Complemented modular lattices with involution and orthogonal geometry
The author considers orthogeometries \((P,\perp)\) where \(P\) is a projective geometry defined in terms of the collinearity condition on the point set (denoted also by \(P\)) and \(\perp\) an orthogonality on \(P\). Let \(\mathbb L(P)\) be the modular lattice of all subspaces of the projective geometry \(P\); for \(X\in \mathbb L(P)\), \(X^\perp\) is defined in the natural way. A complemented modular lattice with involution \(x\rightarrow x'\) is denoted shortly as CMIL. To each orthogeometry \((P,\perp)\) there corresponds the atomic CMIL \(\mathbb L(P,\perp)\) which consists of the \(X\) and \(X^\perp\), where \(X\in\mathbb L(P)\) is finite-dimensional. Further, to each CMIL \(L\) there corresponds an orthogeometry \(\mathbb G(L)\) the points of which are the atoms of \(L\) with \(p,q,r\) collinear iff \(p+q=p+r=q+r\) and \(p\perp q\Leftrightarrow p\leq q'\). The system of all elements of \(L\) having finite dimension or codimension is denoted by \(L_f\). The main results of the paper are Theorems 1.1, 1.2 and 1.3. We quote here Theorem 1.1 as typical: There is an equivalence between orthogeometries and atomic CMILs \(L\) such that \(L=L_f\) given by \(L=\mathbb L(P,\perp)\) and \((P,\perp)=\mathbb G(L)\). In particular, \((P,\perp)\simeq\mathbb{GL}(P,\perp)\) for any orthogeometry and \(L\simeq \mathbb{LG}(L)\) for any finite-dimensional CMIL.
scientific article; zbMATH DE number 5652287
scientific article