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Keywords:
atomic orthomodular lattice; topological orthomodular lattice; almost orthogonal sets of atoms
atomic orthomodular lattice; topological orthomodular lattice; almost orthogonal sets of atoms
Summary:
The set $A$ of all atoms of an atomic orthomodular lattice is said to be almost ortho\-go\-nal if the set $\{b\in A:b\nleq a'\}$ is finite for every $a\in A$. It is said to be strongly almost ortho\-go\-nal if, for every $a\in A$, any sequence $b_1, b_2,\dots $ of atoms such that $a\nleq b'_1, b_1 \nleq b'_2, \dots $ contains at most finitely many distinct elements. We study the relation and consequences of these notions. We show among others that a complete atomic orthomodular lattice is a compact topological one if and only if the set of all its atoms is almost ortho\-go\-nal.
References:
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