
since
(VxIPI
and
(3) We •
We h a v e
Let
~
: jIAI . IZd'P 1:2,
I Z d s I=2
by
[HMTI]6.13.
I
such that
we have
be such that
and
~6 = Then
(RgH)NF=O,
(VjIAI Up~][ :O
Let
IBI>IAI:IBI.
be such that
~
since
since
IZd~4~
~J[ I>23.
(~Up~)
~ Thus
and
let
f(i)
2Ws~ _c UfK.
(i.~)
HNr
(Discussion
HNr
HSK.
SK C Nr
such SO[
# Nr
~ Nr HSK.
HSK.
(1.2)
If we d e l e t e HNr
K c Nr Let
such
that
HNr~@[
is f a l s e
for
hold (2)
HOt
~.
< f(i)/F
elements
:
~
in
Nr~ (2Ws6)
For
_~ Uf UpNr~ K
part
of
(iii),
then
~-O.
_~ N r H ~ . some
of
~_>a
and
for all to
the
some
can be d e l e t e d
By 8.19(ii)
S
as
~~+w
does
proves.
and
K _c CA~,
UfNr~ S U p K
this
equality
would
be
all o c c u r r e n c e s
of
candidates
(I)
UpNr
UpK = Nr
UpK
(to d e l e t e
(II)
UpNr
K
D Nr
UpK
(Up
commutes
o n e way)
(III)
UpNr
K
C Nr
UpK
(Up
commutes
the
(IV)
SUpNr c~ K
(SUp
commutes
one way)
D Nrci S U p K
and
B>~+0)
is not v a l i d
Let
#J[Ecs[egnDc B
K c CA B
this
The o b v i o u s
--
improve
is
Then
from
8.19(i)e
Hence
following.
c can be r e p l a c e d
inequality By
there
the c o n d i t i o n
Ein 8.19(ii)
This
we h a v e
of the
8.19(i)f
B>~+w
all o c c u r r e n c e
K c CA B
because
K=[eA}.
whenever
iff
all
:
by 8 1 9 ( i ) g
6>~+~,
S
8.19.)
. Let
Thus
IB~I~+~.
But
namely
S
--~ Uf UpNr~ UpPK.
be r e p l a c e d
that
here,
obtain
: jEB~w).
we h a v e
has u l t r a -
by the
ultrafilter
zero-dimensional
from
for all
MK C Nr SK
u
and
8.19.)
8.19.8.
be
with
d < dj,i+ j
by
is n o n d i s c r e t e
zero-dimensional be any
follows
Here C cannot
8>~>O
2Ws B
Nr~ U f K --D Nr~ (2WsB)
QED(Theorem
(i)
F
But
UfNr 2 K ~ Nr 2 HSPK,
Remark
many
let
is a s y s t e m
and h e n c e
every
infinitely
O[E2Ws B
every
since
other
way)
= Nr~ SUpK. (I)-(VI) S)
below.
301
8.20.
(V)
SUpNr
(Vl)
UpNr
We p r o v e
K
c Nr S U p K
(SUp c o m m u t e s
the o t h e r way)
SK
= Nr S U p K
(Up
with
that
none
(2.1)
TO d i s p r o v e
8.6,
Ws~
of
commutes
(I)-(VI)
(V),
_c I SNra K.
let
is
valid.
B>~>I
and
By 8.8(ii),
wWsB c K c_ CAB.
By C H M T I ] 8 . 5 -
wWsa --~ U f U p N r CA B _D N r SUpK.
Hence
c~K _(~ Nr a S U p K .
SNr
(2.2)
TO d i s p r o v e
~ Nra UpK.
Let
(IV)
and
B>~+~,
Nr~U~
(II)
a>O.
By 8 . 1 9 ( i ) b
HSP
(2.3)
To d i s p r o v e
(I)
that
UpNr~ K ~ Nr
UpK
(since
UpNr
is
~J~eWs 6
such that
By 8 . 1 9 ( i ) c Choosing
(2.4)
_~ N r U p ~
it is e n o u g h
that
there
K={O'[ }
class
of
and
(Vl)
all
that
there
nondiscrete
one.
Let
obviously
In
moreover
8.19(i)e
Corollary i. (i)
8.20.
Let
Let Then Nr
CA6-s.
Then
and
~I
Let
too,
and
such
_D Nrc~ U p 6 [ .
was desired.
SK ~ Nr SUpK.
