Algebras Of Sets And Combinatorics (translations Of Mathematical Monographs)
by L. S. Grinblat /
2002 / English / PDF, DjVu
15.9 MB Download
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.
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.