# Book:Steven Roman/Lattices and Ordered Sets

## Steven Roman: Lattices and Ordered Sets

Published $\text {2008}$, Springer

ISBN 978-0387789002

### Contents

Preface
Contents
Part I: Basic Theory
1 Partially Ordered Sets
Basic Definitions
Duality
Monotone Maps
Down-Sets and the Down Map
Chain Conditions
Chain Conditions and Finiteness
Dilworth's Theorem
Symmetric and Transitive Closures
Compatible Total Orders
The Poset of Partial Orders
Exercises
2 Well-Ordered Sets
Well-Ordered Sets
Ordinal Numbers
Transfinite Induction
Cardinal Numbers
Ordinal and Cardinal Arithmetic
Complete Posets
Cofinality
Exercises
3 Lattices
Closure and Inheritance
Semilattices
Aribtrary Meets Equivalent to Arbitrary Joins
Lattices
Meet-Structures and Closure Operators
Properties of Lattices
Join-Irreducible and Meet-Irreducible Elements
Sublattices
Denseness
Lattice Homomorphisms
The F-Down Map
Ideals and Filters
Prime and Maximal Ideals
Lattice Representations
Special Types of Lattices
The Dedekind–MacNeille Completion
Exercises
4 Modular and Distributive Lattices
The Definitions
Examples
Characterizations
Modularity and Semimodularity
Partition Lattices and Representations
Distributive Lattices
Irredundant Join-Irreducible Representations
Exercises
5 Boolean Algebras
Boolean Lattices
Boolean Algebras
Boolean Rings
Boolean Homomorphisms
Characterizing Boolean Lattices
Complete and Infinite Distributivity
Exercises
6 The Representation of Distributive Lattices
The Representation of Distributive Lattices with DCC
The Representation of Atomic Boolean Algebras
The Representation of Arbitrary Distributive Lattices
Summary
Exercises
7 Algebraic Lattices
Motivation
Algebraic Lattices
$\cap \overset{\to}{\cup}$-Structures
Algebraic Closure Operators
The Main Correspondence
Subalgebra Lattices
Congruence Lattices
Meet-Representations
Exercises
8 Prime and Maximal Ideals; Separation Theorems
Separation Theorems
Exercises
9 Congruence Relations on Lattices
Congruence Relations on Lattices
The Lattice of Congruence Relations
Commuting Congruences and Joins
Quotient Lattices and Kernels
Congruence Relations and Lattice Homomorphisms
Standard Ideals and Standard Congruence Relations
Exercises
Part II: Topics
10 Duality for Distributive Lattices: The Priestley Topology
The Duality Between Finite Distributive Lattices and Finite Posets
Totally Order-Separated Spaces
The Priestley Prime Ideal Space
The Priestley Duality
The Case of Boolean Algebras
Applications
Exercises
11 Free Lattices
Lattice Identities
Free and Relatively Free Lattices
Constructing a Relatively Free Lattice
Characterizing Equational Classes of Lattices
Canonical Forms
The Free Lattice on Three Generators Is Infinite
Exercises
12 Fixed-Point Theorems
Fixed Point Terminology
Fixed-Point Theorems: Complete Lattices
Fixed-Point Theorems: Complete Posets
Exercises
A1 A Bit of Topology
Topological Spaces
Subspaces
Bases and Subbases
Connectedness and Separation
Compactness
Continuity
The Product Topology
A2 A Bit of Category Theory
Categories
Functors
Natural Transformations

References
Index of Symbols
Index