Arguesian Identities in Linear Lattices

A class of identities in the Grassmann–Cayley algebra which yields a large number of geometric theorems on the incidence of subspaces of projective spaces was found by Hawrylycz (“Geometric Identities in Invariant Theory,” Ph.D. thesis, Massachusetts, Institute of Technology, 1994). In this paper we establish a link between such identities in the Grassmann–Cayley algebra and a class of inequalities in the class of linear lattices, i.e., the lattices of commuting equivalence relations. We prove that a subclass of identities found by Hawrylycz, namely, the Arguesian identities of order 2, can be systematically translated into inequalities holding in linear lattices. As a consequence, we obtain a family of geometric theorems on the incidence of subspaces that are characteristic-free and independent of dimensions.