Hilbert Algebras with Hilbert-Galois Connections II - Celani, Sergio A.; Montagie, Daniela (Bulletin of the Section of Logic, 2024)
Hilbert algebra with a Hilbert-Galois connection, or HilGC-algebra, is a triple \((A,f,g)\) where \(A\) is a Hilbert algebra, and \(f\) and \(g\) are unary maps on \(A\) such that \(f(a)≤ b\) iff \(a≤ g(b)\), and \(g(a→ b)≤ g(a)→ g(b)\) for all \(a,bA\). In this paper, we are going to prove that some varieties of HilGC-algebras are characterized by first-order conditions defined in the dual space and that these varieties are canonical. Additionally, we will also study and characterize the congruences of an HilGC-algebra through specific closed subsets of the dual space. This characterization will be applied to determine the simple algebras and subdirectly irreducible HilGC-algebras.