Arguesian identities in the congruence variety of Abelian groups

A class of identities in the Grassmann-Cayley algebra was found by M. J. Ha wrylyez 1994. "Geometric Identities in Invariant Theory." Ph.D. thesis. Mas sachusetts Institute of Technology which yields a large number of geometric theorems on the incidence of subspaces of projective spaces. In a previous paper we established a link between such identities in the Grassmann Cayle y algebra and a family of inequalities in the class of linear lattices. i.e ., the lattices of commuting equivalence relations, We proved that a subcla ss of identities found by Hawrylyez namely, the Arguesian identities of ord er 2, can be systematically translated into inequalities holding in linear lattices. However, it is not known whether the Arguesian identities of high er orders have such latticial extensions. In this paper, we give an affirma tive answer to the above question in the congruence variety of Abelian grou ps. We prove that every Arguesian identity, regardless or the order. can be systematically translated into ii lattice inequality holding in the congru ence variety of Abelian groups. In particular. such a lattice inequality ho lds in the lattices or subspaces of vector spaces. which lire characteristi c-free and independent of dimensions. As a consequences many classical theo rems of projective geometry. including. Desargues. Bricard. Fontene. and th eir higher dimensional generalizations, can be extended to lattice inequali ties in the general projective spaces. with the variables rc presenting sub spaces of arbitrary dimensions, (C) 2000 Academic Press.