Complemented modular lattices with involution and orthogonal geometry
type: Item
instance of: scholarly article
title: Complemented modular lattices with involution and orthogonal geometry
zbMATH Open document ID: 1192.06008
author: Christian Herrmann
published in: Algebra Universalis
publication date: 2010-01-04T00:00:00Z
review text: 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.
reviewed by: Ján Jakubík
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 06C20
Mathematics Subject Classification ID: 51A50
zbMATH DE Number: 5652287
zbMATH Keywords: complemented modular lattice
zbMATH Keywords: involution
zbMATH Keywords: orthogonal geometry
MaRDI profile type: Publication
full work available at URL: s00012-009-0003-5
OpenAlex ID: W1979304146
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DOI: 10.1007/S00012-009-0003-5
description: scientific article; zbMATH DE number 5652287