Ordered Sets

Bernd Schröder serves as Chair of the Department of Mathematics at The University of Southern Mississippi. In addition to teaching and service, he has published over 40 papers on subjects including ordered sets, graph theory, analysis, computer science, and engineering education.

Preface.- Basics.- Chains, Antichains, and Fences.- Upper and Lower Bounds.- Retractions.- Constraint Satisfaction Problems.- Graphs and Homomorphisms.- Lexicographic Sums.- Lattices.- Truncated Lattices.- Dimension.- Interval Orders.- Sets P^Q = Hom (Q, P) and Products.- Enumeration of Ordered Sets.- Appendix A: Some Algebraic Topology.- Appendix B: Some Discrete Morse Theory.- References.- Index.

"The book is marvelously written with clarity and richness of language and mathematical content being clear hallmarks. ... the book is not just a well crafted textbook but also an invaluable companion to the student, reference to current literature, and a source of inspiration for newcomers and established residents of order theory." (Ittay Weiss, zbMATH 1414.06001, 2019)

"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 whichshould 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