CONTENTS Preface Conventions 1 Preordered Sets and Posets 1.1 Binary Relations 1.2 Equivalence Relations 1.3 Order Relations 1.3.1 Fundamentals of Order Theory 1.3.2 Monotonic Sets 1.3.3 Application: Preorders in Decision Theory 1.3.4 Hasse Diagrams 1.3.5 Atomic Posets
1.4 Preordered Linear Spaces 1.5 Representation through Complete Preorders 1.5.1 Representation of a Complete Binary Relation 1.5.2 Representation of a Preorder
1.5 Extrema 1.6 Parameters of Posets 1.6.1 Height 1.6.2 Graded Posets 1.6.3 Width 1.6.4 Basic Properties of Finite Posets
1.7 Suprema and In…ma 1.7.1 De…nitions and Examples 1.7.2 Conditionally Complete Posets 1.7.3 Chain-Complete Posets 1.7.4 The Bourbaki-Witt Theorem
2 Lattices 2.1 Elements of Lattice Theory 2.1.1 Lattices 2.1.2 Sublattices 2.1.3 Ideals and Filters 2.1.4 Prime Ideals and Prime Filters
2.4 Vector Lattices 2.5 Functions on Lattices 2.5.1 Valuations 2.5.2 Supermodular Maps 2.5.3 Maps with Increasing Di¤erences
3 Order-Preserving Maps and Isomorphisms 3.1 Order-Preserving Maps 2.1.1 De…nitions and Examples 2.1.2
W
- and
V
-Preservation
2.1.3 Order-Embeddability
3.2 Fundamental Isomorphism Theorems for Lattices 3.2.1 The Isomorphism Theorem for Finite Boolean Lattices 3.2.2 The Isomorphism Theorem for Finite Distributive Lattices 3.2.3 Finite Distributive Lattices vs. Finite Posets 3.2.4 The Isomorphism Theorem for Modular Lattices
3.3 Order-Preservation in Vector Lattices 3.3.1 Order-Preserving Linear Maps 3.3.2 Order-Preserving Additive Maps
3.4 Galois Connections 3.4.1 De…nitions and Examples 3.4.2 Properties of Galois Connections 3.4.3
W
-Preservation
3.4.4 Application: The Dedekind-MacNeille Completion
3.5 Order-Preserving Correspondences 3.5.1 Induced Set Orderings 3.5.2 Weakly and Strongly Order-Preserving Correspondences
3.6 An Application to Optimization Theory 4 Möbius Functions 4
4.1 Motivation: Inversion Problems on Posets 4.2 Incidence Algebras 4.2.1 (Real) Algebras 4.2.2 Incidence Algebras on Locally Finite Posets 4.2.9 Convergence in Incidence Algebras 4.2.4 Incidence Algebras on Finite Posets
4.3 Möbius Functions 4.3.1 De…nitions and Examples 4.3.2 Philip Hall’s Chain Theorem 4.3.3 Möbius Inversion 4.3.4 Application: The Riemann Zeta-Function 4.3.5 Application: Random Sets 4.3.6 Application: The Shapley Value
4.4 Möbius Algebras 4.4.1 Möbius Algebras 4.4.2 Möbius Functions on Lattices 4.4.3 Application: Dilworth’s Covering Theorem 4.4.3 Application: The Myerson Value 4.4.4 Total Monotonicity
An Interlude: The Axiom of Choice I.1 The Axiom of Choice I.1.1 An Intuitive Discussion
I.2 Digression: Pathological Consequences of the Axiom of Choice I.2.1 The Vitali Paradox I.2.2 Existence of Non-Measurable Sets I.2.2 The von Neumann Paradox I.2.3 The Banach-Tarski Paradox
5 Zorn’s Lemma and its Applications 5.1 Chains and Antichains, Again 5.1.1 Chains and Antichains of an In…nite Poset 5.1.2 Chain Conditions for Preordered Sets
5.2 The Hausdor¤ Maximal Principle 5.3 An Application to Optimization Theory 5.3.1 Continuity of Preorders 5.3.2 Existence of Maximal Elements 5.3.3 Application: Revealed Preference Theory
5
5.4 Zorn’s Lemma 5.5 Applications of Zorn’s Lemma 5.5.1 Applications to Linear Analysis 5.5.3 Applications to Graph Theory 5.5.4 Applications to Topology
5.6 The Well-Ordering Principle 5.6.1 Well-Ordered Sets 5.6.2 The Well-Ordering Principle 5.6.3 Application: More on Chains and Antichains 5.6.4 The Cardinality Ordering 5.6.5 Ordinals 5.6.6 Set Decomposition Principles
6 Order-Theoretic Fixed Point Theory 6.1 Fixed Point Theory 6.2 Completeness Conditions for Posets, Again 6.2.1 Countably Chain-Complete Posets 6.2.2 Chain-Complete Posets
