{"product_id":"introduction-to-lattice-theory-with-computer-science-applications-9781118914373","title":"Introduction to Lattice Theory with Computer","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eA computational perspective on partial order and lattice theory, focusing on algorithms and their applications   This book provides a uniform treatment of the theory and applications of lattice theory.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\"This nice book on lattices and their applications in computer science is written from the perspective of a computer scientist rather than a mathematician...Given its emphasis on algorithms and their complexity, it seems to be mainly intended for students of computer science and engineering. The author's approach is based on the premise that a student needs to learn the heuristics that guide the proofs, besides the proofs themselves, and to learn ways to extend and analyze theorems...One of the most important and valuable features of the book is its focus on applications of lattice theory. The author intends to treat applications on par with the theory.\" Altogether a \"lovely book\". (\u003ci\u003eMathematical Reviews\/MathSciNet\u003c\/i\u003e April 2017)\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003eList of Figures xiii\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eNomenclature xv\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePreface xvii\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Introduction 1\u003c\/p\u003e \u003cp\u003e1.2 Relations 2\u003c\/p\u003e \u003cp\u003e1.3 Partial Orders 3\u003c\/p\u003e \u003cp\u003e1.4 Join and Meet Operations 5\u003c\/p\u003e \u003cp\u003e1.5 Operations on Posets 7\u003c\/p\u003e \u003cp\u003e1.6 Ideals and Filters 8\u003c\/p\u003e \u003cp\u003e1.7 Special Elements in Posets 9\u003c\/p\u003e \u003cp\u003e1.8 Irreducible Elements 10\u003c\/p\u003e \u003cp\u003e1.9 Dissector Elements 11\u003c\/p\u003e \u003cp\u003e1.10 Applications: Distributed Computations 11\u003c\/p\u003e \u003cp\u003e1.11 Applications: Combinatorics 12\u003c\/p\u003e \u003cp\u003e1.12 Notation and Proof Format 13\u003c\/p\u003e \u003cp\u003e1.13 Problems 15\u003c\/p\u003e \u003cp\u003e1.14 Bibliographic Remarks 15\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Representing Posets 17\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Introduction 17\u003c\/p\u003e \u003cp\u003e2.2 Labeling Elements of The Poset 17\u003c\/p\u003e \u003cp\u003e2.3 Adjacency List Representation 18\u003c\/p\u003e \u003cp\u003e2.4 Vector Clock Representation 20\u003c\/p\u003e \u003cp\u003e2.5 Matrix Representation 22\u003c\/p\u003e \u003cp\u003e2.6 Dimension-Based Representation 22\u003c\/p\u003e \u003cp\u003e2.7 Algorithms to Compute Irreducibles 23\u003c\/p\u003e \u003cp\u003e2.8 Infinite Posets 24\u003c\/p\u003e \u003cp\u003e2.9 Problems 26\u003c\/p\u003e \u003cp\u003e2.10 Bibliographic Remarks 27\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Dilworth’s Theorem 29\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Introduction 29\u003c\/p\u003e \u003cp\u003e3.2 Dilworth’s Theorem 29\u003c\/p\u003e \u003cp\u003e3.3 Appreciation of Dilworth’s Theorem 30\u003c\/p\u003e \u003cp\u003e3.4 Dual of Dilworth’s Theorem 32\u003c\/p\u003e \u003cp\u003e3.5 Generalizations of Dilworth’s Theorem 32\u003c\/p\u003e \u003cp\u003e3.6 Algorithmic Perspective of Dilworth’s Theorem 32\u003c\/p\u003e \u003cp\u003e3.7 Application: Hall’s Marriage Theorem 33\u003c\/p\u003e \u003cp\u003e3.8 Application: Bipartite Matching 34\u003c\/p\u003e \u003cp\u003e3.9 Online Decomposition of Posets 35\u003c\/p\u003e \u003cp\u003e3.10 A Lower Bound on Online Chain Partition 37\u003c\/p\u003e \u003cp\u003e3.11 Problems 38\u003c\/p\u003e \u003cp\u003e3.12 Bibliographic Remarks 39\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Merging Algorithms 41\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Introduction 41\u003c\/p\u003e \u003cp\u003e4.2 Algorithm to Merge Chains in Vector Clock Representation 41\u003c\/p\u003e \u003cp\u003e4.3 An Upper Bound for Detecting an Antichain of Size \u003ci\u003eK\u003c\/i\u003e 47\u003c\/p\u003e \u003cp\u003e4.4 A Lower Bound for Detecting an Antichain of Size \u003ci\u003eK\u003c\/i\u003e 48\u003c\/p\u003e \u003cp\u003e4.5 An Incremental Algorithm for Optimal Chain