 The answer is: without additional assumptions, not very. The lattice \(L(\mathbf{H})\) has several quite special order-theoretic features. First it is atomic—every element is the join of minimal non-zero elements (i.e., one-dimensional subspaces). Second, it is irreducible—it can not be expressed as a non-trivial direct product of simpler  OMLs.[16]  Finally, and most significantly, it satisfies the so-called atomic covering law: if \(p \in L(\mathbf{H})\) is an atom and \(p\nleq q\), then \(p \vee q\) covers \(q\) (no element of \(L(\mathbf{H})\) lies strictly between \(p \vee q\) and \(q)\).