I would like to have a definition for non-modular lattices which clearly sets them appart from their modular counterparts, thereby focusing on their main distinctive feature. Besides, I would be very grateful to you if you could provide me with an explanation about orto-modularity, which Birkhoff and von Neumann allegedly considered to be the basis of quantum logic, as opposed to modular logics. Is there any correlation between the two types of lattices and this issue of orto-modularity?
Thanks in advance.
The definition, if there could be said to be such a thing, would just be the negation of the definition of a modular lattice.
Fortunately there is a rather clear characterization of lattices which aren't modular, and that is that they contain the N5 lattice as a sublattice.
By definition an orthomodular lattice is a complemented lattice satisfying and additional weakened modularity law:
$a\leq b\implies b=a\vee (b\wedge a^\perp)$
The full modular law clearly implies the one we've called weakened:
If $a\leq b\implies a\vee (c\wedge b)=(a\vee c)\wedge b$ for all $c$, then putting $c=a^\perp$ yields the weakened property.
The only relationship is that a complemented modular lattice is orthocomplented. Modular lattices need not be complemented, and complemented lattices need not satisfy even the weakened modular condition.