Full Description
This book provides a clear and accessible introduction to lattice theory for graduate students, researchers, and mathematically mature readers new to the subject. Emphasizing ideas over technical detail, it presents the central concepts and results of lattice theory while avoiding long or intricate proofs.
Diagrams play a central role throughout the text, supporting intuition and visual understanding. Carefully chosen examples clarify definitions and theorems, and optional exercises at the end of each chapter allow readers to deepen their understanding or explore further topics.
Part I develops the foundations of lattice theory, including ordered sets, lattices, homomorphisms, ideals, and congruences. Part II examines some important classes of lattices: distributive, semimodular, modular, and complemented modular lattices. Particular attention is paid to finite congruence lattices. Part III focuses on free lattices, among the most intricate and conceptually rich areas of the theory. Part IV concludes with a brief presentation of several deep and elegant results.
A glossary and a detailed index make the book suitable both for first study and for reference.
Contents
Part I: Shallow Waters.- Ordered Sets.- Lattices.- Morphisms.- Sublattices and Ideals.- Congruences.- Standard Ideals.- Varieties.- Lattice Constructions.-Free Lattices.- Complete Lattices.- Part II: Diving Deeper.- Distributive Lattices.- Congruence Lattices of Finite Lattices.- Semimodular Lattices.- Modular Lattices.- Complemented Modular Lattices.- Part III: The Structure of Free Lattices.- Free Products of Lattices.- Whitman's Theorem.- Free Products and Complements.- Part IV: The Jewels in the Crown.- The Jonsson Lemma.- Finite Sublattices of Free Lattices.- Lattice Varieties of Finite Height.- Congruence Lattices of Algebras.- Congruence Lattices of Lattices.- Glossary of Terms.
