Heyting algebra | planetmath.org

Jump to Navigation
planetmath.org
Math for the people, by the people.

## User login

Username *
Password *

- Create new account
- Request new password

## You are here

Home ›

#

Heyting algebra

## Primary tabs

- View(active tab)
- Coauthors
- PDF
- Source

- Edit

# Heyting algebra

A Heyting lattice L L L is a Brouwerian lattice with a bottom element 0 0 0. Equivalently, L L L is Heyting iff it is relatively pseudocomplemented and pseudocomplemented iff it is bounded and relatively pseudocomplemented.

Let a*superscript a a^{*} denote the pseudocomplement of a a a and a→b normal-→a b a\to b the pseudocomplement of a a a relative to b b b. Then we have the following properties:

1. 1.

   a*=a→0 superscript a a normal-→0 a^{*}=a\to 0 (equivalence of definitions)

2. 2.

   1*=0 superscript 1 0 1^{*}=0 (if c=1→0 c 1 normal-→0 c=1\to 0, then c=c∧1≤0 c c 1 0 c=c\wedge 1\leq 0 by the definition of →normal-→\to.)

3. 3.

   a*=1 superscript a 1 a^{*}=1 iff a=0 a 0 a=0 (1=a→0 1 a normal-→0 1=a\to 0 implies that c∧a≤0 c a 0 c\wedge a\leq 0 whenever c≤1 c 1 c\leq 1. In particular a≤1 a 1 a\leq 1, so a=a∧a≤0 a a a 0 a=a\wedge a\leq 0 or a=0 a 0 a=0. On the other hand, if a=0 a 0 a=0, then a*=0*=0→0=1 superscript a superscript 0 0 normal-→0 1 a^{*}=0^{*}=0\to 0=1.)

4. 4.

   a≤a*⁣*a superscript a a\leq a^{{**}} and a*=a*⁣*⁣*superscript a superscript a a^{*}=a^{{***}} (already true in any pseudocomplemented lattice)

5. 5.

   a*≤a→b superscript a a normal-→b a^{*}\leq a\to b (since a*∧a=0≤b superscript a a 0 b a^{*}\wedge a=0\leq b)

6. 6.

   (a→b)∧(a→b*)=a*fragments fragments normal-(a normal-→b normal-)fragments normal-(a normal-→superscript b normal-)superscript a(a\to b)\wedge(a\to b^{*})=a^{*}

   ###### Proof.

   If c∧a=0 c a 0 c\wedge a=0, then c∧a≤b c a b c\wedge a\leq b so c≤(a→b)fragments c fragments normal-(a normal-→b normal-)c\leq(a\to b), and c≤(a→b*)fragments c fragments normal-(a normal-→superscript b normal-)c\leq(a\to b^{*}) likewise, so c≤(a→b)∧(a→b*)fragments c fragments normal-(a normal-→b normal-)fragments normal-(a normal-→superscript b normal-)c\leq(a\to b)\wedge(a\to b^{*}). This means precisely that a*=(a→b)∧(a→b*)fragments superscript a fragments normal-(a normal-→b normal-)fragments normal-(a normal-→superscript b normal-)a^{*}=(a\to b)\wedge(a\to b^{*}). ∎

7. 7.

   a→b≤b*→a*normal-→a b superscript b normal-→superscript a a\to b\leq b^{*}\to a^{*} (since (a→b)∧b*≤(a→b)∧(a→b*)=a*)fragments fragments normal-(a normal-→b normal-)superscript b fragments normal-(a normal-→b normal-)fragments normal-(a normal-→superscript b normal-)superscript a normal-)(a\to b)\wedge b^{*}\leq(a\to b)\wedge(a\to b^{*})=a^{*})

8. 8.

   a*∨b≤a→b superscript a b a normal-→b a^{*}\vee b\leq a\to b (since b∧a≤b b a b b\wedge a\leq b and a*∧a=0≤b superscript a a 0 b a^{*}\wedge a=0\leq b)

Note that in property 4, a≤a*⁣*a superscript a a\leq a^{{**}}, whereas a*⁣*≤a superscript a a a^{{**}}\leq a is in general not true, contrasting with the equality a=a′′a superscript a′′a=a^{{\prime\prime}} in a Boolean lattice, where ′normal-′{}^{{\prime}} is the complement operator. It is easy to see that if a*⁣*≤a superscript a a a^{{**}}\leq a for all a a a in a Heyting lattice L L L, then L L L is a Boolean lattice. In this case, the pseudocomplement coincides with the complement of an element a*=a′superscript a superscript a normal-′a^{*}=a^{{\prime}}, and we have the equality in property 7: a*∨b=a→b superscript a b a normal-→b a^{*}\vee b=a\to b, meaning that the concept of relative pseudocomplementation coincides with the material implication in classical propositional logic.

A Heyting algebra is a Heyting lattice H H H such that →normal-→\to is a binary operator on H H H. A Heyting algebra homomorphism between two Heyting algebras is a lattice homomorphism that preserves 0, 1 0 1 0,1, and →normal-→\to. In addition, if f f f is a Heyting algebra homomorphism, f f f preserves psudocomplementation: f(a*)=f(a→0)=f(a)→f(0)=f(a)→0=f(a)*fragments f fragments normal-(superscript a normal-)f fragments normal-(a normal-→0 normal-)f fragments normal-(a normal-)normal-→f fragments normal-(0 normal-)f fragments normal-(a normal-)normal-→0 f superscript fragments normal-(a normal-)f(a^{*})=f(a\to 0)=f(a)\to f(0)=f(a)\to 0=f(a)^{*}.

Remarks.

- In the literature, the assumption that a Heyting algebra contains 0 0 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.

Defines:
Heyting lattice
Keywords:
Related:
QuantumTopos, Lattice
Synonym:
pseudo-Boolean algebra
Major Section:
Reference
Type of Math Object:
Definition
Groups audience:
Buddy list for CWoo

## Mathematics Subject Classification

06D20 Heyting algebras
03G10 Lattices and related structures
Add a correction
Attach a problem
Ask a question

## Search form

Search

- Home
- Articles
- Questions
- Problems
- Overview

- Forums
  - Planetary Bugs
  - HS/Secondary
  - University/Tertiary
  - Graduate/Advanced
  - Industry/Practice
  - Research Topics
  - LaTeX help
  - Math Comptetitions
  - Math History
  - Math Humor
  - PlanetMath Comments
  - PlanetMath System Updates and News
  - PlanetMath help
  - PlanetMath.ORG
  - Strategic Communications Development
  - The Math Pub
  - Testing messages (ignore)

## Recent Activity

2013-03-17
new question: Inequality for square of the subgaussian distributions by Fred78
2013-03-16
new question: The Riemann Tensor by Bruce Lee
new question: Cantor' diagonal argument by haideuta
2013-03-15
new question: Finite element method for 3rd order ODE by sakhan
2013-03-14
new question: Deleting my comments by haideuta
2013-03-10
new image: coslaw by drini
new image: SinesLaw by drini
new image: pyth by drini
more

## Info

Submitted by: CWoo
Tue, 2007-01-09 - 20:20
Author(s): CWoo

## Corrections

incorrect properties by yark ✓
Property 2 still incorrect by velochy ✓
aka Pseudo-Boolean algebras by porton ✓

## Versions

(v18) by CWoo 2013-03-10
Powered By: = + + ♡