Brouwerian lattice

Brouwerian lattice

Let

L

be a

lattice

, and

a

,

b

∈

L

. Then

a

is said to be

pseudocomplemented relative to

b

if the set

T

⁢

(

a

,

b

)

:=

{

c

∈

L

∣

c

∧

a

≤

b

}

has a

maximal element

. The maximal element (necessarily unique) of

T

⁢

(

a

,

b

)

is called the

pseudocomplement

of

a

relative to

b

, and is denoted by

a

→

b

. So,

a

→

b

, if exists, has the following property

c

∧

a

≤

b

⁢

iff

⁢

c

≤

a

→

b

.

If

L

has

0

, then the pseudocomplement of

a

relative to

0

is the pseudocomplement of

a

.

An

element

a

∈

L

is said to be

relatively pseudocomplemented

if

a

→

b

exists for every

b

∈

L

. In particular

a

→

a

exists. Since

T

⁢

(

a

,

a

)

=

L

, so

L

has a maximal element, or

1

∈

L

.

A lattice

L

is said to be

relatively pseudocomplemented

, or

Brouwerian

, if every element in

L

is relatively pseudocomplemented. Evidently, as we have just shown, every Brouwerian lattice contains

1

. A Brouwerian lattice is also called an

implicative lattice

.

Here are some other properties of a Brouwerian lattice

L

:

1.

b

≤

a

→

b

(since

b

∧

a

≤

b

)

2.

1

=

a

→

1

(

consequence

of 1)

3.

(Birkhoff-Von Neumann condition)

a

≤

b

iff

a

→

b

=

1

(since

1

∧

a

=

a

≤

b

)

4.

a

∧

(

a

→

b

)

=

a

∧

b

Proof.

On the one hand, by 1,

b

≤

a

→

b

, so

a

∧

b

≤

a

∧

(

a

→

b

)

. On the other hand, by definition,

a

∧

(

a

→

b

)

≤

b

. Since

a

∧

(

a

→

b

)

≤

a

as well,

a

∧

(

a

→

b

)

≤

a

∧

b

, and the proof is

complete

.
∎

5.

a

=

1

→

a

(consequence of 4)

6.

if

a

≤

b

, then

(

c

→

a

)

≤

(

c

→

b

)

(use 4,

c

∧

(

c

→

a

)

=

c

∧

a

≤

a

≤

b

)

7.

if

a

≤

b

, then

(

b

→

c

)

≤

(

a

→

c

)

(use 4,

a

∧

(

b

→

c

)

≤

b

∧

(

b

→

c

)

=

b

∧

c

≤

c

)

8.

a

→

(

b

→

c

)

=

(

a

∧

b

)

→

c

=

(

a

→

b

)

→

(

a

→

c

)

Proof.

We shall use property 4 above a number of times, and the fact that

x

=

y

iff

x

≤

y

and

y

≤

x

.
First

equality

:

(

a

→

(

b

→

c

)

)

∧

(

a

∧

b

)

=

(

a

∧

(

b

→

c

)

)

∧

b

=

(

b

∧

(

b

→

c

)

)

∧

a

=

(

b

∧

c

)

∧

a

≤

c

.

So

a

→

(

b

→

c

)

≤

(

a

∧

b

)

→

c

.

On the other hand,

(

(

a

∧

b

)

→

c

)

∧

a

∧

b

=

a

∧

b

∧

c

≤

c

, so

(

(

a

∧

b

)

→

c

)

∧

a

≤

b

→

c

, and consequently

(

a

∧

b

)

→

c

≤

a

→

(

b

→

c

)

.

Second equality:

(

(

a

∧

b

)

→

c

)

∧

(

a

→

b

)

∧

a

=

(

(

a

∧

b

)

→

c

)

∧

(

a

∧

b

)

=

(

a

∧

b

)

∧

c

≤

c

, so

(

(

a

∧

b

)

→

c

)

∧

(

a

→

b

)

≤

a

→

c

and consequently

(

a

∧

b

)

→

c

≤

(

a

→

b

)

→

(

a

→

c

)

.

