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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.10642 |
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| _version_ | 1866916397090603008 |
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| author | Tabatabai, Amirhossein Akbar Alizadeh, Majid Memarzadeh, Masoud |
| author_facet | Tabatabai, Amirhossein Akbar Alizadeh, Majid Memarzadeh, Masoud |
| contents | A $\nabla$-algebra is a natural generalization of a Heyting algebra, unifying several algebraic structures, including bounded lattices, Heyting algebras, temporal Heyting algebras, and the algebraic representation of dynamic topological systems. In the prequel to this paper [3], we explored the algebraic properties of various varieties of $\nabla$-algebras, their subdirectly-irreducible and simple elements, their closure under Dedekind-MacNeille completion, and their Kripke-style representation.
In this sequel, we first introduce $\nabla$-spaces as a common generalization of Priestley and Esakia spaces, through which we develop a duality theory for certain categories of $\nabla$-algebras. Then, we reframe these dualities in terms of spectral spaces and provide an algebraic characterization of natural families of dynamic topological systems over Priestley, Esakia, and spectral spaces. Additionally, we present a ring-theoretic representation for some families of $\nabla$-algebras. Finally, we introduce several logical systems to capture different varieties of $\nabla$-algebras, offering their algebraic, Kripke, topological, and ring-theoretic semantics, and establish a deductive interpolation theorem for some of these systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10642 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a Generalization of Heyting Algebras II Tabatabai, Amirhossein Akbar Alizadeh, Majid Memarzadeh, Masoud Logic A $\nabla$-algebra is a natural generalization of a Heyting algebra, unifying several algebraic structures, including bounded lattices, Heyting algebras, temporal Heyting algebras, and the algebraic representation of dynamic topological systems. In the prequel to this paper [3], we explored the algebraic properties of various varieties of $\nabla$-algebras, their subdirectly-irreducible and simple elements, their closure under Dedekind-MacNeille completion, and their Kripke-style representation. In this sequel, we first introduce $\nabla$-spaces as a common generalization of Priestley and Esakia spaces, through which we develop a duality theory for certain categories of $\nabla$-algebras. Then, we reframe these dualities in terms of spectral spaces and provide an algebraic characterization of natural families of dynamic topological systems over Priestley, Esakia, and spectral spaces. Additionally, we present a ring-theoretic representation for some families of $\nabla$-algebras. Finally, we introduce several logical systems to capture different varieties of $\nabla$-algebras, offering their algebraic, Kripke, topological, and ring-theoretic semantics, and establish a deductive interpolation theorem for some of these systems. |
| title | On a Generalization of Heyting Algebras II |
| topic | Logic |
| url | https://arxiv.org/abs/2409.10642 |