Path-connected and non path-connected orthomodular lattices
Bull. Korean Math. Soc. 2009 Vol. 46, No. 5, 845-856
Printed September 1, 2009
Eunsoon Park and Wonhee Song
Soongsil University and Soongsil University
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 $S = A\cap B$, where $A$ and $B$ are distinct blocks of $L$. An orthomodular lattice $L$ is called with finite sites if $|A\cap B|<\infty$ 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 $[Com \, L]_\vee$ generated by the commutators of $L$ is finite, then $L$ is path-connected.
Keywords : orthomodular lattice, with finite sites, path-connected, non path-connected, Boolean algebra
MSC numbers : 06C15
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