1 The Basics. - 1. 1 Definition and Examples. - 1. 2 The Diagram. - 1. 3 Order-Preserving Mappings/Isomorphism. - 1. 4 Fixed Points. - 1. 5 Ordered Subsets/The Reconstruction Problem. - Exercises. - Remarks and Open Problems. - 2 Chains Antichains and Fences. - 2. 1 Chains and Zorn's Lemma. - 2. 2 Well-ordered Sets. - 2. 3 A Remark on Duality. - 2. 4 The Rank of an Element. - 2. 5 Antichains and Dilworth's Chain Decomposition Theorem. - 2. 6 Dedekind Numbers. - 2. 7 Fences and Crowns. - 2. 8 Connectivity. - Exercises. - Remarks and Open Problems. - 3 Upper and Lower Bounds. - 3. 1 Extremal Elements. - 3. 2 Covers. - 3. 3 Lowest Upper and Greatest Lower Bounds. - 3. 4 Chain-Completeness and the Abian-Brown Theorem. - Exercises. - Remarks and Open Problems. - 4 Retractions. - 4. 1 Definition and Examples. - 4. 2 Fixed Point Theorems. - 4. 3 Dismantlability. - 4. 4 The Fixed Point Property for Ordered Sets of Width 2 or Height 1. - 4. 5 Li and Milner's Structure Theorem. - 4. 6 Isotone Relations. - Exercises. - Remarks and Open Problems. - 5 Lattices. - 5. 1 Definition and Examples. - 5. 2 Fixed Point Results/The Tarski-Davis Theorem. - 5. 3 Embeddings/The Dedekind-MacNeille Completion. - 5. 4 Irreducible Points in Lattices. - 5. 5 Finite Ordered Sets vs. Distributive Lattices. - 5. 6 More on Distributive Lattices. - Exercises. - Remarks and Open Problems. - 6 Truncated Lattices. - 6. 1 Definition and Examples. - 6. 2 Recognizability and More. - 6. 3 The Fixed Clique Property. - 6. 4 Triangulations of Sn. - 6. 5 Cutsets. - 6. 6 Truncated Noncomplemented Lattices. - Exercises. - Remarks and Open Problems. - 7 The Dimension of Ordered Sets. - 7. 1 (Linear) Extensions of Orders. - 7. 2 Balancing Pairs. - 7. 3 Defining the Dimension. - 7. 4 Bounds on the Dimension. - 7. 5 Ordered Sets of Dimension 2. - Exercises. - Remarks and Open Problems. - 8 Interval Orders. - 8. 1Definition and Examples. - 8. 2 The Fixed Point Property for Interval Orders. - 8. 3 Dedekind's Problem for Interval Orders and Reconstruction. - 8. 4 Interval Dimension. - Exercises. - Remarks and Open Problems. - 9 Lexicographic Sums. - 9. 1 Definition and Examples. - 9. 2 The Canonical Decomposition. - 9. 3 Comparability Invariance. - 9. 4 Lexicographic Sums and Reconstruction. - 9. 5 An Almost Lexicographic Construction. - Exercises. - Remarks and Open Problems. - 10 Sets PQ = Hom(Q P) and Products. - 10. 1 Sets PQ = Hom(Q P). - 10. 2 Finite Products. - 10. 3 Infinite Products. - 10. 4 Hashimoto's Theorem and Automorphisms of Products. - 10. 5 Arithmetic of Ordered Sets. - Exercises. - Remarks and Open Problems. - 11 Enumeration. - 11. 1 Graded Ordered Sets. - 11. 2 The Number of Graded Ordered Sets. - 11. 3 The Asymptotic Number of Graded Ordered Sets. - 11. 4 The Number of Nonisomorphic Ordered Sets. - 11. 5 The Number of Automorphisms. - Exercises. - Remarks and Open Problems. - 12 Algorithmic Aspects. - 12. 1 Algorithms. - 12. 2 Polynomial Efficiency. - 12. 3 NP problems. - 12. 4 NP-completeness. - 12. 5 So It's NP-complete. - 12. 6 A Polynomial Algorithm for the Fixed Point Property in Graded Ordered Sets of Bounded Width. - Exercises. - Remarks and Open Problems. - A A Primer on Algebraic Topology. - A. l Chain Complexes. - A. 2 The Lefschetz Number. - A. 3 (Integer) Homology. - A. 4 A Homological Reduction Theorem. - Remarks and Open Problems. - B Order vs. Analysis. - B. 2 Fixed Point Theorems. - B. 3 An Application. - Remarks and Open Problems. - References. Language: English