**Translations of Mathematical Monographs**

2002;
256 pp;
Hardcover

MSC: Primary 03;
Secondary 28; 54

Print ISBN: 978-0-8218-2765-9

Product Code: MMONO/214

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

**Electronic ISBN: 978-1-4704-4639-0
Product Code: MMONO/214.E**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

# Algebras of Sets and Combinatorics

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*L. Š. Grinblat*

An algebra \(A\) on a set \(X\) is a family of subsets of this
set closed under the operations of union and difference of two subsets. The
main topic of the book is the study of various algebras and families of
algebras on an abstract set \(X\). The author shows how this is related
to famous problems by Lebesgue, Banach, and Ulam on the existence of certain
measures on abstract sets, with corresponding algebras being algebras of
measurable subsets with respect to these measures. In particular it is shown
that for a certain algebra not to coincide with the algebra of all subsets of
\(X\) is equivalent to the existence of a nonmeasurable set with respect
to a given measure.

Although these questions don't seem to be related to mathematical logic,
many results in this area were proved by “metamathematical”
methods, using the method of forcing and other tools related to axiomatic set
theory. However, in the present book, the author uses “elementary”
(mainly combinatorial) methods to study properties of algebras on a set.
Presenting new and original material, the book is written in a clear and
readable style and illustrated by many examples and figures.

The book will be useful to researchers and graduate students working in set
theory, mathematical logic, and combinatorics.

#### Readership

Graduate students and research mathematicians interested in foundations of mathematics and logic.

#### Table of Contents

# Table of Contents

## Algebras of Sets and Combinatorics

- Cover Cover11
- Title page iii4
- Contents v6
- Introduction 18
- Main results 1118
- The main idea 2532
- Finite sequences of algebras (1). Proof of Theorems 2.1 and 2.2 3946
- Countable sequences of algebras (1). Proof of Theorem 2.4 5764
- Proof of the Gitik-Shelah theorem, and more from set theory 7178
- Proof of Theorems 1.17, 2.7, 2.8 8390
- Theorems on almost 𝜎-algebras. Proof of Theorem 2.9 93100
- Finite sequences of algebras (2). The function 𝔤(𝔫) 109116
- A description of the class of functions Ψ_{*}⁷ 145152
- The general problem. Proof of Theorems 2.15 and 2.20 163170
- Proof of Theorems 2.21(1,3), 2.24 175182
- The inverse problem 179186
- Finite sequences of algebras (3). Proof of Theorems 2.27, 2.31, 2.36, 2.38 181188
- Preliminary notions and lemmas 197204
- Finite sequences of algebras (4). Proof of Theorems 2.39(1,2), 2.45(1,2) 207214
- Countable sequences of algebras (2). Proof of Theorems 2.29, 2.32, 2.46 229236
- A refinement of theorems on 𝜎-algebras. Proof of Theorems 2.34, 2.44 235242
- Semistructures and structures of sets. Proof of Theorem 2.48 239246
- Final comments. Generalization of Theorem 2.1 245252
- On a question of Grinblat 247254
- Bibliography 251258
- Index 255262
- Back Cover Back Cover1265