%0 Journal Article
%T Perfect Hilbert algebras
%A González, Luciano J.
%J Reports on Mathematical Logic
%V 2024
%R 10.4467/20842589RM.24.001.20697
%N Number 59
%P 27-48
%K Hilbert algebra, irreducible element, ordered set, representation
%@ 0137-2904
%D 2024
%U https://ejournals.eu/en/journal/reports-on-mathematical-logic/article/perfect-hilbert-algebras
%X In [S. Celani and L. Cabrer. Duality for finite Hilbert algebras. Discrete Math., 305(1-3):74{99, 2005.] the authors proved that every finite Hilbert algebra A is isomorphic to the Hilbert algebra HK(X) = {w ⇒ i v : w ∈ K and v ⊆ w}, where X is a finite poset, K is a distinguished collection of subsets of X, and the implication ⇒i is defined by: w ⇒i  v = {x ∈ X : w ∩ [x) ⊆ v}, where [x) = {y ∈ X : x ≤ y.} The Hilbert implication on HK(X) is the usual Heyting implication ⇒ i  (as just defined) given on the increasing subsets. In the same article, Celani and Cabrer extended this representation to a full categorical duality. The aim of the present article is to obtain an algebraic characterization of the Hilbert algebras HK(X) for all structures (X, ≤, K) defined by Celani and Cabrer but not necessarily finite. Then, we shall extend this  representation to a full dual equivalence generalizing the finite setting given by Celani and Cabrer.