PATH-CONNECTED AND NON PATH-CONNECTED ORTHOMODULAR LATTICES

Abstract

A block of an orthomodular lattice L is a maximal Boolean subalgebra of L. A site is a subalgebra of an orthomodular lattice L of the form S = A boolean AND B, where A and B are distinct blocks of L. An orthomodular lattice L is called with finite sites if vertical bar A boolean AND B vertical bar < infinity for all distinct blocks A, B of L. We prove that there exists a weakly path-connected orthomodular lattice with finite sites which is not path-connected and if L is an orthomodular lattice such that the height of the join-semilattice [Com L]v generated by the commutators of L is finite, then L is pathconnected.