Let
cannot
be
Dind~,
the proof.
e.g.
the
UpNr
We note
following
Then
of 8 . 1 4 ( i ) - ( i i ) . )
(V~AE But
SK ~ Nr S U p K
replaced
since
with
if
HNr c~C~ : I Nr c~'~
hence
for
~~
~-l*(A(ai)~g)
and let
~J[=P ~ / F
49% ~
such that
Let
~
let
be a first o r d e r
Assume
b~n+ipB
Such a
~
and
~(x,y 0 .... ,yn )
and
bki
$%.
on
be the set of i n d i c e s o c c u r r i n g
transformation
V
~eUp(Gs[egnLf~).
be an u l t r a f i l t e r
for some
Let
iEI.
Let
F
in the d i s c o u r s e
E Nr ~0~.
finite
Let
~ C I(Gs[egALf8) "
bo/F,...,bn/F]
Let
and let
is an e l e m e n t a r y
Let
for some
w~B
Then B
~ e
F C
~.
be a p e r m u t a -
and
n,
Let
yESg(~{;F~A){X}.
by 1.3.4(ii),
be such that s ~ F1f
IHi~.
Assume
yESg(]~2F~){Z~. and hence
language
with
9 (~){Z}
is not regular.
O6F
Hence
Then
is
there.
H ~ ~i
{][ ~ ~ ( ~
Z = d(1•
H-regular. Let
Let y
be a term in the discourse = T {)[ (x).
and
~ ~ ~@~.
the hypotheses.
It
in the proof of Prop.3.7
(but not proved)
~=i~I>~
Let
section because
below fills in this gap.
constructed
indeed regular as it was claimed Proposition
3 to the present
y
Let
Let
is T
p e
AylPn ~ f,
for every
ne~.
Let
8.24.
309
new.
Then
pney
Pn' ~
Define
Then
9 a}
and
EH M T I ] 8 . 1 and
similarly
of
x,Z
By
: aEA> with
,~),
and by
: a9
and
s9
p(y)
= 9 : n9
and
Theorem
and
we o b t a i n
and
v(y)
= ~ ( ~ )(~(x)).
Fy
INI=IW~NI=~,
and
b*N
and
and
= M.
definitions
= {g 9
s 9 p(y)AF(y~M),
with
b By
ultrafilter
~ ~ /D
:
E' ~ U
Downward that
rI(Fy)E
d p = < FYN~(a)
{g 9 same
Fy
:
goEM}
argument
N ~ W,
= {g 9
be o n e - o n e
on
some
Let =
and
h~Is( ~ ~ /D,
such
By a p p l y i n g
F(W~N)~(y), : W >~
and
algebraic
p(X)
s
: goeEn }
IYAE' I=IY~E'I=~.
s 9
Let
and by the
: goCE ' }
is
By
by
~ ECs F
By the
d= ~ F x .
s ~ Gn(Y)
by l e t t i n g
there
~
: (g~p~)E
F n (y) = T ( ~ ) (Fn(X))
thus
are
= {geFu
seF(U~E').
~ ECs F
that
= s
and
IMI=IY~MI=w.
such
bos
)
~(x)
Then
C
that
there
EH M T I ] 3 . 1 8 ,
s 9
(p(x))
Let
be a n o n p r i n c i p a l
IFI<w
and
M =d YAE'
(~)
D
--
~9
: aEA) .
(k)
d xN Ln =
and
d •
Does
?
sufficient
O < CoC2-do1
~kw.
-s w h i c h
4
Is
the
I c r s Z~d r e g , I crslreg}.~
eg,
then
and
csregnLf
Problem
{I Crs
~ HCrs O r e g
< i
0.9
are
v) 9 C A
: HK?
Crs
b O < CoCl-dol 9 Zd~5.
K 9
ICrs
that
What
T~(~@
Let or
Note
V c ~U.
with
: xEA}.
"G { <x}UT"
in the
EH M T I ] 3 . 1 4 .
~,
~
E
Cs reg
ext-base-isomorphic
and
let
to b o t h
U[ ~ ~ 0[ a n d
.
Does
d5
?
311
Problem
7
For
any
Crs
~A ,
: iEUF>,~)
-choice
function
introduced
in Def.
3.12.
Let
~Jl,~ E Cs
an
(F,~_>~.
If
is
~ >_
8%
we
(cs[egnLfs)
"the
UIUpWsc~ = UfUpRd Wsg.
8.13
{Ws, Cs r e g , +
151>2 I~1
first
Is
c ICs
uncountable
this
?
measurable
condition
necessary?
