PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
orthomodular lattice (Definition)

Orthogonality Relations

Let $ L$ be an orthocomplemented lattice and $ a,b\in L$. $ a$ is said to be orthogonal to $ b$ if $ a\le b^{\perp}$, denoted by $ a\perp b$. If $ a\le b^{\perp}$, then $ b=b^{\perp\perp}\le a^{\perp}$, so $ \perp$ is a symmetric relation on $ L$. It is easy to see that, for any $ a,b\in L$, $ a\perp b$ implies $ a\wedge b=0$, and $ a\perp a^{\perp}$.

For any $ a\in L$, define $ M(a):=\lbrace c\in L\mid c\perp a$ and $ 1=c \vee a\rbrace$. An element of $ M(a)$ is called an orthogonal complement of $ a$. We have $ a^{\perp}\in M(a)$, and any orthogonal complement of $ a$ is a complement of $ a$.

If we replace the $ 1$ in $ M(a)$ by an arbitrary element $ b\ge a$, then we have the set

$\displaystyle M(a,b):=\lbrace c\in L\mid c\perp a$ and $\displaystyle b=c \vee a\rbrace.$
An element of $ M(a,b)$ is called an orthogonal complement of $ a$ relative to $ b$. Clearly, $ M(a)=M(a,1)$. Also, for $ a,c\le b$, $ c\in M(a,b)$ iff $ a\in M(c,b)$. As a result, we can define a symmetric binary operator $ \oplus$ on $ [0,b]$, given by $ b=a\oplus c$ iff $ c\in M(a,b)$. Note that $ b=b\oplus 0$.

Before the main definition, we define one more operation: $ b-a:=b\wedge a^{\perp}$. Some properties: (1) $ a-a=0$, $ a-0=a$, $ 0-a=0$, $ a-1=0$, and $ 1-a=a^{\perp}$; (2) $ b-a=a^{\perp}-b^{\perp}$; and (3) if $ a\le b$, then $ a\perp (b-a)$ and $ a\oplus (b-a)\le b$.

Definition

A lattice $ L$ is called an orthomodular lattice if
  1. $ L$ is orthocomplemented, and
  2. (orthomodular law) if $ x\le y$, then $ y=x\oplus (y-x)$.

The orthomodular law can be restated as follows: if $ x\le y$, then $ y=x\vee (y\wedge x^{\perp})$. Equivalently, $ x\le y$ implies $ y=(y\wedge x)\vee (y\wedge x^{\perp})$. Note that the equation is automatically true in an arbitrary distributive lattice, even without the assumption that $ x\le y$.

For example, the lattice $ \mathbb{C}(H)$ of closed subspaces of a hilbert space $ H$ is orthomodular. $ \mathbb{C}(H)$ is modular iff $ H$ is finite dimensional. In addition, if we give the set $ \mathbb{P}(H)$ of (bounded) projection operators on $ H$ an ordering structure by defining $ P\le Q$ iff $ P(H)\le Q(H)$, then $ \mathbb{P}(H)$ is lattice isomorphic to $ \mathbb{C}(H)$, and hence orthomodular.

A simple example of an orthocomplemented lattice that is not orthomodular is the benzene:

$\displaystyle \xymatrix { & 1 \ar@{-}[ld] \ar@{-}[rd] & \\ b \ar@{-}[d] & & a^{\perp} \ar@{-}[d] \\ a \ar@{-}[rd] & & b^{\perp} \ar@{-}[ld] \\ & 0 & }$    

Note that $ a\le b$, but $ a\vee (b\wedge a^{\perp})=a\vee 0=a\ne b$.

An nice example of an orthomodular lattice that is not modular can be found in the reference below.

Remarks.

  • Orthomodular lattices were first studied by John von Neumann and Garett Birkhoff, when they were trying to develop the logic of quantum mechanics by studying the structure of the lattice $ \mathbb{P}(H)$ of projection operators on a Hilbert space $ H$. However, the term was coined by Irving Kaplansky, when it was realized that $ \mathbb{P}(H)$, while orthocomplemented, is not modular. Rather, it satisfies a variant of the modular law as indicated above.
  • More generally, an orthomodular poset $ P$ is an orthocomplemented poset such that
    1. given any pair of orthogonal elements $ x,y\in P$ ( $ x\le y^{\perp}$), their greatest lower bound exists ($ x\vee y$ exists). Simply put, $ x\perp y$ implies $ x\vee y\in P$.
    2. for any $ x,y\in P$ such that $ x\le y$, the orthomodular law holds (the right hand side of the orthomodular law exists via the first condition).
    From this definition, we see that an orthomodular lattice is just an orthomodular poset that is also a lattice.

Bibliography

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



"orthomodular lattice" is owned by CWoo.
(view preamble | get metadata)

View style:

See Also: orthocomplemented lattice, lattice of projections

Also defines:  orthomodular poset, orthogonal, orthogonal complement, relative orthogonal complement
Log in to rate this entry.
(view current ratings)

Cross-references: right hand side, greatest lower bound, orthocomplemented poset, modular law, satisfies, term, John von Neumann, reference, benzene, simple, isomorphic, structure, ordering, projection, bounded, finite dimensional, modular, Hilbert space, subspaces, closed, even, distributive lattice, equation, orthocomplemented, lattice, properties, operation, operator, binary, symmetric, iff, complement, implies, easy to see, symmetric relation, orthocomplemented lattice
There are 14 references to this entry.

This is version 7 of orthomodular lattice, born on 2007-01-10, modified 2008-03-08.
Object id is 8735, canonical name is OrthomodularLattice.
Accessed 3220 times total.

Classification:
AMS MSC: 03G12 (Mathematical logic and foundations :: Algebraic logic :: Quantum logic)
 06C15 (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Complemented lattices, orthocomplemented lattices and posets)
 81P10 (Quantum theory :: Axiomatics, foundations, philosophy :: Logical foundations of quantum mechanics; quantum logic)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)