Completely distributive completions of posets

On the free frontal implicative semilattice extension of a frontal Hilbert algebra

“…[15]. In [16] it is proved that it is indeed the (Fil(H), Id(H))completion of (H, ≤) and that the operation ⇒ is the π-extension of the operation → of H to X(H) + .…”

“…The completion ((X(H) + , ⊆), ϕ H ) of the poset (H, ≤) is a ∆ 1 -completion in the sense of [15]. In [16] it is proved that it is indeed the (Fil(H), Id(H))completion of (H, ≤) and that the operation ⇒ is the π-extension of the operation → of H to X(H) + .…”

Variations of the free implicative semilattice extension of a Hilbert algebra

“…It is worth mentioning that ϕ[H] = D(X(H)) ⊆ X(H) + , as defined before Lemma 8, is the logic-based canonical extension of the Hilbert algebra H, as defined in [12]. This fact follows from [14,Corollary 6.26].…”