Exploring Basis Vector Relationships in Incompatible Propositions

[forkosh]

TL;DR Summary

If propositions $p,q\in{\mathscr L}_{\mathcal H}$ (i.e., the lattice of subspaces of $\mathcal H$) are incompatible, then $\hat p\hat q\neq\hat q\hat p$. But since it's a lattice, there exists a unique glb $p\wedge q=q\wedge p$. How are they mathematically related?

If propositions $p,q\in{\mathscr L}_{\mathcal H}$ (i.e., the lattice of subspaces of $\mathcal H$) are incompatible, then $\hat p\hat q\neq\hat q\hat p$. But since it's a lattice, there exists a unique glb $p\wedge q=q\wedge p$. How are they mathematically related?

In particular, I believe (please correct me if I'm wrong) that "incompatible" can also be defined as: there exists no single orthonormal basis $E$ for $\mathcal H$ such that one subset $E_p\subseteq E$ exactly spans $p$, and another subset $E_q\subseteq E$ exactly spans $q$.

Then how are sets of basis vectors (from different bases $E,F,G,\ldots$ as necessary) that exactly span $p,q,\hat p\hat q,\hat q\hat p,p\wedge q$ related? And more particularly, how are basis vectors for $\hat p\hat q,\hat q\hat p$ related to those for $p\wedge q$?

 

[azntoon]

Mathematically, what I'm asking is: is there some way to relate the set of basis vectors $E_p$ (for $p$) to the set of basis vectors $E_{\hat p\hat q},E_{\hat q\hat p},E_{p\wedge q}$ (for $\hat p\hat q,\hat q\hat p,p\wedge q$ respectively) in a way that makes it clear why $\hat p\hat q\neq\hat q\hat p$ and $p\wedge q=q\wedge p$?