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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Lodz University Press
2024-12-01
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| Series: | Bulletin of the Section of Logic |
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| Online Access: | https://czasopisma.uni.lodz.pl/bulletin/article/view/23268 |
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| author | Sergio A. Celani Daniela Montagie |
| author_facet | Sergio A. Celani Daniela Montagie |
| author_sort | Sergio A. Celani |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-15a31542abf94eb794872959d8955071 |
| institution | Kabale University |
| issn | 0138-0680 2449-836X |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Lodz University Press |
| record_format | Article |
| series | Bulletin of the Section of Logic |
| spelling | doaj-art-15a31542abf94eb794872959d89550712025-02-07T07:22:47ZengLodz University PressBulletin of the Section of Logic0138-06802449-836X2024-12-0153453555410.18778/0138-0680.2024.1723575Hilbert Algebras with Hilbert-Galois Connections IISergio A. Celani0https://orcid.org/0000-0003-2542-4128Daniela Montagie1https://orcid.org/0000-0001-8832-6715Universidad Nacional del Centro and CONICET, Departamento de Matemática, ArgentinaInstituto de Investigación en Tecnologías y Ciencias de la Ingeniería; Universidad Nacional del Comahue, Facultad de Economía y Administración, Departamento de Matemática, ArgentinaHilbert 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.https://czasopisma.uni.lodz.pl/bulletin/article/view/23268hilbert algebramodal operatorsgalois connectioncanonical varietiescongruences |
| spellingShingle | Sergio A. Celani Daniela Montagie Hilbert Algebras with Hilbert-Galois Connections II Bulletin of the Section of Logic hilbert algebra modal operators galois connection canonical varieties congruences |
| title | Hilbert Algebras with Hilbert-Galois Connections II |
| title_full | Hilbert Algebras with Hilbert-Galois Connections II |
| title_fullStr | Hilbert Algebras with Hilbert-Galois Connections II |
| title_full_unstemmed | Hilbert Algebras with Hilbert-Galois Connections II |
| title_short | Hilbert Algebras with Hilbert-Galois Connections II |
| title_sort | hilbert algebras with hilbert galois connections ii |
| topic | hilbert algebra modal operators galois connection canonical varieties congruences |
| url | https://czasopisma.uni.lodz.pl/bulletin/article/view/23268 |
| work_keys_str_mv | AT sergioacelani hilbertalgebraswithhilbertgaloisconnectionsii AT danielamontagie hilbertalgebraswithhilbertgaloisconnectionsii |