Quasi-atoms in symmetric orthomodular lattices
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.
scientific article; zbMATH DE number 4177128