commutativity relation in an orthocomplemented lattice

Let $L$ be an orthocomplemented lattice with $a,b\in L$. We say that $a$ commutes with $b$ if $a=(a\wedge b)\vee (a\wedge b^{⟂})$. When $a$ commutes with $b$, we write $a\mathrm{C}b$. Dualize everything, we have that $a$ dually commutes with $b$, written $a\mathrm{D}b$, if $a=(a\vee b)\wedge (a\vee b^{⟂})$.

Some properties. Below are some properties of the commutativity relations $\mathrm{C}$ and $\mathrm{D}$.

1. $\mathrm{C}$ is reflexive.

2. $a\mathrm{C}b$ iff $a\mathrm{C}{b}^{⟂}$.

3. $a\mathrm{C}b$ iff $a^{⟂}\mathrm{D}{b}^{⟂}$.

4. if $a\le b$ or $a\le b^{⟂}$, then $a\mathrm{C}b$.

5. $a$ is said to orthogonally commute with $b$ if $a\mathrm{C}b$ and $b\mathrm{C}a$. In this case, we write $a\mathrm{M}b$. The terminology comes from the following fact: $a\mathrm{M}b$ iff there are $x,y,z,t\in L$, with:

   1. $x⟂y$ ($x$ is orthogonal to $y$, or $x\le y^{⟂}$),

   2. $z⟂t$,

   3. $x⟂z$,

   4. $a=x\vee y$, and

   5. $b=z\vee t$.

6. $\mathrm{C}$ is symmetric iff $\mathrm{D}=\mathrm{C}(=\mathrm{M})$ iff $L$ is an orthomodular lattice.

7. $\mathrm{C}$ is an equivalence relation iff $\mathrm{C}=L\times L$ iff $L$ is a Boolean algebra.

Remark. More generally, one can define commutativity $\mathrm{C}$ on an orthomodular poset $P$: for $a,b\in P$, $a\mathrm{C}b$ iff $a\wedge b$, $a\wedge b^{⟂}$, and $(a\wedge b)\vee (a\wedge b^{⟂})$ exist, and $(a\wedge b)\vee (a\wedge b^{⟂})=a$. Dual commutativity and mutual commutativity in an orthomodular poset are defined similarly (with the provision that the binary operations on the pair of elements are meaningful).