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**Lattices and ordered sets.**
*(English)*
Zbl 1154.06001

New York, NY: Springer (ISBN 978-0-387-78900-2/hbk). xiv, 305 p. (2008).

This book is intended to be a thorough introduction to the subject of order and lattices, with emphasis on the latter. The author presents purely mathematical aspects; he doesn’t discuss the applications of lattice theory to physics, computer science and other disciplines.

The book is divided into two parts: the first part (Chapters 1–9) presents the basic notions of the theory, and the second part (Chapters 10–12) contains independent topics. In the following we present a short description of every chapter.

Chapter 1 describes the basic theory of partially ordered sets, including duality, chain conditions and Dilworth’s theorem.

Chapter 2 is devoted to the basics of well-ordered sets, especially ordinal and cardinal numbers, a topic that is usually presented only in classes on set theory. The author motivates that he has included it here for two reasons: first, he has used the ordinal numbers on a few occasions (in describing conditions that characterize the incompleteness of a lattice and in the discussion of fixed points) and, second, to help those students who do not take a class in set theory.

Chapter 3 starts the study of lattices, introducing the principal properties, such as join-irreducibility, completeness, denseness, lattice homomorphisms, the B-down map, ideals and filters, prime and maximal ideals, and the Dedekind-MacNeille completion.

Chapter 4 is devoted to the basics of modular and distributive lattices, including the many characterizations of these important types of lattices and effect of distributivity on join-irreducibility and prime ideals.

Chapter 5 is devoted to Boolean algebras. The author concentrates on the lattice-theoretic concepts, such as characterizing Boolean algebras as Boolean rings, or in terms of weak forms of modularity or de Morgan laws, also discussing complete and infinite distributivity.

Chapter 6 concerns the representation theory of distributive lattices. Here, the author describes the representation of a distributive lattice with the DCC (descending chain condition) as a power set sublattice, the representation of complete atomic Boolean algebras as power sets, and the representation of an arbitrary distributive lattice as a sublattice of down-sets of prime ideals.

Chapter 7 is devoted to algebraic lattices, a topic that should be of special interest to all those interested in abstract algebra. Here, the author proves that a lattice is algebraic if and only if it is isomorphic to the subalgebra lattice of some algebra.

Chapter 8 concerns the existence of maximal and prime ideals in lattices. The author presents different types of separation theorems and their relationship with distributivity, including the famous Boolean prime ideal theorem.

Chapter 9 is devoted to congruence relations on lattices, quotient lattices, standard congruences and sectionally complemented lattices.

Chapter 10 presents the duality theory for bounded distributive lattices and Boolean algebras. The author treats the finite case first, where topological notions are not required. Then he discusses the necessary topics concerning Boolean and ordered Boolean spaces. Although very little point-set topology is used, a brief appendix is included for readers who are not familiar with the subject. In both the finite and nonfinite cases, the dualities are most elegantly described using the language of category theory. For readers who are not familiar with this language, an appendix is provided with just enough category theory for the discussion in the text.

Finally, Chapter 11 is an introduction to free lattices and Chapter 12 covers fixed-point theorems for monotone and for inflationary functions on complete partially ordered sets and complete lattices.

Concluding, this book is very useful for a course at the graduate or advanced undergraduate level or for independent study. Beside the lucid presentation of the topics, there are more than 240 carefully chosen exercises in every chapter. Prerequisites are kept to a minimum, but an introductory course in abstract algebra is highly recommended, since many of the examples are drawn from this area.

The book is divided into two parts: the first part (Chapters 1–9) presents the basic notions of the theory, and the second part (Chapters 10–12) contains independent topics. In the following we present a short description of every chapter.

Chapter 1 describes the basic theory of partially ordered sets, including duality, chain conditions and Dilworth’s theorem.

Chapter 2 is devoted to the basics of well-ordered sets, especially ordinal and cardinal numbers, a topic that is usually presented only in classes on set theory. The author motivates that he has included it here for two reasons: first, he has used the ordinal numbers on a few occasions (in describing conditions that characterize the incompleteness of a lattice and in the discussion of fixed points) and, second, to help those students who do not take a class in set theory.

Chapter 3 starts the study of lattices, introducing the principal properties, such as join-irreducibility, completeness, denseness, lattice homomorphisms, the B-down map, ideals and filters, prime and maximal ideals, and the Dedekind-MacNeille completion.

Chapter 4 is devoted to the basics of modular and distributive lattices, including the many characterizations of these important types of lattices and effect of distributivity on join-irreducibility and prime ideals.

Chapter 5 is devoted to Boolean algebras. The author concentrates on the lattice-theoretic concepts, such as characterizing Boolean algebras as Boolean rings, or in terms of weak forms of modularity or de Morgan laws, also discussing complete and infinite distributivity.

Chapter 6 concerns the representation theory of distributive lattices. Here, the author describes the representation of a distributive lattice with the DCC (descending chain condition) as a power set sublattice, the representation of complete atomic Boolean algebras as power sets, and the representation of an arbitrary distributive lattice as a sublattice of down-sets of prime ideals.

Chapter 7 is devoted to algebraic lattices, a topic that should be of special interest to all those interested in abstract algebra. Here, the author proves that a lattice is algebraic if and only if it is isomorphic to the subalgebra lattice of some algebra.

Chapter 8 concerns the existence of maximal and prime ideals in lattices. The author presents different types of separation theorems and their relationship with distributivity, including the famous Boolean prime ideal theorem.

Chapter 9 is devoted to congruence relations on lattices, quotient lattices, standard congruences and sectionally complemented lattices.

Chapter 10 presents the duality theory for bounded distributive lattices and Boolean algebras. The author treats the finite case first, where topological notions are not required. Then he discusses the necessary topics concerning Boolean and ordered Boolean spaces. Although very little point-set topology is used, a brief appendix is included for readers who are not familiar with the subject. In both the finite and nonfinite cases, the dualities are most elegantly described using the language of category theory. For readers who are not familiar with this language, an appendix is provided with just enough category theory for the discussion in the text.

Finally, Chapter 11 is an introduction to free lattices and Chapter 12 covers fixed-point theorems for monotone and for inflationary functions on complete partially ordered sets and complete lattices.

Concluding, this book is very useful for a course at the graduate or advanced undergraduate level or for independent study. Beside the lucid presentation of the topics, there are more than 240 carefully chosen exercises in every chapter. Prerequisites are kept to a minimum, but an introductory course in abstract algebra is highly recommended, since many of the examples are drawn from this area.

Reviewer: Florentina Chirteş (Craiova)