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[parent] commutativity relation in an orthocomplemented lattice (Definition)

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\le b$ or $ a\le 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. $ 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. $ \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).

Bibliography

1
L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).



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Also defines:  dually commute, orthogonally commute

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Cross-references: binary operations, orthomodular poset, Boolean algebra, equivalence relation, orthomodular lattice, symmetric, orthogonal, iff, Reflexive, relations, commutativity, properties, orthocomplemented lattice
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This is version 3 of commutativity relation in an orthocomplemented lattice, born on 2007-02-21, modified 2007-03-10.
Object id is 8943, canonical name is CommutativityRelationInAnOrthocomplementedLattice.
Accessed 1216 times total.

Classification:
AMS MSC: 06C15 (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Complemented lattices, orthocomplemented lattices and posets)
 03G12 (Mathematical logic and foundations :: Algebraic logic :: Quantum logic)
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