Geometric Identities in Lattice Theory

An Arguesian identity is an identity in Grassmann–Cayley algebras with certain multi-linear properties of expressions in joins and meets of vectors and covectors. Many classical theorems of projective geometry and their generalizations to higher dimensions can be expressed as simple and elegant Arguesian identities. In a previous work we showed that an Arguesian identity can be unfolded with respect to a vector variable to obtain a lattice inequality, which holds in various lattices. In this paper, we extend this technique to an arbitrary variable. We prove that for any variable v of an Arguesian identity I, a lattice inequality can be obtained by unfolding I with respect to the variable v. This inequality and its dual are valid in the class of linear lattices if the identity is of order 2, and in the congruence variety of Abelian groups if the identity is of a higher order. Consequently, we obtain a family of lattice identities which are self-dual over the class of linear lattices. In particular, all the inequalities obtained by this method are valid in the lattice of subspaces of a vector space, which are characteristic-free and independent of dimensions.