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Semimodular lattices. Dedicated to Garrett Birkhoff. (English) Zbl 0743.06008

Teubner-Texte zur Mathematik. 125. Stuttgart etc.: B. G. Teubner Verlagsgesellschaft. 236 p. (1991).
This clearly written and well organized monograph consists of four chapters. In chapters I, II and III semimodular lattices (sml) of finite length are dealt with; in chapter IV some particular types of sml having infinite length are investigated.
Chapter I gives the description of basic properties of sml. Next, the Theorems of Kurosh-Ore, Schmidt-Ore, Dilworth’s covering Theorem and Borůvka’s result on the greedy algorithm are presented. The relations between sml and some notions from poset theory and, in particular, lattice theory (geometric lattices, matroids, greedoids, shellable posets, Cohen-Macaulay posets) are studied.
The contents of chapters II and III are characterized by their titles, namely “Strong balanced and consistent lattices”, and “Strong semimodular lattices”. In these chapters a series of the author’s results is presented; some of them were obtained in cooperation with U. Faigle, G. Richter and O. Tamaschke.
By studying sml of infinite length in chapter IV, emphasis is laid on atomistic lattices satisfying a certain neighborhood condition. Also, the notions of standard elements and standard ideals (which go back to G. Grätzer) are dealt with. Most of the results of this chapter are due to the author, F. Maeda and S. Maeda, and M. F. Janowitz, respectively.
The style is lucid and insightful, much effort is devoted to motivation. The book may serve as a text for the area mentioned in the title.

MSC:

06C10 Semimodular lattices, geometric lattices
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
06A11 Algebraic aspects of posets
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