Year: 2024
Type: article
Abstract: 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.
Source: Reports on Mathematical Logic
Author: Luciano J. González
Institution: Universidad Nacional de La Pampa
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
Awards: G339417742, G5779069361, G7622340535
OpenAlex ID: W4405380479