Quasi-atoms in symmetric orthomodular lattices
type: Item
instance of: scholarly article
title: Quasi-atoms in symmetric orthomodular lattices
zbMATH Open document ID: 0715.06007
DOI: 10.1007/BF01188991
author: Richard J. Greechie
author: Louis Herman
published in: Algebra Universalis
publication date: 1990-01-01T00:00:00Z
review text: Quasi-atoms in an orthomodular lattice L are studied. Let \(A^{(0)}L\) be the set of elements \(a\in L\) such that the interval [0,a] is Boolean, and \(A^{(1)}L\subseteq A^{(0)}L\) be the set of quasi-atoms a satisfying the modular pair-condition xMa for all \(x\in L\), and \(0\in A^{(1)}L\). If L is symmetric then \(A^{(1)}L\) is an order ideal closed under perspectivity. \(a\in L\) is an atom iff \(a\in A^{(0)}L\), a dominates a quasi-atom and for every p-ideal \(I\subseteq L\) either \(a\perp I\) or \(a\in I\). Among other results it is shown that a simple, symmetric orthomodular lattice L with \(A^{(1)}L\neq <0\}\) is either 2, a nontrivial horizontal sum of copies of \(2^ 2\) or a projective geometry.
Mathematics Subject Classification ID: 06C15
zbMATH DE Number: 4177128
zbMATH Keywords: Quasi-atoms
zbMATH Keywords: order ideal
zbMATH Keywords: symmetric orthomodular lattice
zbMATH Keywords: projective geometry
reviewed by: Gudrun Kalmbach
MaRDI profile type: Publication
cites work: Q3666897
cites work: Q5552761
cites work: Q5559561
cites work: Commutator-finite orthomodular lattices
cites work: Separation conditions in relatively complemented lattices
cites work: Nearly Modular Orthocomplemented Lattices
cites work: Q5623714
full work available at URL: https://doi.org/10.1007/bf01188991
OpenAlex ID: W2039568460
description: scientific article; zbMATH DE number 4177128