Arguesian identities in the congruence variety of Abelian groups
The author continues the work begun with \textit{M. Mainetti} [Adv. Math. 144, 50-93 (1999; Zbl 0933.06004)], treating the following problem posed by \textit{M. Hawrylycz} [Adv. Math. 122, 1-48 (1996; Zbl 0858.15019)]: Translate Arguesian identities in Grassmann-Cayley algebras into inequalities holding in linear lattices (i.e. lattices of commuting equivalence relations) or in the congruence variety of abelian groups. The problem has been solved for Arguesian identities of order 2 in the joint paper cited above. The main result (Thm. 4.2) of the paper under review is a general solution for Arguesian identities \(P=Q\) of arbitrary order by transforming these identities into lattice inequalities in the congruence variety of abelian groups. The transformation is carried out by applying the so-called unfolding procedure to the polynomials \(P\), \(Q\) on either side of the identity. Whereas many Arguesian identities fail to hold when viewed as lattice identities in a projective geometry, their unfoldings are valid inequalities in the lattice of subspaces of any vector subspace as a consequence of (4.2). This phenomenon is demonstrated by generating various theorems of synthetic projective geometry and their generalizations to higher dimensions (Desargues, Bricard, Fontené) by the unfolding procedure.
scientific article; zbMATH DE number 1422998