Hilbert Algebras with Hilbert-Galois Connections II
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 \(a,b\in A\). I...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Lodz University Press
2024-12-01
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| Series: | Bulletin of the Section of Logic |
| Subjects: | |
| Online Access: | https://czasopisma.uni.lodz.pl/bulletin/article/view/23268 |
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| Summary: | 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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| ISSN: | 0138-0680 2449-836X |