Orthomodular lattices from 3-dimensional quadratic spaces

Abstract

If E is a vector space over a field K, then any regular symmetric bilinear form φ on E induces a polarity \(M \mapsto M^ \bot \) on the lattice of all subspaces of E. In the particular case where E is 3-dimensional, the set of all subspaces M of E such that both M and \(M^ \bot \) are not N-subspaces (which, in most cases, is equivalent to saying that M is nonisotropic), ordered by inclusion and endowed with the restriction of the above polarity, is an orthomodular lattice T(E, φ). We show that if K ′ is a proper subfield of K, with K ′ ≠ F2, and E ′ a 3-dimensional K ′-subspace of E such that the restriction of φ to E ′ × E ′ is, up to multiplicative constant, a bilinear form φ ′ on the K ′-space E ′, then T(E ′, φ ′) is isomorphic to an irreducible 3-homogeneous proper subalgebra of T(E, φ). Our main result is a structure theorem stating that, when K is not of characteristic 3, the converse is true, i.e., any irreducible 3-homogeneous proper subalgebra of T(E, φ) is of this form. As a corollary, we construct infinitely many finite orthomodular lattices which are minimal in the sense that all their proper subalgebras are modular. In fact, this last result was our initial aim in this paper.