On geometric representations of modular ortholattices (Q2450845)

The authors study special symmetric binary relations \(\bot \) on a projective space \(P\). Such a relation is called a \textit{pre-orthogonality} iff \(p^\bot :=\{q\in P : q\bot p\}\) is a subspace of \(P\) for all points \(p\in P\). It is called \textit{orthogonality} iff \(p^\bot\) is a hyperplane for all \(p\in P\). A pre-orthogonality is called \textit{anisotropic} iff \(p\bot p\) never holds. The subgeometries of any projective geometry \(P\) give rise to an algebraic complemented modular lattice \(L(P)\). For a subgeometry \(Q\) of \(P\) there is an evident join semilattice embedding \(\varphi : L(Q) \rightarrow L(P)\). \(Q\) is called a \textit{Baer subgeometry} iff \(\varphi\) is even a lattice embedding. The main results comprise the following: If \(\bot\) is an anisotropic pre-orthogonality on \(P\), then \(Q:= \{p\in P : p + p^\bot = P \}\) is a Baer subgeometry of \(P\) and the induced pre-orthogonality on \(Q\) is an anisotropic orthogonality (\(+\) denotes the join of subspaces) (Theorem 5.1). Let \(L\) be a modular lattice equipped with an orthocomplementation \(^\prime\), which is 0-1-embedded by \(\varphi\) into a modular 0-1-lattice \(M\), and let \(M\) carry an anisotropic pre-orthogonality \(\bot\), such that \(\varphi(a)\bot\varphi(a^\prime)\) for all \(a\in L\). If there is a non-void ``Baer subgeometry'' \(Q\) (defined as in Theorem 5.1) of \(M\), then \(\eta(a):=\{ p\in Q : p\leq \varphi(a)\}\) defines a representation \(\eta : L \rightarrow L(Q) \) of \((L,^\prime)\) in the anisotropic orthogeometry \((Q,\bot)\) (Theorem 6.6).