Brouwerian lattice

Let L be a latticeMathworldPlanetmath, and a,bL. Then a is said to be pseudocomplemented relative to b if the set


has a maximal elementMathworldPlanetmath. The maximal element (necessarily unique) of T(a,b) is called the pseudocomplement of a relative to b, and is denoted by ab. So, ab, if exists, has the following property

cab iff cab.

If L has 0, then the pseudocomplement of a relative to 0 is the pseudocomplement of a.

An element aL is said to be relatively pseudocomplemented if ab exists for every bL. In particular aa exists. Since T(a,a)=L, so L has a maximal element, or 1L.

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. 1.

    bab (since bab)

  2. 2.

    1=a1 (consequence of 1)

  3. 3.

    (Birkhoff-Von Neumann condition) ab iff ab=1 (since 1a=ab)

  4. 4.



    On the one hand, by 1, bab, so aba(ab). On the other hand, by definition, a(ab)b. Since a(ab)a as well, a(ab)ab, and the proof is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. ∎

  5. 5.

    a=1a (consequence of 4)

  6. 6.

    if ab, then (ca)(cb) (use 4, c(ca)=caab)

  7. 7.

    if ab, then (bc)(ac) (use 4, a(bc)b(bc)=bcc)

  8. 8.



    We shall use property 4 above a number of times, and the fact that x=y iff xy and yx. First equality:

    (a(bc))(ab) = (a(bc))b
    = (b(bc))a
    = (bc)ac.

    So a(bc)(ab)c.

    On the other hand, ((ab)c)ab=abcc, so ((ab)c)abc, and consequently (ab)ca(bc).

    Second equality: ((ab)c)(ab)a=((ab)c)(ab)=(ab)cc, so ((ab)c)(ab)ac and consequently (ab)c(ab)(ac).

    On the other hand,

    ((ab)(ac))(ab) = ((ab)(ac))(a(ab))
    = ((ab)(ac))a
    = (ab)(ac)
    = b(ac)c,

    so (ab)(ac)(ab)c. ∎

  9. 9.

    By the propositionPlanetmathPlanetmath found in entry distributive inequalities, it is enough to show that


    To see this: note that ab(ab)(ac), so ba((ab)(ac)). Similarly, ca((ab)(ac)). So bca((ab)(ac)), or a(bc)(ab)(ac). ∎

If a Brouwerian lattice were a chain, then relative pseudocomplentation can be given by the formulaMathworldPlanetmathPlanetmath: ab=1 if ab, and ab=b otherwise. From this, we see that the real interval (,r] is a Brouwerian lattice if xy 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.


  • Brouwerian lattice is named after the Dutch mathematician L. E. J. Brouwer, who rejected classical logic and proof by contradictionMathworldPlanetmathPlanetmath 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 categoryMathworldPlanetmath of Brouwerian lattices, a morphismMathworldPlanetmath between a pair of objects is a lattice homomorphismMathworldPlanetmath f that preserves relative pseudocomplementation:


    As f(1)=f(aa)=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 spaceMathworldPlanetmath. Then L(X) is Brouwerian. For any open sets A,BX, AB=(AcB), the interior of the union of B and the complementPlanetmathPlanetmath of A.


  • 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