Cylindric Set Algebras

327 Pages·1981·4.457 MB·English

by  Leon Henkin, J. Donald Monk, Alfred Tarski, Hajnalka Andréka, István Németi (auth.)

Lecture Notes ni
Mathematics
Edited yb .A Dold dna .B Eckmann
883
Cylindric Set Algebras
Cylindric Set Algebras and Related Structures
yB .L Henkin, .J .D Monk, dna .A iksraT
On Cylindric-Relativized Set Algebras
yB .H Andreka and .I Nemeti
galreV-regnirpS
Berlin Heidelberg
weN kroY 1891
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 51
ISBN 3-540-10881-5 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-10881-5 Springer-Verlag NewYork Heidelberg Berlin
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(cid:14)9 by Springer-Verlag Berlin Heidelberg 1891
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Introduction
This volume is devoted to a comprehensive treatment of certain set-
theoretical structures which consist of fields of sets enhanced by addi-
tional fundamental operations and distinguished elements. The treatment
is largely self-contained.
Each of these structures has an associated dimension ~ , a finite or
infinite ordinal; their basic form is well illustrated in the case ~ = 3 .
Let R be an arbitrary set, and let ~ be a field of subsets of the set
3
R of all triples of elements of R . Thus ~ is a non-empty collection
of subsets of 3R closed under union, intersection, and complementatlen
relative to 3R . We shall assume that ~ is closed under the three opera-
tions CO, Cl, C 2 of cylindriflcation, where C 0 , for example, is the
operation given by:
CoX = {<x,y,z> : (u,y,z> ~ X for some ,u with xER};
X0C is the cylinder formed by moving X parallel to the first axis. CIX
and C2X are similarly related to the second and third axes. We also assume
that the diagonal planes D01, D02, 21D are in ~ ; here, for example,
:>y,x,x<{=10D
x,y~R} .
Similarly D02 (resp., DI2 ) consists of all triples of 3R whose first
and third (resp., second and third) coordinates coincide. A collection
satisfying all of these conditions is called a cylindric field of sets
(of dimension 3). Cylindric fields of sets and certain closely related
structures are the objects of study in this volume. Considered not merely
as collections of sets, hut as algebraic objects endowed with fundamental
Vi
Operations and distinguished elements, cylindric fields of sets are called
cylindric set algebras. Cs is the class of all cylindric set algebras
of dimension ~ , and ICs 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 3R and isomorphism to algebras of subsets of
3S with 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 eht 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
proceeded, they learned of the closely related results obtained by Andreka l
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. The second, by Andrlka and Nemeti, is organized
parallel to the first. 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 of contents
First Part: Cylindric set algebras and related structures, by
L. Henkin, J.D~ Monk, and A. Tarski .................. i
1. Various set algebras .............................. 4
2. Relativization .................................... 12
3. Change of base .................................... 33
4. Subalgebras ....................................... 56
5. Homomorphisms ..................................... 66
6. Products .......................................... 73
7. Ultraproducts ..................................... 86
8. Reducts ........................................... 122
9. Problems .......................................... 126
References ............................................ 128
Second Part: On cylindric-relativized set algebras, by
H Andr~ka and I. N~meti .............................. 131
O Basic concepts and notations ....................... 132
i Regular cylindric set algebras ..................... 145
2 Relativization ..................................... 153
3 Change of base ..................................... 155
4 Subalgebras ........................................ 185
5 Homomorphisms ...................................... 209
6 Products ........................................... 222
7 Ultraproducts ...................................... 229
8 Reducts ............................................ 261
9 Problems ........................................... 310
References ............................................ 314
Index of symbols ............................................... 317
Index of defined terms ......................................... 322
I)
Cylindric set algebras and related structures
by L. Henkln, J.D. Monk, and A. Tarskl
The abstract theory of cylindric algebras is extensively developed in
the book HMT by the authors. Several kinds of special set algebras were
mentioned, primarily for motivational purposes, in that book. It is the
purpose of this article to begin the examination of these set algebras in
more detail. The simplest and most important kind of set algebras are
the cylindric set algebras introduced in 1.1.5. (Throughout this article
we shall refer to items from HMT by number without explicitly mentioning
that book). Recall that the unit element of any ~ - dimensional cylindric
set algebra (Cs) ~ is the Cartesian power ~U of a set U (the base),
and the other elements of A are subsets of ~U . The diagonal element
DKk of ~ is the set ix E ~U: x K = x k } for each K,h < ~ " the
fundamental Boolean operations of ~ are union, intersection, and comple-
