Finite ordered sets. Concepts, results and uses.

*(English)*Zbl 1238.06001
Encyclopedia of Mathematics and its Applications 144. Cambridge: Cambridge University Press (ISBN 978-1-107-01369-8/hbk). xi, 337 p. (2012).

The book exclusively deals with finite ordered sets, which have significant applications in computer science, statistics, biology, and social sciences. The book is recommended for graduate students and researchers. Each chapter begins with fundamental definitions and theorems, many remarks, examples and results of key concepts such as Dilworth’s theorem and Sperner’s theorem, interval and semiorders, Galois connection, duality with distributive lattices, coding, and dimension theory. In every chapter the authors present “Further topics and references”. Exercises are included at the end of each chapter with helpful hints. The last and largest chapter focuses on the notion of a preference, which concerns cognitive science, microeconomics, operations research, artificial intelligence, and databases. Moreover, the problem of the aggregation of several preference relations into a global preference relation is considered and a number of Arrowian theorems are established.

The book is well organized and provides a very good survey over the theory and its applications. The authors also point to further topics of research.

The book is well organized and provides a very good survey over the theory and its applications. The authors also point to further topics of research.

Reviewer: Renata Majovská (Horní Suchá)

##### MSC:

06-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures |

68-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science |

90-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming |

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |