Let $L$ be an atomic ortholattice. We say that two elements $a$ and $b$ of $L$ are orthogonal if $a\leq b^\perp$. If $L$ is orthomodular then every element of $L$ can be written as a join of pairwise orthogonal atoms of $L$. Is the converse also true?



Are there atomistic ortholattices which are not modular?
Yes there are. Any non-modular finite orthomodular lattice is an example.


A concrete example: take the Boolean algebras $2^2$,$2^3$ and identify their top and bottom elements. This is clearly an atomistic ortholattice, which is not modular, because it contains a pentagon as a sublattice.


