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Interval orders and interval graphs. A study of partially ordered sets. (English) Zbl 0551.06001
Wiley-Interscience Series in Discrete Mathematics. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. XI, 215 p. £40.50 (1985).
The author’s book has the pleasant relatively small size of a book such as G. Grätzer’s book on lattice theory [General lattice theory (1978; Zbl 0385.06014)]. This makes it attractive for use as an introduction to a useful and very interesting subject in an inviting manner which does not overwhelm the beginner with an excess of information.
Although both Fishburn’s book and Grätzer’s book can be interpreted as dealing with aspects of the theory of posets, there is in fact very little overlap. Furthermore, several other such small books could be written on closely related topics without a serious amount of duplication with either volume.
The topics discussed in Fishburn’s book are very much topics with which the author’s name is associated, although some of them are of very respectable antiquity as these matters go in mathematics. Nevertheless, the book suits the author, especially in its mix of pure mathematics with those applied problems which provided original motivations for the topics under consideration.
Much of what Fishburn writes about can, e.g., be recovered from F. S. Roberts’ more encyclopedic volume on ”Measurement theory” (1979; Zbl 0432.92001) which refers to Fishburn 46 times according to its author index.
The practical reasons for introducing interval orders and other types of posets discussed here are connected precisely with the analysis of events which may be placed on the real line (or on other linearly ordered sets, but not always in the form of points. Thus, the intervals may be those of time (duration) or uncertainty related to the imprecision of various measurements. While Fishburn’s book does not primarily emphasize these topics over the theoretical ones discussed, it is clear from the entire text that these practical reasons are of great importance to the author’s views in the introduction of materials. Thus, although this is not precisely an applied mathematics book, parts of it could well serve the mathematically inclined social scientist or engineer make better sense out of what has by now become another rather enormous field which is expanding at rates not yet in conflict with unlimited natural growth.
The Chapters included are: Introduction (to posets and other generalities); Interval Orders; Interval Graphs; Betweenness; Dimensionality and Other Parameters; Embedded Semiorders and Indifference Graphs; Real Representations; Bounded Interval Orders; Numbers of Length; Extremization Problems.
The author has not sought to construct exercises to introduce complicated subjects not covered in the text. Thus that which needs explaining has largely been explained; and very carefully. Readers will benefit, whether they are being introduced to the subject, or whether they are more experienced and especially interested in topics presented in later chapters which tend to be a bit more specialized than the earlier ones. One does not need to refer to quantities of outside results. This is not the type of book that in the end (or considerably much sooner) discourages the student or prospective student from further exploration. To this reviewer’s taste the chapter on Dimensionality and Other Parameters went down best and as a short (25 page) survey of that subject it will be used by him in the appropriate courses. If all advanced subjects in mathematics were introduced in this careful and elegant manner, all would be served very well.
Although this reviewer might have selected some different materials for some of the presentations, it is not the reviewer’s wish to find any fault with a labor which has obviously been executed with great care and by a skilled hand. If haste make waste, this small and excellent volume demonstrates how the negative of that old saw can sometimes be invoked with excellent reason also.

MSC:
06A05 Total orders
06A06 Partial orders, general
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
92-02 Research exposition (monographs, survey articles) pertaining to biology
94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory
05Cxx Graph theory