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**Basic posets.**
*(English)*
Zbl 0928.06001

Singapore: World Scientific. viii, 178 p. (1998).

This book is an introduction to the theory of partially ordered sets. Its level of sophistication should be easily accessible to advanced undergraduate students in mathematics, computer science, and the engineering disciplines. The tone of the text is informal with plenty of examples illustrating the concepts and definitions, and Hasse diagrams building graphical intuition for the structure of finite partially ordered sets.

Adjacency matrices, interval orders, angle orders, and circle orders are also introduced. A short chapter on order-preserving maps and isomorphisms investigates Harris diagrams and Harris maps, and concludes with an introduction of a simple poset and the classification problem of simple posets.

In a chapter on constructions of new posets from old posets, the authors discuss ordinal sums as well as product orders, lexicographic orders, and exponential orders, and illustrate all concepts with well-chosen examples.

The last chapter includes the discussion of linear extensions and the dimension of a partially ordered set and an application of linear extensions to scheduling problems.

A small number of lemmas, propositions, and theorems are collected in an appendix and rigorous proofs are given at this point. A reader of the book might want to study them separately but concurrently. This is a well-written general introduction to partially ordered sets, that touches on most of the important ideas in the field without going into depth. An extensive bibliography is provided to direct the reader who wants to know more about partially ordered sets to other books and research papers.

Adjacency matrices, interval orders, angle orders, and circle orders are also introduced. A short chapter on order-preserving maps and isomorphisms investigates Harris diagrams and Harris maps, and concludes with an introduction of a simple poset and the classification problem of simple posets.

In a chapter on constructions of new posets from old posets, the authors discuss ordinal sums as well as product orders, lexicographic orders, and exponential orders, and illustrate all concepts with well-chosen examples.

The last chapter includes the discussion of linear extensions and the dimension of a partially ordered set and an application of linear extensions to scheduling problems.

A small number of lemmas, propositions, and theorems are collected in an appendix and rigorous proofs are given at this point. A reader of the book might want to study them separately but concurrently. This is a well-written general introduction to partially ordered sets, that touches on most of the important ideas in the field without going into depth. An extensive bibliography is provided to direct the reader who wants to know more about partially ordered sets to other books and research papers.

Reviewer: M.Höft (Dearborn)