Definable relations in finite-dimensional subspace lattices with involution

Abstract

For a large class of finite dimensional inner product spaces V, over division $*$-rings F, we consider definable relations on the subspace lattice $\mathsf{L}(V)$ of V, endowed with the operation of taking orthogonals. In particular, we establish translations between the relevant first order languages, in order to associate these relations with definable and invariant relations on F—focussing on the quantification type of defining formulas. As an intermediate structure we consider the $*$-ring $\mathsf{R}(V)$ of endomorphisms of V, thereby identifying $\mathsf{L}(V)$ with the lattice of right ideals of $\mathsf{R}(V)$, with the induced involution. As an application, model completeness of F is shown to imply that of $\mathsf{R}(V)$ and $\mathsf{L}(V)$.