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**Lattice theory: Foundation.**
*(English)*
Zbl 1233.06001

Basel: Birkhäuser (ISBN 978-3-0348-0017-4/pbk; 978-3-0348-0018-1/ebook). xxix, 613 p. (2011).

This book is a new version of the author’s famous book on lattice theory, which appeared in several editions [General lattice theory. Berlin: Akademie-Verlag (1978; Zbl 0385.06014); Basel-Stuttgart: Birkhäuser (1978; Zbl 0385.06015); 2nd edition. Basel: Birkhäuser (1998; Zbl 0909.06002) and (2003; Zbl 1152.06300)]. In the 1998 edition, the contents were updated by eight appendices, where the first one reviewed developments of the years 1978–1998, and the other seven were surveys on new fields (written by a team of experts), whereas the 2003 edition is only a paperback reprint of the 1998 edition.

In the present new version, on the one hand large parts of the text were taken from the old versions and the chapter titles remained almost the same (only IV is new): I First concepts; II Distributive lattices; III Congruences; IV Lattice constructions; V Modular and semimodular lattices; VI Varieties of lattices; VII Free products.

On the other hand, also a lot of changes were made. The most significant is that there are no appendices now, but 44 so-called “diamond sections”, where 16 of them are written by 12 other authors. Furthermore, it is intended to publish a companion volume with the title “Lattice theory: Special topics and applications” (in the afterword, the author states that this project was started at the end of 2010). The 44 diamond-sections – together with the almost one thousand exercises – should “help the reader to acquire a wider view and provide brief glimpses into research fields beyond the horizon of this book” (as stated in the preface). The bibliography was updated and now contains 742 items.

In the reviewer’s opinion, the integration of the former appendices into the main body of the book was a very good idea, as it will help the reader to get a clearer picture of the current state of the art in lattice theory. As a final statement, the last sentence of T. Katriňák’s review of the 1998 edition still can serve: “As the first edition of this book had a deep influence on the development of lattice theory, I expect that the new edition will continue to hold its leading position among the books on lattice theory.”

In the present new version, on the one hand large parts of the text were taken from the old versions and the chapter titles remained almost the same (only IV is new): I First concepts; II Distributive lattices; III Congruences; IV Lattice constructions; V Modular and semimodular lattices; VI Varieties of lattices; VII Free products.

On the other hand, also a lot of changes were made. The most significant is that there are no appendices now, but 44 so-called “diamond sections”, where 16 of them are written by 12 other authors. Furthermore, it is intended to publish a companion volume with the title “Lattice theory: Special topics and applications” (in the afterword, the author states that this project was started at the end of 2010). The 44 diamond-sections – together with the almost one thousand exercises – should “help the reader to acquire a wider view and provide brief glimpses into research fields beyond the horizon of this book” (as stated in the preface). The bibliography was updated and now contains 742 items.

In the reviewer’s opinion, the integration of the former appendices into the main body of the book was a very good idea, as it will help the reader to get a clearer picture of the current state of the art in lattice theory. As a final statement, the last sentence of T. Katriňák’s review of the 1998 edition still can serve: “As the first edition of this book had a deep influence on the development of lattice theory, I expect that the new edition will continue to hold its leading position among the books on lattice theory.”

Reviewer: G. Eigenthaler (Wien)

### MSC:

06-02 | Research exposition (monographs, survey articles) pertaining to ordered structures |

06Bxx | Lattices |

06Cxx | Modular lattices, complemented lattices |

06Dxx | Distributive lattices |

06Exx | Boolean algebras (Boolean rings) |