Page 18 - Textos de Matemática Vol. 41

4 Chapter 1. Preliminaries
Proof.(1)a∧a∗ =0,hencea≤(a∗)∗.
(2) is trivial.
(3) follows from (1) and (2): a∗ ≤ (a∗)∗∗ and a∗ ≥ (a∗∗)∗.
(4)x≤0∗ iffx∧0=0,thatis,always,andx≤1∗ iffx=x∧1=0.
(5) follows from 1.3(-distr).
1.4.2. Complements and Boolean algebras. An element b is a complement of a in a distributive lattice (and we shall denote it by ¬a) if
a∧b=0 and a∨b=1.
Note that
if a complement b of a exists, it is the pseudocomplement a∗ (indeed, if a ∧ x = 0 then x = (a ∨ b) ∧ x = b ∧ x and hence x ≤ b).
A Boolean algebra is a distributive lattice in which every element has a complement.
Proposition. Boolean algebras are precisely the Heyting algebras in which a∗∗ = (a→0)→0 = a for every a.
Proof. If a∗∗ = a for all a then, by (5) and (4) of Proposition 1.4.1, a∨a∗ =(a∨a∗)∗∗ =(a∗ ∧a∗∗)∗ =0∗ =1.
Now it suffices to prove that a Boolean algebra possesses a Heyting oper- ation. Set a→b = ¬a ∨ b.
1.5. Frames. Complete lattices L satisfying (f-distr) for any collection {ai | i ∈ J} ⊆ L are called frames. By 1.2.1 any frame has a Heyting operation, and since such an operation (if it exists) is obviously uniquely determined, frames are essentially the same as complete Heyting algebras. But the interest is here focused on the distributivity, not so much on the operation, and also the choice of the privileged mappings in the two contexts differ (see 2.1 below).
1.6.Completelyprimefiltersandmeet-irreducibility. AfilterFinadistributive latticeL(thatis,asubsetF ⊆Lsuchthata,b∈F impliesa∧b∈F and b≥a∈F impliesb∈F)willbealwaysassumedproper,thatis,0∈/F.Itis prime if a∨b ∈ F implies that either a ∈ F or b ∈ F, and completely prime if there is an aj ∈ F for some j ∈ J whenever (i∈J ai exists and) i∈J ai ∈ F.
An element a ∈ L, a ̸= 1, is meet-irreducible if x ∧ y ≤ a implies that either x ≤ a or y ≤ a (which condition is, of course, equivalent to requiring that x ∧ y = a implies that either x = a or y = a).