Geometric identities in lattice theory
The authors continue the work started in Adv. Math. 144, 50-93 (1999; Zbl 0933.06004) and Adv. Math. 150, 36-79 (2000; Zbl 0957.06007), giving further evidence to a conjecture of Hawrylycz and Rota's saying that any identity in a Grassmann-Cayley algebra translates into a related identity holding in linear lattices (i.e. lattices of commuting equivalence relations) or in modular lattices. The problem has been solved for Arguesian identities of order 2 in the first paper cited above, and for Arguesian identities of arbitrary order in the second paper by transforming these identities into lattice inequalities in the congruence variety of Abelian groups. The main result (3.2) of the paper under review is to define and carry out the unfolding procedure of an Arguesian identity for covector variables, a procedure which hitherto had only been done for vector variables [loc. cit.]. The resulting lattice inequality and its dual are valid for linear lattices for the case of order 2, and (4.3) in the congruence variety \({\mathbf A}\) of Abelian groups in the case of identities of higher-order. Interpretations and generalizations of classical theorems of projective geometry are given. Finally, the authors present two examples of a non-Arguesian identity whose unfolding is valid in \({\mathbf A}\). Throughout this paper the authors reveal a beautiful interplay of algebra and geometry on the level of results (Grassmann-Cayley-algebra -- latticial projective geometry) as well as on the level of methods (graphs as a heuristic tool for translating equations from Grassmann-Cayley-algebras to lattices).
scientific article; zbMATH DE number 1525317
scientific article