# Book:Steven Roman/Lattices and Ordered Sets

## Steven Roman: *Lattices and Ordered Sets*

Published $2008$, **Springer**

- ISBN 978-0387789002.

### Subject Matter

### 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

- 1 Partially Ordered Sets

- 2 Well-Ordered Sets
- Well-Ordered Sets
- Ordinal Numbers
- Transfinite Induction
- Cardinal Numbers
- Ordinal and Cardinal Arithmetic
- Complete Posets
- Cofinality
- Exercises

- 2 Well-Ordered Sets

- 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

- 3 Lattices

- 4 Modular and Distributive Lattices
- Quadrilaterals
- The Definitions
- Examples
- Characterizations
- Modularity and Semimodularity
- Partition Lattices and Representations
- Distributive Lattices
- Irredundant Join-Irreducible Representations
- Exercises

- 4 Modular and Distributive Lattices

- 5 Boolean Algebras
- Boolean Lattices
- Boolean Algebras
- Boolean Rings
- Boolean Homomorphisms
- Characterizing Boolean Lattices
- Complete and Infinite Distributivity
- Exercises

- 5 Boolean Algebras

- 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

- 6 The Representation of Distributive Lattices

- 7 Algebraic Lattices
- Motivation
- Algebraic Lattices
- $\cap \overset{\to}{\cup}$-Structures
- Algebraic Closure Operators
- The Main Correspondence
- Subalgebra Lattices
- Congruence Lattices
- Meet-Representations
- Exercises

- 7 Algebraic Lattices

- 8 Prime and Maximal Ideals; Separation Theorems
- Separation Theorems
- Exercises

- 8 Prime and Maximal Ideals; Separation Theorems

- 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

- 9 Congruence Relations on Lattices

- 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

- 10 Duality for Distributive Lattices: The Priestley Topology

- 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

- 11 Free Lattices

- 12 Fixed-Point Theorems
- Fixed Point Terminology
- Fixed-Point Theorems: Complete Lattices
- Fixed-Point Theorems: Complete Posets
- Exercises

- 12 Fixed-Point Theorems

- A1 A Bit of Topology
- Topological Spaces
- Subspaces
- Bases and Subbases
- Connectedness and Separation
- Compactness
- Continuity
- The Product Topology

- A1 A Bit of Topology

- A2 A Bit of Category Theory
- Categories
- Functors
- Natural Transformations

- A2 A Bit of Category Theory

- References
- Index of Symbols
- Index