# 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. 1.

$\operatorname{C}$ is reflexive.

2. 2.

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

3. 3.

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

4. 4.

if $a\leq b$ or $a\leq b^{\perp}$, then $a\operatorname{C}b$.

5. 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. 6.

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

7. 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 CommutativityRelationInAnOrthocomplementedLattice 2013-03-22 16:43:22 2013-03-22 16:43:22 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 06C15 msc 03G12 dually commute orthogonally commute