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Natural dualities for the working algebraist. (English) Zbl 0910.08001
Cambridge Studies in Advanced Mathematics. 57. Cambridge: Cambridge University Press. xii, 356 p. (1998).
The book presents the theory of dualities for (quasi)varieties of algebras, which has been developed over the past 20 years. The two authors are among the principal creators of this theory. The basic problem is to understand the dual of the quasivariety \(\mathcal Q_A\) generated by a given finite algebra \(A\). This framework generalizes classical situations like Stone duality for Boolean algebras or Priestley duality for distributive lattices. Moreover, this framework is very convenient because \(A\), being a (regular) generator, provides a natural forgetful functor \(\text{hom}(-,A):\mathcal Q_A^{op}\to \text{Set}\) which can be used for analyzing \(\mathcal Q_A^{op}\). There is always a natural topology on \(\text{hom}(X,A)\) and one tries to describe \(\mathcal Q_A^{op}\) as a suitable category of structured topological spaces. Here, a structured topological space is a topological space equipped with a set of finitary operations and relations. The weakest duality, which the authors call a natural duality, means that \(\mathcal Q_A^{op}\) is equivalent to a full reflective subcategory of a suitable category \(\mathcal X\) of all structured topological spaces of a given type. A full duality is a duality \(\mathcal Q_A^{op}\) which is equivalent to \(\mathcal X\). All existing full dualities need the additional assumption that \(A\) is a regular injective in \(\mathcal Q_A\). Under this assumption, a full duality is called a strong duality. In fact, under this assumption, \(\mathcal Q_A^{op}\) can always be described by infinitary operations and relations. Strong duality means that there are no other infinitary data except a topology.
The book is divided into ten chapters. The first one is introductory, the second one deals with natural dualities and the third chapter presents the theory of strong dualities, culminating in two major theorems giving conditions for strong dualities. The second and the third chapter yield the theoretical core of the subject. The forth chapter gives examples of strong dualities and the rest of the book consists of applications and special topics. The book is self-contained and well written. It contains a wealth of material not only about dualities but about structural properties of quasivarieties in general. Basic duality concepts are category-theoretic but the amount of category theory is very small. My feeling is that, at some places, there could be given more credit to category theory.
I conclude by recommending the book to all algebraists interested to learn about a new field of dualities. There still remain many fundamental problems to be answered.

08-02 Research exposition (monographs, survey articles) pertaining to general algebraic systems
08C15 Quasivarieties
08B99 Varieties
18-02 Research exposition (monographs, survey articles) pertaining to category theory
18B99 Special categories
54H99 Connections of general topology with other structures, applications
08C05 Categories of algebras