Complemented modular lattices with involution and orthogonal geometry

Abstract

With each orthogeometry (P, ⊥) we associate ${{\mathbb {L}}(P, \bot)}$, a complemented modular lattice with involution (CMIL), consisting of all subspaces X and X ^⊥ such that dim X < ℵ₀, and we study its rôle in decompositions of (P, ⊥) as directed (resp., disjoint) union. We also establish a 1–1 correspondence between ∃-varieties ${\mathcal {V}}$ of CMILs with ${\mathcal {V}}$ generated by its finite dimensional members and ‘quasivarieties’ ${\mathcal {G}}$ of orthogeometries: ${\mathcal {V}}$ consists of the CMILs representable within some geometry from ${\mathcal {G}}$ and ${\mathcal {G}}$ of the (P, ⊥) with ${{\mathbb {L}}(P, \bot) \in {\mathcal {V}}}$. Here,${\mathcal {V}}$ is recursively axiomatizable if and only if so is ${\mathcal {G}}$. It follows that the equational theory of ${\mathcal {V}}$ is decidable provided that the equational theories of the ${\{{\mathbb {L}}(P, \bot)\, |\, (P, \bot) \in \mathcal {G}, {\rm{dim}} P = n\}}$ are uniformly decidable.