161:1~1 ~- (UfUpKa = UfUpRd~ KB)
have
WsC~Lf, c s r e g m D c } ,
so the
question
Cf.8.10(4).
concerns
for
the
By
K e
case
6>I~I
Problem
21
By
Uf Up (Cs nLf
EAN8]
and the
) : Uf Up (Cs NDc
= Uf Rd~ Uf Up (CsBNDc6)
for
proof
of
8.13
and
8.4 we
) : Uf UpR% (CsBNLfs) 6~eAw.
Can
Cs
have
=
be r e p l a c e d
with
Ws
or
Cs reg here? Note
that
it can be r e p l a c e d
CraxUtr~{CO-C7}
Problem and
22
in the p r o o f
Does
8.13.4
~
:
What
is the
Problem ADcB)
23
Cf.
of
with
for
free
nCw
8.13,
8.13
~ Ccs~eg}?
: m~>~, that
answer
8.13.
3zVi~(i,ext(z,i))3
~(i,V,Xm,Um, Sm,Jm Crs-structures
of
CA,
or the p r o o f
(Sl)-(S9)U@p{str(~)
Ex ~ { E V i 3 ! v ~ ( i , v )
with
is the
measurable by the
proof
the c o n d i t i o n
Problem
Uf UpLf 5
25
extent
cardinal method
l~i:w
Dc B
the c a r d i n a l i t y
condition
In 8.21(vi)
nor w i t h
are
if
I~I=~
needed?
implies
of
8.13
Ws
but
can
be o m i t t e d
UfLf 5
cannot
6~+(~
(by
conditions The
existence
~ U f R d Ws B
for
it is c o n s i s t e n t from
some
with
Then
neither what
are
in
of an
8.~4.
be r e p l a c e d EAN83).
needed
with the
ZFC
314
necessary conditions? Problem 26
In 8.14(iii)
the new condition
the condition
"lel=w '' can be replaced with
"there is no uncountable measurable cardinal
~ e".
Is this new condition necessary? Problem 27
Let
= SNrK ~ ?
K9
Cf. 8.18.1
Problem 28
Let
reg
such that
Problem 29
Let
Let
Let
8>~>~.
Is
K
=
~EGwsreg.
Does there exist
~ C
rs~ 9 I s ( ~ , ~ ) ?
6>~>O.
is regular in
reg}.
and 8.18(iii)
8>~>O.
eSNr~ uws B
(VxeB)[x
Cs reg, IGws c~
Let ~
3.
~eGws B Is then
and ~( ~)B
~
~
Assume
%UL.
regular?
REFERENCES
[A~ Andr~ka,H., Universal Algebraic Logic, Dissertation, Sci.Budapest 1975. (In Hungarian)
Hungar. Acad.
[AGN13 Andr~ka,H. Gergely,T. and N~meti,I., Purely algebraical construction of first order loqics, Publications of Central Res.Inst. for Physics, Hungar. Acad. Sci., No KFKI-73-71, Budapest, 1973. [AGN23 Andr~ka,H. G e r g e l y , T and N~meti,I., On universal algebraic constructions of logics, Studia Logica XXXVI, i-2(1977), pp.9-47. [ANI3 Andr~ka,H. and N~meti,I., A simple, purely algebraic proof of the completeness of some first order logics, Algebra Universalis 5 (1975), pp.8-15. EAN23 Andr~ka,H. and N~meti,I., On universal algebraic logic and cylindric algebras, Bulletin of the Section of Logic, Vol.7, No.4 (Wroclaw, Dec. 1978), pp.152-158. JAN33 1978.
Andr~ka,H.
and N~meti,I.,
O_~nuniversal algebraic
logic,Preprint,
EAN43 Andr~Ka,H. and N~meti,I., The class of neat-reducts of cylindric al~ebras is not a variety but is c~losed W.r.t. HP, Math. Inst. Hungar. Acad. Sci., Preprint NO 14/1979, Budapest 1979 ~. Submitted to The J. of Symb. Logic. [AN5~ Andr~ka,H. and N~meti,I., Neat reducts of varieties, Math. Hungar. 13(1978), pp.47-51.
Studia Sci.
[AN6~ Andr~ka,H. and N~meti,I., ICrs is a variety and ICrs reg is a quasivariety but not a variety , Preprint Math. Inst. Hungar. Acad. Sci.,
315
Budapest,
1980.
