Book:Steven Roman/Lattices and Ordered Sets

Subject Matter

 * Lattice Theory
 * Order Theory

Contents

 * Preface
 * Contents


 * Part I: Basic Theory


 * 1 Partially Ordered Sets
 * Basic Definitions
 * Duality
 * Monotone Maps
 * Down-Sets and the Down Map
 * Height and Graded Posets
 * 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
 * Quadrilaterals
 * 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
 * The Word Problem for Free 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