Hilbert Algebras with Hilbert-Galois Connections II

Abstract

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, b ∈ A. 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.