Decomposition 50\u003c\/p\u003e \u003cp\u003e4.6 Problems 50\u003c\/p\u003e \u003cp\u003e4.7 Bibliographic Remarks 51\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Lattices 53\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Introduction 53\u003c\/p\u003e \u003cp\u003e5.2 Sublattices 54\u003c\/p\u003e \u003cp\u003e5.3 Lattices as Algebraic Structures 55\u003c\/p\u003e \u003cp\u003e5.4 Bounding The Size of The Cover Relation of a Lattice 56\u003c\/p\u003e \u003cp\u003e5.5 Join-Irreducible Elements Revisited 57\u003c\/p\u003e \u003cp\u003e5.6 Problems 59\u003c\/p\u003e \u003cp\u003e5.7 Bibliographic Remarks 60\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Lattice Completion 61\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Introduction 61\u003c\/p\u003e \u003cp\u003e6.2 Complete Lattices 61\u003c\/p\u003e \u003cp\u003e6.3 Closure Operators 62\u003c\/p\u003e \u003cp\u003e6.4 Topped ∩-Structures 63\u003c\/p\u003e \u003cp\u003e6.5 Dedekind–Macneille Completion 64\u003c\/p\u003e \u003cp\u003e6.6 Structure of Dedekind--Macneille Completion of a Poset 67\u003c\/p\u003e \u003cp\u003e6.7 An Incremental Algorithm for Lattice Completion 69\u003c\/p\u003e \u003cp\u003e6.8 Breadth First Search Enumeration of Normal Cuts 71\u003c\/p\u003e \u003cp\u003e6.9 Depth First Search Enumeration of Normal Cuts 73\u003c\/p\u003e \u003cp\u003e6.10 Application: Finding the Meet and Join of Events 75\u003c\/p\u003e \u003cp\u003e6.11 Application: Detecting Global Predicates in Distributed Systems 76\u003c\/p\u003e \u003cp\u003e6.12 Application: Data Mining 77\u003c\/p\u003e \u003cp\u003e6.13 Problems 78\u003c\/p\u003e \u003cp\u003e6.14 Bibliographic Remarks 78\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Morphisms 79\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Introduction 79\u003c\/p\u003e \u003cp\u003e7.2 Lattice Homomorphism 79\u003c\/p\u003e \u003cp\u003e7.3 Lattice Isomorphism 80\u003c\/p\u003e \u003cp\u003e7.4 Lattice Congruences 82\u003c\/p\u003e \u003cp\u003e7.5 Quotient Lattice 83\u003c\/p\u003e \u003cp\u003e7.6 Lattice Homomorphism and Congruence 83\u003c\/p\u003e \u003cp\u003e7.7 Properties of Lattice Congruence Blocks 84\u003c\/p\u003e \u003cp\u003e7.8 Application: Model Checking on Reduced Lattices 85\u003c\/p\u003e \u003cp\u003e7.9 Problems 89\u003c\/p\u003e \u003cp\u003e7.10 Bibliographic Remarks 90\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Modular Lattices 91\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Introduction 91\u003c\/p\u003e \u003cp\u003e8.2 Modular Lattice 91\u003c\/p\u003e \u003cp\u003e8.3 Characterization of Modular Lattices 92\u003c\/p\u003e \u003cp\u003e8.4 Problems 98\u003c\/p\u003e \u003cp\u003e8.5 Bibliographic Remarks 98\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Distributive Lattices 99\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Introduction 99\u003c\/p\u003e \u003cp\u003e9.2 Forbidden Sublattices 99\u003c\/p\u003e \u003cp\u003e9.3 Join-Prime Elements 100\u003c\/p\u003e \u003cp\u003e9.4 Birkhoff’s Representation Theorem 101\u003c\/p\u003e \u003cp\u003e9.5 Finitary Distributive Lattices 104\u003c\/p\u003e \u003cp\u003e9.6 Problems 104\u003c\/p\u003e \u003cp\u003e9.7 Bibliographic Remarks 105\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Slicing 107\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Introduction 107\u003c\/p\u003e \u003cp\u003e10.2 Representing Finite Distributive Lattices 107\u003c\/p\u003e \u003cp\u003e10.3 Predicates on Ideals 110\u003c\/p\u003e \u003cp\u003e10.4 Application: Slicing Distributed Computations 116\u003c\/p\u003e \u003cp\u003e10.5 Problems 117\u003c\/p\u003e \u003cp\u003e10.6 Bibliographic Remarks 118\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Applications of Slicing to Combinatorics 119\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Introduction 119\u003c\/p\u003e \u003cp\u003e11.2 Counting Ideals 120\u003c\/p\u003e \u003cp\u003e11.3 Boolean Algebra and Set Families 121\u003c\/p\u003e \u003cp\u003e11.4 Set Families of Size \u003ci\u003ek\u003c\/i\u003e 122\u003c\/p\u003e \u003cp\u003e11.5 Integer Partitions 123\u003c\/p\u003e \u003cp\u003e11.6 Permutations 127\u003c\/p\u003e \u003cp\u003e11.7 Problems 129\u003c\/p\u003e \u003cp\u003e11.8 Bibliographic Remarks 129\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Interval