Geometric identities in lattice theory
type: Item
instance of: scholarly article
title: Geometric identities in lattice theory
zbMATH Open document ID: 0974.06002
author: Matteo Mainetti
author: Catherine Yan
published in: Journal of Combinatorial Theory. Series A
publication date: 2001-12-18T00:00:00Z
review text: 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).
reviewed by: Horst Szambien
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 08B99
Mathematics Subject Classification ID: 51A05
zbMATH DE Number: 1525317
zbMATH Keywords: Grassmann-Cayley algebra
zbMATH Keywords: linear lattices
zbMATH Keywords: modular lattices
zbMATH Keywords: Arguesian identities
zbMATH Keywords: congruence variety of Abelian groups
zbMATH Keywords: unfolding procedure
zbMATH Keywords: covector variables
zbMATH Keywords: lattice inequality
zbMATH Keywords: projective geometry
zbMATH Keywords: non-Arguesian identity
MaRDI profile type: Publication
full work available at URL: jcta.2000.3104
OpenAlex ID: W1969850881
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DOI: 10.1006/JCTA.2000.3104
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description: scientific article; zbMATH DE number 1525317