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Ordered sets. An introduction. (English) Zbl 1010.06001

Boston, MA: Birkhäuser. xvii, 391 p. (2003).
This book is one more general introduction to the theory of partially ordered sets or posets, which no longer presents the subject merely as a smallish part of discrete mathematics, subset of graph theory or, via the order-complex, a combinatorial structure of algebraic-topological interest. Instead it treats the subject as an area of study in its own right and not in the form of a specialized monograph, no matter how brilliantly conceived or how perfectly executed such might be. Of course, in attempting to provide an initial oversight of a renovated field and incorporating the hope that this text may prove foundational for a generation of young poset theorists, the author has shown high ambition and should be praised for incorporating as many results as he has at a level which not only makes it accessible to the mature undergraduate curious about this subject but the graduate student interested not only in learning about the field but looking for problem areas to plant useful thesis claims with an eye to completing successful research topics. Even a veteran, but one specialized in a particular area, may browse the text with pleasure and advantage, noting that much information is hidden in the exercises which permit the apprentice and journeyman to acquire necessary skills and insights either through direct effort or via a certain amount of work topped-off by consulting the literature provided in a substantial but by no means complete set of references. Certainly the author has done valiant battle with space limitations so as not to end up with the proverbial “bible” which would serve to chase prospective students away rather than attract them. As a result he had to make choices to eliminate or underemphasize certain topics, e.g., those associated with locally finite or Eulerian posets, the Möbius function and associated combinatorics or the area opened up by Stanley in his dissertation example and followed up by students in his school, the chain-meets-antichain theorem of Rival-Zaguia etc. Fortunately there are other classics to cover these areas ranging from Berge to Stanley himself. And, as we have already recognized, books have to be finite in a strong sense to remain digestible. 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.

MSC:

06-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
06A06 Partial orders, general
06A07 Combinatorics of partially ordered sets
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