6.3 Iterative Fixed Point Theorems 6.3.1 A Prelude 6.3.2 The Tarski-Kantorovich Fixed Point Theorem 6.3.3 An Application to Fractal Geometry 6.3.4 Application: The Contraction Mapping Theorem
6.4 Tarski’s Fixed Point Theorems 6.4.1 The Knaster-Tarski Theorem 6.4.2 Application: The Schröder-Bernstein Theorem 6.4.3 Application: Variational Inequalities
6.5 Converse of the Knaster-Tarski Theorem 6.5.1 Davis’Converse 6.5.2 Retractions of Posets 6.5.3 Proof of Davis’Converse
6.6 The Abian-Brown Fixed Point Theorem 6.6.1 The Abian-Brown Theorem 6.6.2 An Application to Fixed Set Theory 6.6.3 An Application to Game Theory: Rationalizability 6.6.4 An Application to Markov Processes
6.7 Fixed Points of Order-Preserving Corrrespondences 6.7.1 Zhou’s Fixed Point Theorem
6
6.7.2 An Application to Game Theory: Supermodular Games
7 The Brézis-Browder Ordering Principle and its Applications 7.1 A Selection of Ordering Principles 7.1.1 The Brézis-Browder Ordering Principle 7.1.2 Altman’s Ordering Principle
7.2 Applications to Fixed Point Theory 7.2.1 Order-Theoretic Approach to Fixed Point Theory 7.2.2 The Contraction Mapping Theorem 7.2.3 Caristi’s Fixed Point Theorem 7.2.4 Nadler’s Fixed Point Theorem
7.3 Applications to Variational Analysis 7.3.1 Existence of Maxima on Noncompact Domains 7.3.2 Ekeland’s Variational Principle
7.4 An Application to Convex Analysis 7.4.1 The Bishop-Phelps Theorem 7.4.2 A Proof of the Bishop-Phelps Theorem
****************************** (Very Incomplete) 8 Completions and Decompositions of Preordered Sets 7.1 Completions of Preorders 7.1.1 Szpilrajn’s Theorem 7.1.2 Representation of a Preorder through its Completions 7.1.3 Order-Dimension Theory
7.2 Decompositions of Preordered Sets 7.2.1 The Dilworth Decomposition Theorem 7.2.2 Applications of Dilworth’s Theorem 7.2.2 König’s Theorem, Hall’s Matching Principle, etc.
****************************** 9 Functional (Utility) Representation of Preorders 8.1 Preliminaries 8.1.1 Order-Preserving Real Functions 8.1.2 Losets and Q 8.1.3 The Utility Representation Problem 8.2 Representation through Order-Separability 8.3 Representation through Semicontinuity 8.4 The Open Gap Lemma 7
8.5 The Debreu-Eilenberg Representation Theorems 8.6 Multi-Utility Representation 8.8 Continuous Multi-Utility Representation 8.8.1 The Nachbin Extension Theorem 8.8.2 Continuous Multi-Utility Representation 8.8 Finite Multi-Utility Representation ************************************* (Not Complete) 10 Advances in Lattice Theory Subgroup Lattices, Whitney’s Embedding Theorem etc.
Representation of Distributive Lattices, The Prime Ideal Theorem Ordered Stone Spaces Preordered Topological Spaces, Ordered Stone Spaces etc.
Representation of Distributive Lattices, Again The Priestley Isomorphism Theorem Bounded Distributive Lattices vs. Ordered Stone Spaces
************************************* Appendix: A Primer on Topological Spaces A.1 Topological Spaces A.1.1 De…nitions and Examples A.1.2 Basis for a Topology A.1.3 The Subspace Topology A.1.4 The Product Topology A.1.5 Connectedness A.1.6 Compactness A.1.7 Continuity A.1.8 Semicontinuity A.1.9 The Quotient Topology
A.2 Metric Spaces A.2.1 De…nitions and Examples A.2.2 Topology of Metric Spaces A.2.3 Separable Metric Spaces A.2.4 Complete Metric Spaces A.2.5 Compact Metric Spaces A.2.6 The Arzela-Ascoli Theorem A.2.7 The Stone-Weierstrass Theorem A.2.8 Lipschitz Continuity
8
A.3 The Hausdor¤ Metric A.3.1 The Distance of a Point from a Set A.3.2 The Hausdor¤ Metric
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