Hilbert Algebras with Hilbert-Galois Connections II

Sergio A. Celani; Daniela Montagie

doi:10.18778/0138-0680.2024.17

Authors

Sergio A. Celani Universidad Nacional del Centro and CONICET, Departamento de Matemática, Argentina Author https://orcid.org/0000-0003-2542-4128
Daniela Montagie Instituto de Investigación en Tecnologías y Ciencias de la Ingeniería; Universidad Nacional del Comahue, Facultad de Economía y Administración, Departamento de Matemática, Argentina Author https://orcid.org/0000-0001-8832-6715

DOI:

https://doi.org/10.18778/0138-0680.2024.17

Keywords:

Hilbert algebra, modal operators, Galois connection, canonical varieties, congruences

Abstract

Hilbert algebra with a Hilbert-Galois connection, or HilGC-algebra, is a triple \(\left(A,f,g\right)\) where \(A\) is a Hilbert algebra, and \(f\) and \(g\) are unary maps on \(A\) such that \(f(a)\leq b\) iff \(a\leq g(b)\), and \(g(a\rightarrow b)\leq g(a)\rightarrow g(b)\) for
all \(a,b\in A\). 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.

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