Introduction to lattice theory with computer science applications

Content: List of Figures xiii    Nomenclature xv    Preface xvii    1 Introduction 1    1.1 Introduction 1    1.2 Relations 2    1.3 Partial Orders 3    1.4 Join and Meet Operations 5    1.5 Operations on Posets 7    1.6 Ideals and Filters 8    1.7 Special Elements in Posets 9    1.8 Irreducible Elements 10    1.9 Dissector Elements 11    1.10 Applications: Distributed Computations 11    1.11 Applications: Combinatorics 12    1.12 Notation and Proof Format 13    1.13 Problems 15    1.14 Bibliographic Remarks 15    2 Representing Posets 17    2.1 Introduction 17    2.2 Labeling Elements of The Poset 17    2.3 Adjacency List Representation 18    2.4 Vector Clock Representation 20    2.5 Matrix Representation 22    2.6 Dimension-Based Representation 22    2.7 Algorithms to Compute Irreducibles 23    2.8 Infinite Posets 24    2.9 Problems 26    2.10 Bibliographic Remarks 27    3 Dilworth   s Theorem 29    3.1 Introduction 29    3.2 Dilworth   s Theorem 29    3.3 Appreciation of Dilworth   s Theorem 30    3.4 Dual of Dilworth   s Theorem 32    3.5 Generalizations of Dilworth   s Theorem 32    3.6 Algorithmic Perspective of Dilworth   s Theorem 32    3.7 Application: Hall   s Marriage Theorem 33    3.8 Application: Bipartite Matching 34    3.9 Online Decomposition of Posets 35    3.10 A Lower Bound on Online Chain Partition 37    3.11 Problems 38    3.12 Bibliographic Remarks 39    4 Merging Algorithms 41    4.1 Introduction 41    4.2 Algorithm to Merge Chains in Vector Clock Representation 41    4.3 An Upper Bound for Detecting an Antichain of Size K 47    4.4 A Lower Bound for Detecting an Antichain of Size K 48    4.5 An Incremental Algorithm for Optimal Chain Decomposition 50    4.6 Problems 50    4.7 Bibliographic Remarks 51    5 Lattices 53    5.1 Introduction 53    5.2 Sublattices 54    5.3 Lattices as Algebraic Structures 55    5.4 Bounding The Size of The Cover Relation of a Lattice 56    5.5 Join-Irreducible Elements Revisited 57    5.6 Problems 59    5.7 Bibliographic Remarks 60    6 Lattice Completion 61    6.1 Introduction 61    6.2 Complete Lattices 61    6.3 Closure Operators 62    6.4 Topped    -Structures 63    6.5 Dedekind   Macneille Completion 64    6.6 Structure of Dedekind--Macneille Completion of a Poset 67    6.7 An Incremental Algorithm for Lattice Completion 69    6.8 Breadth First Search Enumeration of Normal Cuts 71    6.9 Depth First Search Enumeration of Normal Cuts 73    6.10 Application: Finding the Meet and Join of Events 75    6.11 Application: Detecting Global Predicates in Distributed Systems 76    6.12 Application: Data Mining 77    6.13 Problems 78    6.14 Bibliographic Remarks 78    7 Morphisms 79    7.1 Introduction 79    7.2 Lattice Homomorphism 79    7.3 Lattice Isomorphism 80    7.4 Lattice Congruences 82    7.5 Quotient Lattice 83    7.6 Lattice Homomorphism and Congruence 83    7.7 Properties of Lattice Congruence Blocks 84    7.8 Application: Model Checking on Reduced Lattices 85    7.9 Problems 89    7.10 Bibliographic Remarks 90    8 Modular Lattices 91    8.1 Introduction 91    8.2 Modular Lattice 91    8.3 Characterization of Modular Lattices 92    8.4 Problems 98    8.5 Bibliographic Remarks 98    9 Distributive Lattices 99    9.1 Introduction 99    9.2 Forbidden Sublattices 99    9.3 Join-Prime Elements 100    9.4 Birkhoff   s Representation Theorem 101    9.5 Finitary Distributive Lattices 104    9.6 Problems 104    9.7 Bibliographic Remarks 105    10 Slicing 107    10.1 Introduction 107    10.2 Representing Finite Distributive Lattices 107    10.3 Predicates on Ideals 110    10.4 Application: Slicing Distributed Computations 116    10.5 Problems 117    10.6 Bibliographic Remarks 118    11 Applications of Slicing to Combinatorics 119    11.1 Introduction 119    11.2 Counting Ideals 120    11.3 Boolean Algebra and Set Families 121    11.4 Set Families of Size k 122    11.5 Integer Partitions 123    11.6 Permutations 127    11.7 Problems 129    11.8 Bibliographic Remarks 129    12 Interval Orders 131    12.1 Introduction 131    12.2 Weak Order 131    12.3 Semiorder 133    12.4 Interval Order 134    12.5 Problems 136    12.6 Bibliographic Remarks 137    13 Tractable Posets 139    13.1 Introduction 139    13.2 Series   Parallel Posets 139    13.3 Two-Dimensional Posets 142    13.4 Counting Ideals of a Two-Dimensional Poset 145    13.5 Problems 146    13.6 Bibliographic Remarks 147    14 Enumeration Algorithms 149    14.1 Introduction 149    14.2 BFS Traversal 150    14.3 DFS Traversal 154    14.4 LEX Traversal 154    14.5 Uniflow Partition of Posets 160    14.6 Enumerating Tuples of Product Spaces 163    14.7 Enumerating All Subsets 163    14.8 Enumerating All Subsets of Size k 165    14.9 Enumerating Young   s Lattice 166    14.10 Enumerating Permutations 167    14.11 Lexical Enumeration of All Order Ideals of a Given Rank 168    14.12 Problems 172    14.13 Bibliographic Remarks 173    15 Lattice of Maximal Antichains 159    15.1 Introduction 159    15.2 Maximal Antichain Lattice 161    15.3 An Incremental Algorithm Based on Union Closure 163    15.4 An Incremental Algorithm Based on BFS 165    15.5 Traversal of the Lattice of Maximal Antichains 166    15.6 Application: Detecting Antichain-Consistent Predicates 168    15.7 Construction and Enumeration of Width Antichain Lattice 169    15.8 Lexical Enumeration of Closed Sets 171    15.9 Construction of Lattices Based on Union Closure 174    15.10 Problems 174    15.11 Bibliographic Remarks 175    16 Dimension Theory 177    16.1 Introduction 177    16.2 Chain Realizers 178    16.3 Standard Examples of Dimension Theory 179    16.4 Relationship Between the Dimension and the Width of a Poset 180    16.5 Removal Theorems for Dimension 181    16.6 Critical Pairs in the Poset 182    16.7 String Realizers 184    16.8 Rectangle Realizers 193    16.9 Order Decomposition Method and Its Applications 194    16.10 Problems 196    16.11 Bibliographic Remarks 197    17 Fixed Point Theory 215    17.1 Complete Partial Orders 215    17.2 Knaster   Tarski Theorem 216    17.3 Application: Defining Recursion Using Fixed Points 218    17.4 Problems 226    17.5 Bibliographic Remarks 227    Bibliography 229    Index 235