General Lattice Theory

✍ Scribed by Grätzer, George

• Publisher: Birkhäuser, Birkhäuser Basel, Imprint
• Year: 1978
• Tongue: English
• Leaves: 396
• Series: Mathematische Reihe Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften 52
• Edition: 1ed.
• Category: Library

✦ Synopsis

In the first half of the nineteenth century, George Boole's attempt to formalize propositional logic led to the concept of Boolean algebras. While investigating the axiomatics of Boolean algebras at the end of the nineteenth century, Charles S. Peirce and Ernst Schröder found it useful to introduce the lattice concept. Independently, Richard Dedekind's research on ideals of algebraic numbers led to the same discov ery. In fact, Dedekind also introduced modularity, a weakened form of distri butivity. Although some of the early results of these mathematicians and of Edward V. Huntington are very elegant and far from trivial, they did not attract the attention of the mathematical community. It was Garrett Birkhoff's work in the mid-thirties that started the general develop ment of lattice theory. In a brilliant series of papers he demonstrated the importance of lattice theory and showed that it provides a unifying framework for hitherto unrelated developments in many mathematical disciplines. Birkhoff himself, Valere Glivenko, Karl Menger, John von Neumann, Oystein Ore, and others had developed enough of this new field for Birkhoff to attempt to "seIl" it to the general mathematical community, which he did with astonishing success in the first edition of his Lattice Theory. The further development of the subject matter can best be followed by com paring the first, second, and third editions of his book (G. Birkhoff [1940], [1948], and [1967]). Read more...

Abstract: In the first half of the nineteenth century, George Boole's attempt to formalize propositional logic led to the concept of Boolean algebras. While investigating the axiomatics of Boolean algebras at the end of the nineteenth century, Charles S. Peirce and Ernst Schröder found it useful to introduce the lattice concept. Independently, Richard Dedekind's research on ideals of algebraic numbers led to the same discov ery. In fact, Dedekind also introduced modularity, a weakened form of distri butivity. Although some of the early results of these mathematicians and of Edward V. Huntington are very elegant and far from trivial, they did not attract the attention of the mathematical community. It was Garrett Birkhoff's work in the mid-thirties that started the general develop ment of lattice theory. In a brilliant series of papers he demonstrated the importance of lattice theory and showed that it provides a unifying framework for hitherto unrelated developments in many mathematical disciplines. Birkhoff himself, Valere Glivenko, Karl Menger, John von Neumann, Oystein Ore, and others had developed enough of this new field for Birkhoff to attempt to "seIl" it to the general mathematical community, which he did with astonishing success in the first edition of his Lattice Theory. The further development of the subject matter can best be followed by com paring the first, second, and third editions of his book (G. Birkhoff [1940], [1948], and [1967])

✦ Table of Contents

Content: I. First Concepts --
1. Two Definitions of Lattices --
2. How to Describe Lattices --
3. Some Algebraic Concepts --
4. Polynomials, Identities, and Inequalities --
5. Free Lattices --
6. Special Elements --
Further Topics and References --
Problems --
II. Distributive Lattices --
1. Characterization Theorems and Representation Theorems --
2. Polynomials and Freeness --
3. Congruence Relations --
4. Boolean Algebras R-generated by Distributive Lattices --
5. Topological Representation --
6. Distributive Lattices with Pseudocomplementation --
Further Topics and References --
Problems --
III. Congruences and Ideals --
1. Weak Projectivity and Congruences --
2. Distributive, Standard, and Neutral Elements --
3. Distributive, Standard, and Neutral Ideals --
4. Structure Theorems --
Further Topics and References --
Problems --
IV. Modular and Semimodular Lattices --
1. Modular Lattices --
2. Semimodular Lattices --
3. Geometric Lattices --
4. Partition Lattices --
5. Complemented Modular Lattices --
Further Topics and References --
Problems --
V. Equational Classes of Lattices --
1. Characterizations of Equational Classes --
2. The Lattice of Equational Classes of Lattices --
3. Finding Equational Bases --
4. The Amalgamation Property --
Further Topics and References --
Problems --
VI. Free Products --
1. Free Products of Lattices --
2. The Structure of Free Lattices --
3. Reduced Free Products --
4. Hopfian Lattices --
Further Topics and References --
Problems --
Concluding Remarks --
Table of Notation.

✦ Subjects

Science (General)