Perfect Hilbert algebras

• Published in: Reports on mathematical logic Vol. 59; no. 59; pp. 27 - 48
• Main Author: González, Luciano J
• Format: Journal Article
• Language: English
• Published: Kraków Wydawnictwo Uniwersytetu Jagiellońskiego 01.01.2024; Jagiellonian University Press; Jagiellonian University-Jagiellonian University Press
• Subjects: Algebra; Logic; Representations; Set theory; irreducible element; representation; ordered set; Hilbert algebra
• ISSN: 0137-2904; 2084-2589
• DOI: 10.4467/20842589RM.24.001.20697

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.