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

Some properties. Below are some properties of the commutativity relations $\com$ and $\dcom$ .

1. $\com$ is reflexive.
2. $a \com b$ iff $a \com b^{\perp}$ .
3. $a \com b$ iff $a^{\perp} \dcom b^{\perp}$ .
4. if $a\le b$ or $a\le b^{\perp}$ , then $a\com b$ .
5. $a$ is said to orthogonally commute with $b$ if $a \com b$ and $b\com a$ . In this case, we write $a \mcom b$ . The terminology comes from the following fact: $a \mcom b$ iff there are $x,y,z,t\in L$ , with:
   1. $x\perp y$ ($x$ is orthogonal to $y$ , or $x\le y^{\perp}$ ),
   2. $z\perp t$ ,
   3. $x\perp z$ ,
   4. $a=x\vee y$ , and
   5. $b=z\vee t$ .
6. $\com$ is symmetric iff $\dcom=\com (=\mcom)$ iff $L$ is an orthomodular lattice.
7. $\com$ is an equivalence relation iff $\com=L\times L$ iff $L$ is a Boolean algebra.

Remark. More generally, one can define commutativity $\com$ on an orthomodular poset $P$ : for $a,b\in P$ , $a \com b$ iff $a\wedge b$ , $a\wedge b^{\perp}$ , and $(a\wedge b)\vee (a\wedge b^{\perp})$ exist, and $(a\wedge b)\vee (a\wedge b^{\perp})=a$ . 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).

Bibliography

1

L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).