Heiko Reppe Three Generalisations of Lattice Distributivity: An FCA Perspective ISBN: 9783844000375 Preis: 48,80 € / 97,60 SFR 

Rezension 

T. Katrińak (Bratislava) 
The publication under review is the author´s Ph.D. Thesis. It deals with the subject of formal concept analysis. The author presents a thorough exposition of three generalizations of lattice distributivity, namely, ndistributivity, introduced by A. P. Huhn [Acta Sci. Math. 52, No. 12, 3545 (1988; Zbl 668.06003)], nmodularity, defined by G. Grätzer and F. Wehrung [Algebra Univers. 41, No. 2, 87114 (1999; Zbl 965.06008)], and kjoin semidistributivity, proposed by W. Geyer [Order 10, No. 1, 77{92 (1993; Zbl 813.06007)]. The study is divided into five chapters. The first three of them present results concerning the mentioned subject. Here are some of them: (1) Let K = (G;M; I) be a doublyfounded formal context. The following statements are equivalent: (i) The concept lattice of K is ndistributive; (ii) Every implication A → A* Ε pp(K) satisfies: either A has at most n elements or all elements of A* are reducible. (2) The question is, how can we determine the smallest n for which a lattice is nmodular? We need the balanced triples, a special sort of 3element antichains. Then, it can be shown that the set of all balanced triples of a finite lattice forms a lattice. The author gives the reduced formal context of this lattice. (3) He proves that ndistributivity, nmodularity and kjoin semidistributivity are independent to a considerable degree. A large part of this book has been published separately in three papers [B. Ganter and H. Reppe, \Base points, nonunit implications, and convex geometries", Lect. Notes Comput. Sci. 4390, 210220 (2007; Zbl 1187.68582); H. Reppe, \An FCA perspective on ndistributivity", Lect. Notes Comput. Sci. 4604, 255268 (2007; Zbl 1213.06008); \Attribute exploration using implications with proper premises", Lect. Notes Comput. Sci. 5113, 161174 (2008; Zbl 5315555)]. 
Quelle: Zentralblatt MATH 1226  1  
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