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Combinatorics and partially ordered sets: dimension theory. (English) Zbl 0764.05001
Johns Hopkins Series in the Mathematical Sciences. Baltimore: The Johns Hopkins University Press. xiv, 307 p. (1992).
Although the author of this admirable volume claims a 1971 mathematical birth, it was a rebirth and perhaps it is not surprising that his preference and ultimate development has been recorded in the book he has written. After all, the four color conjecture is an excellent paradise in which to have spent at least part of a previous existence. Anyway, in this work, graph theory is strongly present in the combinatorial form as practised by many workers in that area during the period prior to its settlement.
These days, the subject of ‘poset theory’ is no longer just presented in preliminary chapters in books or monographs on various related or more specialized topics in algebra, topology or analysis as has been the case. Although other books, e.g., P. C. Fishburn’s volume on Interval Orders and Interval Graphs (1985; Zbl 0551.06001) have dwelled on the study of posets per se, it was not yet true that the theory has been discussed from a poset centered viewpoint in the manner accomplished here, even though the author is restrictive in the sense that the guiding principle here is the notion of dimension of posets (various versions) and that the methods employed restrict the class under discussion mostly to finite posets, rather than to other classes such as locally finite or finitary posets, where other combinatorial principles tend to dominate the usual arguments. Various important foundational and review articles have made important first steps in the direction of providing poset theory an independent pavilion at the fair, but nevertheless this book is a first of its kind in the area it deals with.
When the idea of dimension of posets is considered, then the author’s name, because of his valuable contributions to the development of this theory over the past several decades, comes up immediately and it would be difficult to imagine a more natural book on this subject. For this reason, among others perhaps, the author has been very successful in including a personal commentary within the technical discussions as well as the introductions and overviews that will prove to be of historical interest. Certainly one of the reasons for this success is that to quite a large extent, unusual in many other subjects, the commentary concerns aspects of research with which the author has had very direct contact through some form of authorship of many of the articles visited by him in this book, as is clear from the bibliography, if it wouldn’t be obvious in other ways. What it does to the book is to give it a very special style that succeeds in making it more interesting then it might be if it were presented as either a mere textbook or a mere research monograph. Instead, it succeeds in being all three.
As a textbook it proceeds through a sequence of eleven chapters, beginning with an Introduction to Dimension through a chapter on Large Minimal Realizers, while passing through intermediate terrain consisting of theorems and results proven in detail to illustrate techniques, theorems and results stated to provide a more complete survey of the subject and exercises designed to test the reader, with references in the last cases given to direct the frustrated to provided proofs and the interested to a further supply of nutritious morsels to discover. There is a very adequate supply in the text and any student of the subject, upon going through it in detail, will find that he is ready for research in the areas discussed. As a monograph the author also makes it possible for readers interested in specific topics only to limit themselves to introductory material and then to proceed directly to the chapter in question.
Thus, e.g., chapter nine “Greedy Dimension, Back-tracking and Depth First Search” starts with an overview, including relevant definitions whose implications for the topic under discussion are outlined in a compact informal manner which enables the reader to decide in short order whether the right choice of chapter has been made. Next, the Jump Number Problem is introduced in a similar manner in its own section, inclusive of references to the literature. The next sections provide a general Linear Extension Algorithm and other versions via tie breaking rules for Greedy Linear Extensions and the associated Greedy Dimension along with inequalities, examples and open questions concerning this dimension. A further step leads to Super-Greedy extensions, an algorithm and inequalities; examples and open questions as before and according to an analogous treatment.
Other topics covered in a similar manner not yet mentioned by us include: Chapter 2: Crowns, Splits, Stacks, Suns and Products; Chapter 3: Characterizations Problems for Posets, Lattices, Graphs, and Families of Sets; Chapter 4: Hypergraph Coloring, Computational Complexity, and Irreducible Posets; Chapter 5: Planar Posets and Trees; Chapter 6: Planar Graphs, Planar Maps, and Convex Polytopes; Chapter 7: Probabilistic Methods in Dimension Theory; Chapter 8: Interval and Geometric Containment Orders; Chapter 10: Products of Chains of bounded Length.
Obviously the contents indicate that the book is quite comprehensive in its area. As we have also said above it is well written. It has been produced quite carefully, with very few misprints. As it stands this volume should become known as ‘Trotter’ to those in the area and it is hoped that its currency will be maintained via updating appendices and reasonably frequent re-editing, to keep it as useful a text as it is now already. Hopefully, it will go a ways to become a frequently consulted text which will serve to train a generation or so of students in the area, covered admirably by an author ambitious to write a text which will make a difference, who has certainly succeeded at that task according to this reviewer.

05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
05Cxx Graph theory
06A07 Combinatorics of partially ordered sets
06A06 Partial orders, general