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

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 𝖫(V) , endowed with the involution induced by orthogonality, admit quantifier-free descriptions within F , also in terms of Grassmann–Plücker coordinates. In the latter setting, homogeneous descriptions are obtained if one allows quantification type Σ _1 . In absence of involution, these results remain valid.