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 $a=(a\wedge b)\vee (a\wedge {b}^{\u27c2})$. 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 $a=(a\vee b)\wedge (a\vee {b}^{\u27c2})$.
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 $a\mathrm{C}{b}^{\u27c2}$.

3.
$a\mathrm{C}b$ iff ${a}^{\u27c2}\mathrm{D}{b}^{\u27c2}$.

4.
if $a\le b$ or $a\le {b}^{\u27c2}$, 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:

(a)
$x\u27c2y$ ($x$ is orthogonal to $y$, or $x\le {y}^{\u27c2}$),

(b)
$z\u27c2t$,

(c)
$x\u27c2z$,

(d)
$a=x\vee y$, and

(e)
$b=z\vee t$.

(a)

6.
$\mathrm{C}$ is symmetric iff $\mathrm{D}=\mathrm{C}\phantom{\rule{veryverythickmathspace}{0ex}}(=\mathrm{M})$ 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 $a\wedge b$, $a\wedge {b}^{\u27c2}$, and $(a\wedge b)\vee (a\wedge {b}^{\u27c2})$ exist, and $(a\wedge b)\vee (a\wedge {b}^{\u27c2})=a$. 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).
References
 1 L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).
Title  commutativity relation in an orthocomplemented lattice 

Canonical name  CommutativityRelationInAnOrthocomplementedLattice 
Date of creation  20130322 16:43:22 
Last modified on  20130322 16:43:22 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06C15 
Classification  msc 03G12 
Defines  dually commute 
Defines  orthogonally commute 