Ordered Sets: An Introduction

An introduction to the basic tools of the theory of (partially) ordered sets such as visualization via diagrams, subsets, and homomorphisms. Using a thematic approach, the author presents open or recently solved problems to motivate the development of constructions and investigations for new classes of ordered sets. The text can be used as a focused follow-up or companion to a first proof (set theory and relations) or graph theory course. Rich in exercises, the book will be a valuable text for undergraduate/graduate students and an excellent resource for interested researchers.

Table of Contents:
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.

Review :
"The author has done the field a service by producing an excellent text strong in the presentation of certain topological aspects of the underlying diagrams, e.g., which should serve the developing community and field well and which can be recommended without reservations as one of the volumes which should grace a poseteer's library whether she is interested only or mainly in the theory of these objects or has directed her gaze towards applications. As a sourcebook of ideas and understanding it will make its mark. And deservedly so." -ZENTRALBLATT MATH "This book is a most successful introduction to the theory of ordered sets. The range of topics covered is wide yet the exposition is self-contained ... The focus on open problems makes for enthusiastic writing and provides continuity among the diverse topics that are discussed ... in each chapter the author proves a solid collection of basic results, presents one or more major theorems with detailed and nontrivial proofs, and states without proof one or two other important results. At the end of every chapter is an ample collection of exercises ... A novel aspect of [the twelfth] chapter is the presentation of much of the material in terms of constraint satisfaction problems." -MATHEMATICAL REVIEWS "...surprisingly, despite the extremely basic nature of the subject, this monograph has essentially no competition. But fortunately the first entry proves a gem. Undergraduate mathematics and computer science majors will find the first chapters offering background that will serve them well in many courses. The rest of the book, which features many open problems, constitutes an accessible and stimulating invitation to research ... Highly recommended." -CHOICE "There are two groups of people who will primarily profit from the book. First, researchers in the field of ordered sets and lattices and related fields like algebra, graphs, and combinatorics. The book provides an excellent look at the field with numerous remarks including historical remarks and open problems. Second, students who are looking for a PhD topic. The author presents the field of ordered sets in an attractive way and the many open problems presented in the book are invaluable. Mathematicians and computer scientists could use the book as a reference book, too...[T]he book is a success, it presents an in depth and up to date carefully written coverage of ordered sets." -SIGACT NEWS