mentatlon; and for each K < ~ the fundamental operation C K consists
of cyllndrlflcatlon by translation parallel to the Kth axis of the space.
Several other kinds of set algebras were briefly considered in HMT, and
their definition is similar to that of a cylindric set algebra: weak
cylindric set algebras (cf. 2.2.11), generalized cylindric set algebras
(of. 1.1.13), and what we shall now call generalized weak cylindric set
algebras (cf. 2.2.11). The algebras of each of these kinds have for their
unit elements 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 sup-
ported in part by NSF grants MPS 75-03583j MCS 77-22913.
the fundamental operations of any such algebra are obtained from those of
a Cs , with unit element ~U , by relativization (in the sense of 2.2)
to the unit element of the algebra discussed. To unify our treatment of
these several classes of cylindric set algebras we use here as the most
general class of set algebras that of the cylindric-relativized set
algebras, in which the unit elements may be arbitrary subsets of a Cartesian
space. These algebras are simply subalgebras of those algebras that are
obtained by arbitrary relativizations from full cylindric set algebras.
We shall not discuss here, however, the class of all cylindric-relativized
set algebras in any detail, restricting ourselves to the aspects of
relativization directly relevant to our discussion of those set algebras
which are CA's
Much of the importance of cylindric set algebras stems from the
following construction. Given any relational structure ~ and any
first-order discourse language A for ~ , the collection
~: ~ a formula of A forms an ~- dimensional cylindric field of sets,
where ~ is the length of the sequence of variables of A (for the
notation used here, see the Preliminaries of HMT). Thus the above
collection is the universe of a cylindric set algebra ~ . This algebra
is locally finite dimensional in the sense of .i Ii.i , since CK~ ~ =
except possibly for the finitely many K < ~ such that the Kth variable of
occurs free in ~ Furthermore, ~ has an additional property of
regularity: if x E B , f E x , g E C , and Ax~f = ~xlg , then
g E x (Here xg~ , the dimension set of x , is ~K E ~ : cKx f x} ;
see 1.6.1 ). Regular set algebras will be discussed extensively later.
The article has nine sections. In section 1 we give formal defini-
tions of the classes of set algebras which are studied in this article
and we state the simplest relationships between them; the proofs are
found in later sections of the paper. In section 2 some deeper relation-
ships are established, using the notion of relativization. Section 3
is concerned with change of base, treating the question of conditions
under which a set algebra with base U is isomorphic to one with a
different base W ; the main results are algebraic versions of the
L~wenheim, Skolem, Tarski theorems (some results on change of base
are also found in section 7). In section 4 the algebraic notion of
subalgebra is investigated for our various set algebras, paying parti-
cular attention to the problem about the minimum number of generators
for a set algebra. Homomorphisms of set algebras are discussed in sec-
tion 5, and products, along with the related indecomposability notions,
are studied in their application to set algebras in section .6 Section
,7 devoted to ultraproducts of set algebras, gives perhaps the deepest
results in the paper. In particular, it is in this section that the
less trivial of the relationships between the classes of set algebras
described in section 1 are established. Reducts and neat embeddings of
set algebras are discussed in section .8 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 to describe the status
of the problems stated in HMT .
For reference 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. Other major results were obtained
in Henkin, Monk HM . In preparing the present comprehensive discus-
sion of set algebras many natural questions arose. 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, and we have indicated that the results are
theirs. Many of their longer solutions will be found in the paper AN3
4 Various set algebras I.i.i
following this one, which is organized parallel to our paper; a few
of their related results are found in JAN2, JAN4 , or N . In the
course of our article we shall have occasion to mention explicitly most
of their related results. We are indebted to Andr~ka and N~meti for
their considerable help in preparing this paper for publication,
The following set-theoretical notation not in HMT will be useful.
If f E ~U , ~ < ~ , and u E U , then fK is the member of ~U such
u
that (fK)>~ = fk if k # < , while (fK)K = u . For typographical
u u
reasons we sometimes write f(K/u) in place of fK
u
.1 y~arious set al$ebr~as
Definition I.I.I. (i) Let U be a set, ~ an ordinal, and
V ~ ~U . For all K,~ < ~ we set
DV = y EV : YK = Yk}
and we let C IV be the mapping from SbV into SbV such that, for
every X ~- V ,
C Vx = yEV : Yu K E X for some u E U
(When V is implicitly understood we shall write simply D or C ).
(ii) A is an - dimensional cylindric-relativized field of sets
iff there is a set U and a set V ~ ~U such that A is a non-empty
family of subsets of V closed under all the operations U, N,
V
and C IV (for each K < ~ ), and containing as elements the subsets
K
D V (for all <,~ < ~ ). The base of A is the set UxEvRgx .
(iii) ~ is a cylindric-relativized set al~ebra of dimension ~C
with base U iff there is a set V ~ ~U such that