On the other hand,

(

(

a

→

b

)

→

(

a

→

c

)

)

∧

(

a

∧

b

)

=

(

(

a

→

b

)

→

(

a

→

c

)

)

∧

(

a

∧

(

a

→

b

)

)

=

(

(

a

→

b

)

∧

(

a

→

c

)

)

∧

a

=

(

a

∧

b

)

∧

(

a

→

c

)

=

b

∧

(

a

∧

c

)

≤

c

,

so

(

a

→

b

)

→

(

a

→

c

)

≤

(

a

∧

b

)

→

c

.
∎

9.

L

is a

distributive lattice

.

Proof.

By the

proposition

found in entry

distributive inequalities

, it is enough to show that

a

∧

(

b

∨

c

)

≤

(

a

∧

b

)

∨

(

a

∧

c

)

.

To see this: note that

a

∧

b

≤

(

a

∧

b

)

∨

(

a

∧

c

)

, so

b

≤

a

→

(

(

a

∧

b

)

∨

(

a

∧

c

)

)

. Similarly,

c

≤

a

→

(

(

a

∧

b

)

∨

(

a

∧

c

)

)

. So

b

∨

c

≤

a

→

(

(

a

∧

b

)

∨

(

a

∧

c

)

)

, or

a

∧

(

b

∨

c

)

≤

(

a

∧

b

)

∨

(

a

∧

c

)

.
∎

If a Brouwerian lattice were a chain, then relative pseudocomplentation can be given by the

formula

:

a

→

b

=

1

if

a

≤

b

, and

a

→

b

=

b

otherwise. From this, we see that the real

interval

(

∞

,

r

]

is a Brouwerian lattice if

x

→

y

is defined according to the formula just mentioned (with

∨

and

∧

defined in the obvious way). Incidentally, this lattice has no bottom, and is therefore not a

Heyting algebra

.

Remarks

.

•

Brouwerian lattice is named after the Dutch mathematician L. E. J. Brouwer, who rejected

classical logic

and

proof by contradiction

in particular. The lattice was invented as the algebraic counterpart to the Brouwerian intuitionistic (or constructionist) logic, in contrast to the

Boolean lattice

, invented as the algebraic counterpart to the classical

propositional logic

.

•

In the literature, a Brouwerian lattice is sometimes defined to be synonymous as a Heyting algebra (and sometimes even a

complete Heyting algebra

). Here, we shall distinguish the two related

concepts

, and say that a Heyting algebra is a Brouwerian lattice with a bottom.

•

In the

category

of Brouwerian lattices, a

morphism

between a pair of objects is a

lattice homomorphism

f

that preserves relative pseudocomplementation:

f

(

a

→

b

)

=

f

(

a

)

→

f

(

b

)

.

As

f

(

1

)

=

f

(

a

→

a

)

=

f

(

a

)

→

f

(

a

)

=

1

, this morphism preserves the top elements as well.

Example

. Let

L

⁢

(

X

)

be the lattice of

open sets

of a

topological space

. Then

L

⁢

(

X

)

is Brouwerian. For any open sets

A

,

B

∈

X

,

A

→

B

=

(

A

c

∪

B

)

∘

, the

interior

of the union of

B

and the

complement

of

A

.

References

1

G. Birkhoff,

Lattice Theory

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

2

R. Goldblatt,

Topoi, The Categorial

Analysis

of Logic

, Dover Publications (2006).

Title

Brouwerian lattice

Canonical name

BrouwerianLattice

Date of creation

2013-03-22 16:32:59

Last modified on

2013-03-22 16:32:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

21

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06D20

Classification

msc 06D15

Synonym

relatively pseudocomplemented

Synonym

pseudocomplemented relative to

Synonym

Brouwerian algebra

Synonym

implicative lattice

Related topic

PseudocomplementedLattice

Related topic

Pseudocomplement

Related topic

RelativeComplement

Defines

relative pseudocomplement

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