Издательство North-Holland, 1989, -1437 pp.
The genesis of the motion of a Boolean algebra (BA) is, of course, found in the works of George Boole; but his works are now only of historical interest - cf. Hailperin [1981] in the bibliography (elementary part). The notions of Boolean algebra were developed by many people in the early part of this century- Schroder, Lowenheim, etc. usually working with the concrete operations union, intersection, and complementation. But the abstract notion also appeared early, in the works of Huntington and others. Despite these early developments, the modern theory of BAs can only be considered to have started in the 1930s with works of M.H. Stone and A. Tarski. Since then there has been a steady development of the subject. .
The present Handbook treats those parts of the theory of Boolean algebras of most interest to pure mathematicians: the set-theoretical abstract theory and applications and relationships to measure theory, topology, and logic. Aspects of the subject not treated here are discussion of axiom systems for BAs, finite Boolean algebras and switching circuits, Boolean functions, Boolean matrices, Boolean algebras with operators - including cylindric algebras and related algebraic forms of logic – and the role of BAs in ring theory and in functional analysis.Part I. General Theory of Boolean Algebras.
Elementary arithmetic.
Algebraic theory.
Topological duality.
Free constructions.
Infinite operations.
Special classes of Boolean algebras.
Metamathematics.
Undecidability of the first order theory of Boolean algebras with a distinguished subalgebra.
Part II. Topics in the theory of Boolean algebras.
Section A. Arithmetical properties of Boolean algebras.
Distributive laws.
Disjoint refinement.
Section B. Algebraic properties of Boolean algebras.
Subalgebras.
Cardinal functions on Boolean spaces.
The number of Boolean algebras.
Endomorphisms of Boolean algebras.
Automorphism groups.
On the reconstruction of Boolean algebras from their automorphism groups.
Embeddings and automorphisms.
Rigid Boolean algebras.
Homogeneous Boolean algebras.
Section C. Special classes of Boolean algebras.
Superatomic Boolean algebras.
Projective Boolean algebras.
Countable Boolean algebras.
Measure algebras.
Section D. Logical questions.
Decidable extensions of the theory of Boolean algebras.
Undecidable extensions of the theory of Boolean algebras.
Recursive Boolean algebras.
Lindenbaum-Tarski algebras.
Boolean-valued models.
Appendix on set theory.
Appendix on general topology.
