Definable relations in finite dimensional subspace lattices with involution. Part II: Quantifier-free and homogeneous descriptions

Abstract

For finite dimensional hermitean inner product spaces V, over \(*\)-fields F, and in the presence of orthogonal bases providing form elements in the prime subfield of F, we show that quantifier-free definable relations in the subspace lattice \(\mathsf{L}(V)\), endowed with the involution induced by orthogonality, admit quantifier-free descriptions within F, also in terms of Grassmann–Plucker coordinates. In the latter setting, homogeneous descriptions are obtained if one allows quantification type \(\Sigma _1\). In absence of involution, these results remain valid.