© 2016

The Congruences of a Finite Lattice

A "Proof-by-Picture" Approach


Table of contents

  1. Front Matter
    Pages i-xxxiv
  2. A Brief Introduction to Lattices

    1. Front Matter
      Pages 1-1
    2. George Grätzer
      Pages 3-17
    3. George Grätzer
      Pages 19-34
    4. George Grätzer
      Pages 35-45
    5. George Grätzer
      Pages 47-53
  3. Some Special Techniques

    1. Front Matter
      Pages 55-55
    2. George Grätzer
      Pages 57-65
    3. George Grätzer
      Pages 67-79
    4. George Grätzer
      Pages 81-85
  4. Congruence Lattices of Finite Lattices

    1. Front Matter
      Pages 87-87
    2. George Grätzer
      Pages 89-100
    3. George Grätzer
      Pages 101-111
    4. George Grätzer
      Pages 113-122
    5. George Grätzer
      Pages 123-126
    6. George Grätzer
      Pages 127-139
    7. George Grätzer
      Pages 141-153
  5. Congruence Lattices and Lattice Extensions

    1. Front Matter
      Pages 155-155
    2. George Grätzer
      Pages 157-164
    3. George Grätzer
      Pages 165-172

About this book


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


Lattices Congruence Boolean Triples Minimal Representation Automorphism Group Principal Congruences Prime Interval

Authors and affiliations

  1. 1.TorontoCanada

About the authors

George A. Grätzer is a Hungarian-Canadian mathematician, specializing in lattice theory and universal algebra. He is also known for his books on LaTeX and his proof with E. Tamás Schmidt of the Grätzer-Schmidt theorem.

Bibliographic information