General lattice theory. New appendices with B. A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H. A. Priestley, H. Rose, E. T. Schmidt, S. E. Schmidt, F. Wehrung, R. Wille. 2nd ed. (English) Zbl 0909.06002

Grätzer, George

Basel: Birkhäuser. xix, 663 p. (1998).

This is the second edition of the book first published in 1978 and edited simultaneously by three publishers: Academic Press (Zbl 0436.06001), Akademie-Verlag (Zbl 0385.06014) and Birkhäuser Verlag (Zbl 0385.06015). The author included the text of the first edition unchanged (correcting some minor misprints and changing notation in two cases: \(\text{Con} L\) is used instead of \(C(L)\) and \(\text{Id} L\) instead of \(I(L))\). The real novelty of the new edition consists in eight appendices (140 pages) and an additional bibliography with 530 entries.
Appendix A (“Retrospective”) surveys the development of the last 20 years and reviews the problems posed in the first edition. Appendix B (“Distributive lattices and duality”), by B. A. Davey and H. A. Priestley, outlines Priestley duality for distributive lattices and its application to classes of distributive lattices with additional operators, and indicates the role of natural duality theory. In Appendix C (“Congruence lattices”), the author and E. T. Schmidt present a survey of results concerning representations of finite distributive lattices as congruence lattices. In Appendix D (“Continuous geometry”), F. Wehrung gives an abbreviated review of J. von Neumann’s work on continuous geometry. In Appendix E (“Projective lattice geometries”), M. Greferath and S. E. Schmidt present their approach to projective geometry. In Appendix F (“Varieties of lattices”), P. Jipsen and H. Rose give an abbreviated version of their monograph (Lect. Notes Math. 1533) (1992; Zbl 0779.06005) on lattice varieties. Similarly, R. Freese, in Appendix G (“Free lattices”), gives an account of free lattices, and, finally, in Appendix H (“Applied lattice theory: formal concept analysis”), B. Ganter and R. Wille outline concept lattices and their applications.
Despite the large number of coauthors the style is uniform and the book is well written. 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: T.Katriňák (Bratislava)

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)

Keywords:

free lattices; complemented modular lattices; free products of lattices; concept lattices; projective lattice geometries; varieties of lattices; Priestley duality; distributive lattices; representations; congruence lattices; continuous geometry

Citations:

Zbl 0436.06001; Zbl 0385.06015; Zbl 0385.06014; Zbl 0518.06001; Zbl 0779.06005