orthomodular lattice

orthomodular lattice

Orthogonality Relations

Let

L

be an

orthocomplemented lattice

and

a

,

b

∈

L

.

a

is said to be

orthogonal

to

b

if

a

≤

b

⟂

, denoted by

a

⟂

b

. If

a

≤

b

⟂

, then

b

=

b

⟂

⟂

≤

a

⟂

, so

⟂

is a

symmetric relation

on

L

. It is easy to see that, for any

a

,

b

∈

L

,

a

⟂

b

implies

a

∧

b

=

0

, and

a

⟂

a

⟂

.

For any

a

∈

L

, define

M

⁢

(

a

)

:=

{

c

∈

L

∣

c

⟂

a

⁢

and

⁢

1

=

c

∨

a

}

. An element of

M

⁢

(

a

)

is called an

orthogonal complement

of

a

. We have

a

⟂

∈

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

≥

a

, then we have the set

M

⁢

(

a

,

b

)

:=

{

c

∈

L

∣

c

⟂

a

⁢

and

⁢

b

=

c

∨

a

}

.

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

≤

b

,

c

∈

M

⁢

(

a

,

b

)

iff

a

∈

M

⁢

(

c

,

b

)

. As a result, we can define a

symmetric

binary operator

⊕

on

[

0

,

b

]

, given by

b

=

a

⊕

c

iff

c

∈

M

⁢

(

a

,

b

)

. Note that

b

=

b

⊕

0

.

Before the main definition, we define one more

operation

:

b

-

a

:=

b

∧

a

⟂

. Some properties: (1)

a

-

a

=

0

,

a

-

0

=

a

,

0

-

a

=

0

,

a

-

1

=

0

, and

1

-

a

=

a

⟂

; (2)

b

-

a

=

a

⟂

-

b

⟂

; and (3) if

a

≤

b

, then

a

⟂

(

b

-

a

)

and

a

⊕

(

b

-

a

)

≤

b

.

Definition

A

lattice

L

is called an

orthomodular lattice

if

1.

L

is orthocomplemented, and

2.

(

orthomodular law

) if

x

≤

y

, then

y

=

x

⊕

(

y

-

x

)

.

The orthomodular law can be restated as follows: if

x

≤

y

, then

y

=

x

∨

(

y

∧

x

⟂

)

. Equivalently,

x

≤

y

implies

y

=

(

y

∧

x

)

∨

(

y

∧

x

⟂

)

. Note that the equation is automatically true in an arbitrary

distributive lattice

, even without the

assumption

that

x

≤

y

.

For example, the lattice

ℂ

⁢

(

H

)

of closed subspaces of a hilbert space

H

is orthomodular.

ℂ

⁢

(

H

)

is

modular

iff

H

is finite dimensional. In

addition

, if we give the set

ℙ

⁢

(

H

)

of (

bounded

) projection operators on

H

an

ordering

structure

by defining

P

≤

Q

iff

P

⁢

(

H

)

≤

Q

⁢

(

H

)

, then

ℙ

⁢

(

H

)

is lattice

isomorphic

to

ℂ

⁢

(

H

)

, and hence orthomodular.

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

benzene

:

\xymatrix

⁢

&

⁢

1

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

⁢

a

⟂

⁢

\ar

⁢

@

-

[

d

]

⁢

a

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

b

⟂

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

0

⁢

&

Note that

a

≤

b

, but

a

∨

(

b

∧

a

⟂

)

=

a

∨

0

=

a

≠

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 (

http://planetmath.org/QuantumLogic

) by studying the structure of the lattice

ℙ

⁢

(

H

)

of projection operators on a Hilbert space

H

. However, the term was coined by Irving Kaplansky, when it was realized that

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

(a)

given any pair of orthogonal elements

x

,

y

∈

P

(

x

≤

y

⟂

), their

greatest lower bound

exists (

x

∨

y

exists). Simply put,

x

⟂

y

implies

x

∨

y

∈

P

.

(b)

for any

x

,

y

∈

P

such that

x

≤

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.

References

1

L. Beran,

Orthomodular Lattices, Algebraic Approach

, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).

Title

orthomodular lattice

Canonical name

OrthomodularLattice

Date of creation

2013-03-22 16:33:06

Last modified on

2013-03-22 16:33:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06C15

Classification

msc 81P10

Classification

msc 03G12

Related topic

OrthocomplementedLattice

Related topic

LatticeOfProjections

Defines

orthomodular poset

Defines

orthogonal

Defines

orthogonal complement

Defines

relative orthogonal complement

Generated on Fri Feb 9 09:59:51 2018 by

LaTeXML