orthocomplemented lattice

orthocomplemented lattice

An

orthocomplemented lattice

is a

complemented lattice

in which every

element

has a distinguished

complement

, called an

orthocomplement

, that behaves like
the

complementary subspace

of a

subspace

in a

vector space

.

Formally, let

L

be a complemented lattice and denote

M

the set of complements of elements of

L

.

M

is clearly a subposet of

L

, with

≤

inherited from

L

. For each

a

∈

L

, let

M

a

⊆

M

be the set of complements of

a

.

L

is said to be

orthocomplemented

if there is a

function

:

⟂

L

→

M

, called an

orthocomplementation

, whose image is written

a

⟂

for any

a

∈

L

, such that

1.

a

⟂

∈

M

a

,

2.

(

a

⟂

)

⟂

=

a

, and

3.

⟂

is

order-reversing

; that is, for any

a

,

b

∈

L

,

a

≤

b

implies

b

⟂

≤

a

⟂

.

The element

a

⟂

is called an

orthocomplement

of

a

(via

⟂

).

Examples

. In

addition

to the example of the

lattice

of vector subspaces of a vector space cited above, let’s look at the

Hasse diagrams

of the two finite

complemented

lattices below,

\xymatrix

⁢

&

⁢

1

⁢

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@

-

[

l

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\ar

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-

[

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]

⁢

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-

[

r

⁢

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]

⁢

&

⁢

a

⁢

\ar

⁢

@

-

[

r

⁢

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]

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

0

⁢

&

\xymatrix

⁢

&

⁢

&

⁢

1

⁢

\ar

⁢

@

-

[

l

⁢

l

⁢

d

]

⁢

\ar

⁢

@

-

[

l

⁢

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]

⁢

\ar

⁢

@

-

[

r

⁢

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]

⁢

\ar

⁢

@

-

[

r

⁢

r

⁢

d

]

⁢

&

⁢

&

⁢

a

⁢

\ar

⁢

@

-

[

r

⁢

r

⁢

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]

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

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⁢

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]

⁢

&

⁢

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⁢

c

⁢

\ar

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[

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⁢

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⁢

d

⁢

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⁢

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0

⁢

&

⁢

&

the one on the right is orthocomplemented, while the one on the left is not. From this one deduces that orthcomplementation is not unique, and that the

cardinality

of any finite orthocomplemented lattice is even.

Remarks

.

•

From the first condition above, we see that an orthocomplementation

⟂

is a

bijection

. It is

one-to-one

: if

a

⟂

=

b

⟂

, then

a

=

(

a

⟂

)

⟂

=

(

b

⟂

)

⟂

=

b

. And it is onto: if we pick

a

∈

M

⊆

L

, then

(

a

⟂

)

⟂

=

a

. As a result,

M

=

L

, every element of

L

is an orthocomplement. Furthermore, we have

0

⟂

=

1

and

1

⟂

=

0

.

•

Let

L

′

be the dual lattice of

L

(a lattice having the same underlying set, but with meet and join

operations

switched). Then any orthocomplementation

⟂

can be viewed as a

lattice isomorphism

between

L

and

L

′

.

•

From the above conditions, it follows that elements of

L

satisfy

the de Morgan’s laws: for

a

,

b

∈

L

, we have

a

⟂

∧

b

⟂

=

(

a

∨

b

)

⟂

,

(1)

a

⟂

∨

b

⟂

=

(

a

∧

b

)

⟂

.

(2)

To derive the first equation, first note

a

≤

a

∨

b

. Then

(

a

∨

b

)

⟂

≤

a

⟂

.
Similarly,

(

a

∨

b

)

⟂

≤

b

⟂

. So

(

a

∨

b

)

⟂

≤

a

⟂

∧

b

⟂

. For the other inequality, we start with

a

⟂

∧

b

⟂

≤

a

⟂

. Then

a

≤

(

a

⟂

∧

b

⟂

)

⟂

. Similarly,

b

≤

(

a

⟂

∧

b

⟂

)

⟂

.
Therefore,

a

∨

b

≤

(

a

⟂

∧

b

⟂

)

⟂

, which implies that

a

⟂

∧

b

⟂

≤

(

a

∨

b

)

⟂

.

•

Conversely

, any of two equations in the previous remark can replace the third condition in the definition above. For example, suppose we have the second equation

a

⟂

∨

b

⟂

=

(

a

∧

b

)

⟂

. If

a

≤

b

, then

a

=

a

∧

b

, so

a

⟂

=

(

a

∧

b

)

⟂

=

a

⟂

∨

b

⟂

, which shows that

b

⟂

≤

a

⟂

.

•

From the example above, one sees that orthocomplementation need not be unique. An orthocomplemented lattice with a unique orthocomplementation is said to be

uniquely orthocomplemented

. A

uniquely complemented lattice

that is also orthocomplemented is uniquely orthocomplemented.

•

Orthocomplementation can be more generally defined over a

bounded poset

P

by requiring the orthocomplentation

operator

⟂

to satisfy conditions 2 and 3 above, and a weaker version of condition 1:

a

∧

a

⟂

exists and

=

0

. Since

⟂

is an order reversing bijection on

P

,

0

⟂

=

1

and

1

⟂

=

0

. From this, one deduces that

a

∧

a

⟂

=

0

iff

a

∨

a

⟂

=

1

. A bounded poset in which an orthocomplementation is defined is called an

orthocomplemented poset

.

•

In the

category

of orthocomplemented lattices, the

morphism

between a pair of objects is a

{

0

,

1

}

-

lattice homomorphism

(

http://planetmath.org/LatticeHomomorphism

)

f

that preserves orthocomplementation:

f

⁢

(

a

⟂

)

=

f

⁢

(

a

)

⟂

.

References

1

G. Birkhoff,

Lattice Theory

, AMS Colloquium Publications, Vol. XXV, 3rd Ed. (1967).

Title

orthocomplemented lattice

Canonical name

OrthocomplementedLattice

Date of creation

2013-03-22 15:50:36

Last modified on

2013-03-22 15:50:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

20

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G12

Classification

msc 06C15

Synonym

ortholattice

Synonym

uniquely orthocomplemented

Related topic

ComplementedLattice

Related topic

OrthomodularLattice

Defines

orthocomplement

Defines

orthocomplemented

Defines

orthocomplementation

Defines

orthocomplemented poset

Defines

uniquely orthocomplemented lattice

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