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 $a=(a\wedge b)\vee(a\wedge b^{\perp})$ . When $a$ commutes with $b$ , we write $a\operatorname{C}b$ . Dualize everything, we have that $a$ dually commutes with $b$ , written $a\operatorname{D}b$ , if $a=(a\vee b)\wedge(a\vee b^{\perp})$ .

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

1.

$\operatorname{C}$ is reflexive.
2.

$a\operatorname{C}b$ iff $a\operatorname{C}b^{\perp}$ .
3.

$a\operatorname{C}b$ iff $a^{\perp}\operatorname{D}b^{\perp}$ .
4.

if $a\leq b$ or $a\leq b^{\perp}$ , then $a\operatorname{C}b$ .
5.
$a$ is said to orthogonally commute with $b$ if $a\operatorname{C}b$ and $b\operatorname{C}a$ . In this case, we write $a\operatorname{M}b$ . The terminology comes from the following fact: $a\operatorname{M}b$ iff there are $x,y,z,t\in L$ , with:
1. (a)
  
  $x\perp y$ ( $x$ is orthogonal to $y$ , or $x\leq y^{\perp}$ ),
2. (b)
  
  $z\perp t$ ,
3. (c)
  
  $x\perp z$ ,
4. (d)
  
  $a=x\vee y$ , and
5. (e)
  
  $b=z\vee t$ .
6.

$\operatorname{C}$ is symmetric iff $\operatorname{D}=\operatorname{C}(=\operatorname{M})$ iff $L$ is an orthomodular lattice.
7.

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

Remark. More generally, one can define commutativity $\operatorname{C}$ on an orthomodular poset $P$ : for $a,b\in P$ , $a\operatorname{C}b$ 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).

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	2013-03-22 16:43:22
Last modified on	2013-03-22 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