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Title
The congruences of a finite lattice [electronic resource] : a "Proof-by-Picture" approach / George Grätzer.
Edition
2nd ed.
ISBN
9783319387987 (electronic book)
3319387987 (electronic book)
9783319387963
Published
Cham : Birkhäuser, 2016.
Language
English
Description
1 online resource (xxxiv, 346 pages) : illustrations.
Item Number
10.1007/978-3-319-38798-7 doi
Call Number
QA171.5
Dewey Decimal Classification
511.3/3
Summary
This is a self-contained exposition by one of the leading experts in lattice theory, George Grätzer, presenting the major results of the last 70 years on congruence lattices of finite lattices, featuring the author's signature Proof-by-Picture method. Key features: * Insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions * Contains complete proofs, an extensive bibliography and index, and over 140 illustrations * This new edition includes two new parts on Planar Semimodular Lattices and The Order of Principle Congruences, covering the research of the last 10 years The book is appropriate for a one-semester graduate course in lattice theory, and it is a practical reference for researchers studying lattices. Reviews of the first edition: "There exist a lot of interesting results in this area of lattice theory, and some of them are presented in this book. [This] monograph...is an exceptional work in lattice theory, like all the contributions by this author. ... The way this book is written makes it extremely interesting for the specialists in the field but also for the students in lattice theory. Moreover, the author provides a series of companion lectures which help the reader to approach the Proof-by-Picture sections." (Cosmin Pelea, Studia Universitatis Babes-Bolyai Mathematica, Vol. LII (1), 2007) "The book is self-contained, with many detailed proofs presented that can be followed step-by-step. [I]n addition to giving the full formal details of the proofs, the author chooses a somehow more pedagogical way that he calls Proof-by-Picture, somehow related to the combinatorial (as opposed to algebraic) nature of many of the presented results. I believe that this book is a much-needed tool for any mathematician wishing a gentle introduction to the field of congruences representations of finite lattices, with emphasis on the more 'geometric' aspects." --Mathematical Reviews.
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Digital File Characteristics
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Available in Other Form
Print version: 9783319387963
I: A Brief Introduction to Lattices
Basic Concepts
Special Concepts
Congruences
Planar Semimodular Lattices
II: Some Special Techniques
Chopped Lattices
Boolean Triples
Cubic Extensions
III: Congruence Lattices of Finite Lattices
The Dilworth Theorem
Minimal Representations
Semimodular Lattices
Rectangular Lattices
Modular Lattices
Uniform Lattices
IV: Congruence Lattices and Lattice Extensions
Sectionally Complemented Lattices
Semimodular Lattices
Isoform Lattices
The Congruence Lattice and the Automorphism Group
Magic Wands
V: Congruence Lattices of Two Related Lattices
Sublattices
Ideals
Tensor Extensions
VI The Ordered Set of Principal Congruences
Representation Theorems
Isotone Maps
VII: Congruence Structure
Prime Intervals and Congruences
Some Applications of the Swing Lemma.