Orders 131\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Introduction 131\u003c\/p\u003e \u003cp\u003e12.2 Weak Order 131\u003c\/p\u003e \u003cp\u003e12.3 Semiorder 133\u003c\/p\u003e \u003cp\u003e12.4 Interval Order 134\u003c\/p\u003e \u003cp\u003e12.5 Problems 136\u003c\/p\u003e \u003cp\u003e12.6 Bibliographic Remarks 137\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Tractable Posets 139\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Introduction 139\u003c\/p\u003e \u003cp\u003e13.2 Series–Parallel Posets 139\u003c\/p\u003e \u003cp\u003e13.3 Two-Dimensional Posets 142\u003c\/p\u003e \u003cp\u003e13.4 Counting Ideals of a Two-Dimensional Poset 145\u003c\/p\u003e \u003cp\u003e13.5 Problems 146\u003c\/p\u003e \u003cp\u003e13.6 Bibliographic Remarks 147\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Enumeration Algorithms 149\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Introduction 149\u003c\/p\u003e \u003cp\u003e14.2 BFS Traversal 150\u003c\/p\u003e \u003cp\u003e14.3 DFS Traversal 154\u003c\/p\u003e \u003cp\u003e14.4 LEX Traversal 154\u003c\/p\u003e \u003cp\u003e14.5 Uniflow Partition of Posets 160\u003c\/p\u003e \u003cp\u003e14.6 Enumerating Tuples of Product Spaces 163\u003c\/p\u003e \u003cp\u003e14.7 Enumerating All Subsets 163\u003c\/p\u003e \u003cp\u003e14.8 Enumerating All Subsets of Size \u003ci\u003ek\u003c\/i\u003e 165\u003c\/p\u003e \u003cp\u003e14.9 Enumerating Young’s Lattice 166\u003c\/p\u003e \u003cp\u003e14.10 Enumerating Permutations 167\u003c\/p\u003e \u003cp\u003e14.11 Lexical Enumeration of All Order Ideals of a Given Rank 168\u003c\/p\u003e \u003cp\u003e14.12 Problems 172\u003c\/p\u003e \u003cp\u003e14.13 Bibliographic Remarks 173\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Lattice of Maximal Antichains 159\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Introduction 159\u003c\/p\u003e \u003cp\u003e15.2 Maximal Antichain Lattice 161\u003c\/p\u003e \u003cp\u003e15.3 An Incremental Algorithm Based on Union Closure 163\u003c\/p\u003e \u003cp\u003e15.4 An Incremental Algorithm Based on BFS 165\u003c\/p\u003e \u003cp\u003e15.5 Traversal of the Lattice of Maximal Antichains 166\u003c\/p\u003e \u003cp\u003e15.6 Application: Detecting Antichain-Consistent Predicates 168\u003c\/p\u003e \u003cp\u003e15.7 Construction and Enumeration of Width Antichain Lattice 169\u003c\/p\u003e \u003cp\u003e15.8 Lexical Enumeration of Closed Sets 171\u003c\/p\u003e \u003cp\u003e15.9 Construction of Lattices Based on Union Closure 174\u003c\/p\u003e \u003cp\u003e15.10 Problems 174\u003c\/p\u003e \u003cp\u003e15.11 Bibliographic Remarks 175\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Dimension Theory 177\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Introduction 177\u003c\/p\u003e \u003cp\u003e16.2 Chain Realizers 178\u003c\/p\u003e \u003cp\u003e16.3 Standard Examples of Dimension Theory 179\u003c\/p\u003e \u003cp\u003e16.4 Relationship Between the Dimension and the Width of a Poset 180\u003c\/p\u003e \u003cp\u003e16.5 Removal Theorems for Dimension 181\u003c\/p\u003e \u003cp\u003e16.6 Critical Pairs in the Poset 182\u003c\/p\u003e \u003cp\u003e16.7 String Realizers 184\u003c\/p\u003e \u003cp\u003e16.8 Rectangle Realizers 193\u003c\/p\u003e \u003cp\u003e16.9 Order Decomposition Method and Its Applications 194\u003c\/p\u003e \u003cp\u003e16.10 Problems 196\u003c\/p\u003e \u003cp\u003e16.11 Bibliographic Remarks 197\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Fixed Point Theory 215\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 Complete Partial Orders 215\u003c\/p\u003e \u003cp\u003e17.2 Knaster–Tarski Theorem 216\u003c\/p\u003e \u003cp\u003e17.3 Application: Defining Recursion Using Fixed Points 218\u003c\/p\u003e \u003cp\u003e17.4 Problems 226\u003c\/p\u003e \u003cp\u003e17.5 Bibliographic Remarks 227\u003c\/p\u003e \u003cp\u003e\u003cb\u003eBibliography 229\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eIndex 235\u003c\/b\u003e\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49406945821015,"sku":"9781118914373","price":71.96,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781118914373.jpg?v=1730497650","url":"https:\/\/bookcurl.com\/products\/introduction-to-lattice-theory-with-computer-science-applications-9781118914373","provider":"Book Curl","version":"1.0","type":"link"}