Brouwerian lattice (original) (raw)

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

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

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. 1. b≤a→b (since b∧a≤b)
3. 1. (Birkhoff-Von Neumann condition) a≤b iff a→b=1 (since 1∧a=a≤b)
4. 1. a∧(a→b)=a∧b
5. 1. a=1→a (consequence of 4)
6. 1. if a≤b, then (c→a)≤(c→b) (use 4, c∧(c→a)=c∧a≤a≤b)
7. 1. if a≤b, then (b→c)≤(a→c) (use 4, a∧(b→c)≤b∧(b→c)=b∧c≤c)
8. 1. 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:

Proof.

By the proposition found in entry distributive inequalities, it is enough to show that
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 ifx→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.

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=(Ac∪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).