Introduction to Lattice Theory with Computer Science Applications

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 intersection -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