First of all, we have the following observation: Proposition 6.1. For any monotone function fA:
A→B and its extension fA
δ
1. (fA)σ ≤fAδ iff the following closed Esakia condition holds: fAδ(V{
ci :i∈I}) =V{fA δ
(ci) :
i∈I} for any downward-directed collection {ci :i∈I} ⊆K(Aδ);
2. fAδ ≤(fA)π iff the following open Esakia condition holds: fAδ(W
{oi :i ∈I}) =
W
{fAδ(o
i) :
i∈I} for any upward-directed collection {oi :i∈I} ⊆O(Aδ).
Proof. We prove 1, statement 2 being an order-variant of 1.
⇐: Assume the closed Esakia condition holds for fAδ. To show that (fA)σ ≤ fAδ, we have to show that for all u ∈ Aδ, W
{V
{fA(a) : x ≤ a ∈
A} : u ≥ x ∈ K(Aδ)} ≤ fA
δ
(u), i.e., for all
u ∈ Aδ, all x ∈ K(
Aδ) such that x ≤ u, V{fA(a) : x ≤ a ∈ A} ≤ fAδ(u). By the mono-
tonicity of fAδ and that fA and fAδ coincide on
A, it suffices to show that for all x ∈ K(Aδ),
V
{fAδ(a) :x≤a∈
A} ≤fA
δ
(x), which follows easily from the closed Esakia condition.
⇒: Suppose we have (fA)σ ≤fAδ, i.e., for allu∈
Aδ,W{V{fA(a) :x≤a∈A}:u≥x∈K(Aδ)} ≤
fAδ(u). Then let us preliminarily show that V
{fAδ(a) :x≤a∈
A}=fA
δ
(x) for all x∈K(Aδ): For any x ∈ K(Aδ), we have V
{fA(a) : x ≤ a ∈
A} = W{V{fA(a) : x0 ≤ a ∈ A} : x ≥ x0 ∈
K(Aδ)} ≤fAδ(x). Since fA and fAδ coincide on
A, we have V{fA
δ
(a) :x≤a∈A} ≤fAδ(x). The converse inequality follows from the monotonicity of fAδ.
Let us prove thatfAδ satisfies the closed Esakia condition, i.e., for any downward-directed collection
{ci :i∈I} ⊆K(Aδ), fA δ (V{ ci :i∈I}) = V{fA δ (ci) :i∈I}: The inequality fAδ(V {ci : i ∈ I}) ≤ V{fA δ
(ci) : i ∈ I} straightforwardly follows from the mono-
tonicity of fAδ. For the converse direction V{ fAδ(c
i) : i ∈I} ≤fA δ
(V{
ci : i∈ I}), let x =V{ci :
i ∈ I} ∈ K(Aδ). By the preliminary fact shown above, we have that fAδ(x) = V
{fAδ(a) : a ∈
A
and x ≤ a}. Hence, it is enough to show that V
{fAδ(c
i) : i ∈ I} ≤ V{fA δ
(a) : a ∈ A and
x ≤ a}, i.e., we need to show that for each a ∈ A, if x ≤ a, then there exists an i0 ∈ I such that
fAδ(c i0)≤fA δ (a). By compactness, from V {ci :i∈I}=x≤a we get that x0 = V {ci :i∈I0} ≤a
for some finite I0 ⊆I. By the downward-directedness of {ci :i ∈ I}, we get ci0 ≤ x
0 ≤a for some i0 ∈I. Therefore, fA δ (ci0)≤fA δ (a).
From the proposition above, we can see that the essential property for a term to be σ-contracting (resp. π-expanding) is the closed (resp. open) Esakia condition. By checking the proof details, we can see that the fact that fA maps elements in
A to clopen elements in B plays no role. Therefore
the result still holds if we allow fA to map elements in
A to non-clopen elements in Bδ, so we can
work in a setting in which terms in the expanded signature L++ can also be accounted for. There-
fore, we can define the generalized canonical extension fAδ :
Aδ → Bδ for functions fA : A → Bδ,
which we will give below.
The following definitions are very similar to those in Chapter 2, the only difference being the codomain of the functions, but for the sake of clarity we repeat them in full.
Definition 6.2 (Generalized Canonical Extension for Maps). For any mapfA:
A→Bδ, for
(fA)σ(u) =_{^{f(a) :a ∈
A, x≤a≤y}:K(Aδ)3x≤u≤y∈O(Aδ)}
(fA)π(u) = ^{_{f(a) :a∈
A, x≤a≤y}:K(Aδ)3x≤u≤y∈O(Aδ)}.
For example, we can consider ♦Aδ; let ♦A : A→ Aδ be its restriction to A. Then the generalized canonical extension of♦Ais (♦A)λ, whereλ∈ {σ, π}. Then we will show that (♦A)σ = (♦A)π =♦Aδ (see Section 7.1.2).
For order-preserving maps, we also have the following: Theorem 6.3. If fA :
A→Bδ is order-preserving, then for all u∈Aδ,
(fA)σ(u) = _ {^{fA(a) :x≤a∈ A}:u≥x∈K(Aδ)} (fA)π(u) =^ {_{fA(a) :y≥a∈ A}:u≤y∈O(Aδ)}.
In the following, we are mainly working with order-preserving maps.
Many properties aboutσ- and π-extensions still hold in the generalized setting. Here we state some of them without proof.
Proposition 6.4. For any order-preserving map fA :A→Bδ,
1. Both (fA)σ and (fA)π are order-preserving;
2. (fA)σ(x) =V {fA(a) :x≤a∈ A} for x∈K(Aδ); 3. (fA)π(y) =W {fA(a) :y≥a∈ A} for y ∈O(Aδ); 4. (fA)σ(a) = (fA)π(a) =fA(a) for all a∈A; 5. (fA)σ(u)≤(fA)π(u) for all u∈
Aδ, and “ = ” holds for u∈K(Aδ)∪O(Aδ).
Proof. The items 1.-4. follows straightforwardly from the definition. The proof of item 5. is very similar to the proof of Theorem 3.1 in [15].
We can also generalize the definition of stable, expanding and contracting maps to this new setting: Definition 6.5 (Generalized Stable, Expanding and Contracting Maps). For any order- preserving maps fA : A → Bδ and fA δ : Aδ → Bδ where fA δ extends fA, λ ∈ {σ, π}, we say that • fAδ isλ-stable if fAδ = (fA)λ; • fAδ isλ-expanding if fAδ ≤(fA)λ; • fAδ isλ-contracting if fAδ ≥(fA)λ.