Heyting algebra

Heyting algebra

A

Heyting lattice

L

is a

Brouwerian lattice

with a bottom

element

0

. Equivalently,

L

is Heyting iff it is relatively pseudocomplemented and pseudocomplemented iff it is

bounded

and relatively pseudocomplemented.

Let

a

*

denote the

pseudocomplement

of

a

and

a

→

b

the pseudocomplement of

a

relative to

b

. Then we have the following properties:

1.

a

*

=

a

→

0

(equivalence of definitions)

2.

1

*

=

0

(if

c

=

1

→

0

, then

c

=

c

∧

1

≤

0

by the definition of

→

.)

3.

a

*

=

1

iff

a

=

0

(

1

=

a

→

0

implies that

c

∧

a

≤

0

whenever

c

≤

1

. In particular

a

≤

1

, so

a

=

a

∧

a

≤

0

or

a

=

0

. On the other hand, if

a

=

0

, then

a

*

=

0

*

=

0

→

0

=

1

.)

4.

a

≤

a

*

*

and

a

*

=

a

*

⁣

*

*

(already true in any pseudocomplemented lattice)

5.

a

*

≤

a

→

b

(since

a

*

∧

a

=

0

≤

b

)

6.

(

a

→

b

)

∧

(

a

→

b

*

)

=

a

*

Proof.

If

c

∧

a

=

0

, then

c

∧

a

≤

b

so

c

≤

(

a

→

b

)

, and

c

≤

(

a

→

b

*

)

likewise, so

c

≤

(

a

→

b

)

∧

(

a

→

b

*

)

. This means precisely that

a

*

=

(

a

→

b

)

∧

(

a

→

b

*

)

.
∎

7.

a

→

b

≤

b

*

→

a

*

(since

(

a

→

b

)

∧

b

*

≤

(

a

→

b

)

∧

(

a

→

b

*

)

=

a

*

)

8.

a

*

∨

b

≤

a

→

b

(since

b

∧

a

≤

b

and

a

*

∧

a

=

0

≤

b

)

Note that in property 4,

a

≤

a

*

*

, whereas

a

*

*

≤

a

is in general not true, contrasting with the

equality

a

=

a

′′

in a

Boolean lattice

, where

′

is the

complement

operator

. It is easy to see that if

a

*

*

≤

a

for all

a

in a Heyting lattice

L

, then

L

is a Boolean lattice. In this case, the pseudocomplement coincides with the complement of an element

a

*

=

a

′

, and we have the equality in property 7:

a

*

∨

b

=

a

→

b

, meaning that the

concept

of relative pseudocomplementation (

http://planetmath.org/RelativelyPseudocomplemented

) coincides with the

material implication

in classical

propositional logic

.

A

Heyting algebra

is a Heyting lattice

H

such that

→

is a

binary operator

on

H

. A Heyting algebra

homomorphism

between two Heyting algebras is a

lattice homomorphism

that preserves

0

,

1

, and

→

. In

addition

, if

f

is a Heyting algebra homomorphism,

f

preserves psudocomplementation:

f

(

a

*

)

=

f

(

a

→

0

)

=

f

(

a

)

→

f

(

0

)

=

f

(

a

)

→

0

=

f

(

a

)

*

.

Remarks

.

•

In the literature, the

assumption

that a Heyting algebra contains

0

is sometimes dropped. Here, we call it a Brouwerian lattice instead.

•

Heyting algebras are useful in modeling

intuitionistic logic

. Every intuitionistic propositional logic can be modelled by a Heyting algebra, and every intuitionistic predicate logic can be modelled by a

complete Heyting algebra

.

Title

Heyting algebra

Canonical name

HeytingAlgebra

Date of creation

2013-03-22 16:33:03

Last modified on

2013-03-22 16:33:03

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

20

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06D20

Classification

msc 03G10

Synonym

pseudo-Boolean algebra

Related topic

QuantumTopos

Related topic

Lattice

Defines

Heyting lattice

Generated on Fri Feb 9 10:01:01 2018 by

LaTeXML