Brouwerian lattice
Let be a lattice, and . Then is said to be pseudocomplemented relative to if the set
has a maximal element. The maximal element (necessarily unique) of is called the pseudocomplement of relative to , and is denoted by . So, , if exists, has the following property
If has , then the pseudocomplement of relative to is the pseudocomplement of .
An element is said to be relatively pseudocomplemented if exists for every . In particular exists. Since , so has a maximal element, or .
A lattice is said to be relatively pseudocomplemented, or Brouwerian, if every element in is relatively pseudocomplemented. Evidently, as we have just shown, every Brouwerian lattice contains . A Brouwerian lattice is also called an implicative lattice.
Here are some other properties of a Brouwerian lattice :
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1.
(since )
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2.
(consequence of 1)
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3.
(Birkhoff-Von Neumann condition) iff (since )
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4.
Proof.
On the one hand, by 1, , so . On the other hand, by definition, . Since as well, , and the proof is complete. ∎
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5.
(consequence of 4)
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6.
if , then (use 4, )
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7.
if , then (use 4, )
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8.
Proof.
On the other hand, , so , and consequently .
Second equality: , so and consequently .
On the other hand,
so . ∎
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9.
is a distributive lattice.
Proof.
By the proposition found in entry distributive inequalities, it is enough to show that
To see this: note that , so . Similarly, . So , or . ∎
If a Brouwerian lattice were a chain, then relative pseudocomplentation can be given by the formula: if , and otherwise. From this, we see that the real interval is a Brouwerian lattice if 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.
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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.
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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.
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In the category of Brouwerian lattices, a morphism between a pair of objects is a lattice homomorphism that preserves relative pseudocomplementation:
As , this morphism preserves the top elements as well.
Example. Let be the lattice of open sets of a topological space. Then is Brouwerian. For any open sets , , the interior of the union of and the complement of .
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 |