JAN73 Andr~ka,H. and N~meti,I., On the number of generators of cylindric algebras, Preprint, Math. Inst. Hungar. Acad. Sci., Budapest 1979. [AN83 Andr~ka,H. and N~meti,I., Dimension complemented and locally finite dimensional cylindric algebras a r e elementarily equivalent, Algebra Universalis, to appear. JAN93 Andr~ka,H. and N~meti,I., Varieties definable by schemes of equations, Algebra Universalis Ii(1980), pp. 105-116. [CK3
Chang,C.C.
and Keisler,H.J., Model Theory, North-Holland,
1973.
[G3 Gergely,T., Algebraic representation of language hierarchies, Working Paper, Research Inst. for Applied Computer Sciences (Hungary), Budapest 1981. To appear in Acta Cybernetica. [H3 Hausdorff,F., Uber Zwei S~tze yon G. Fichtenholz und L. Kantorovitch, Studia Math. 6(1936), pp. i8-19. [HMT3 Henkin,L. Monk,J.D. North-Holland, 1971.
and Tarski,A.,
Cylindric Algebras Part I,
[HMTI3 Henkin,L. Monk. J.D. and Tarski,A., Cylindric set algebras and related structures, this volume. EM3
Monk,J.D., Mathematical Logic, Springer Verlag,
1976.
[M13 Monk,J.D., Nonfinitizability of classes of representable c[lindric algebras, The J. Symb. Logic 34(1969), pp. 331-343. [N3 N~meti,I., Connections between cylindric algebras and initial algebra semantics of CF languages, In: Mathematical Logic in Computer Science (D~m~iki,B. Gergely,T. eds.) Colloq.Math. Soc. J.Bolyai Voi.26, North-Holland, 1981, pp. 561-606. [N13 N~meti,I., Some constructions of cylindric algebra theory applied to d~namic algebras Of programs, Comput. Linguist. Comput. Lang. (Budapest) VoI.XIV(1980), pp.43-65. [p3~ P~Ify,P.P., On the chromatic number of certain graphs, Math. Inst. Hungar. Acad. Sci. Preprint NO 17/1980. To appear in Discrete Mathematics. [$3
Sikorski,R.,
Boolean Algebras,
Springer Verlag,
1960.
317
INDEX
This book
list
EHMT].
should For
be used
EHMT]
OF
SYMBOLS
together
with
the index of symbols
see any one of the lists
of r e f e r e n c e s
in the
in this
volume.
~x,
Ax
[iE~
AEV]x,
A(U)x
: c l x#x}
dimension
2, 133,
;
set of
x;
EHMT]
132-3
fxu
{< x , u > } U ( D o f ~ { ~ } ) I f
f()~lu)
f•u ;
4
fEH/g]
H1g U
(DofNH)If
~U
set of f u n c t i o n s
from m to U;
~u(P)
{qE~U
;
DEV]
diagonal
cEV]
cylindrification;
Crs
, Cse,
Ws
, Gs
, Gws
: lq~pl
members
distributed
{~b ~
rlw,
of
class of w i d e l y K; 138(O.5)
Rl K, Rs K
rl~w , rl A, rlA(w),
members
134
: ~CK,
universe
; of
~(~W)rlw;'~A
beA}
;
;
6
73(I.6.1) 153(2.1)
P/W[~; ;
base-isomorphism 155(3.1)
153(2.1(ii))
153(2.1(ii)) induced
by f; 37(I.3.5),
318
v~
ANB + c
{aeA
: a~B}
;
;
[HMT3
[HMT3
d e f i n e d if c is an ( F , U , ~ ) - c h o i c e tion; 86(1.7.1)
Rep(c),
Rep(F,U,e,A,c),
Rep
r e p r e s e n t i n g f u n c t i o n of u l t r a p r o d u c t s of Crs -s; 86(I.7.1) RePc
Rep c A ud F , ud F
~ Rep(c)
;
244,
diagonal ultrapower 162(3.5.1)
onto
function;
sub-base-isomorphism;
132
>~
one-one
function;
>~
one-one
and o n t o
c
"finite
subset
V
{X
: X ~cW
Sb V
powerset
~
full
v
universe
Zd
{xeA
the
of V;
of
132
relation;
132
132 [HMT3 unit
V; of
132
132
;
133,
~ ) Zd~3% ; 133,
corresponding 133
132 4)t ;
~(~);
: A(~)x=O}
~(~O etc.
function;
of"
;
with
132
subalgebra
Mn(~)
AtA,
V}
Crs
minimal
ZdA,
86
d i a g o n a l u l t r a p o w e r h o m o m o r p h i s m if c is an ~,I