Duality Theory for Bounded Lattices: A Comparative Study
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909845501771776 |
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| author | Bezhanishvili, Guram Carai, Luca Morandi, Patrick |
| author_facet | Bezhanishvili, Guram Carai, Luca Morandi, Patrick |
| contents | There are numerous generalizations of the celebrated Priestley duality for bounded distributive lattices to the non-distributive setting. The resulting dualities rely on an earlier foundational work of such authors as Nachbin, Birkhoff-Frink, Bruns-Lakser, Hofmann-Mislove-Stralka, and others. We undertake a detailed comparative study of the existing dualities for arbitrary bounded (non-distributive) lattices, including supplying the dual description of bounded lattice homomorphisms where it was lacking. This is achieved by working with relations instead of functions. As a result, we arrive at a landscape of categories that provide various generalizations of the category of Priestley spaces. We provide explicit descriptions of the functors yielding equivalences of these categories, together with explicit equivalences with the category of bounded lattices and bounded lattice homomorphisms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_21307 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Duality Theory for Bounded Lattices: A Comparative Study Bezhanishvili, Guram Carai, Luca Morandi, Patrick Logic 06B15, 06B35, 06A12, 06A15, 06E15, 06F30, 18F70 There are numerous generalizations of the celebrated Priestley duality for bounded distributive lattices to the non-distributive setting. The resulting dualities rely on an earlier foundational work of such authors as Nachbin, Birkhoff-Frink, Bruns-Lakser, Hofmann-Mislove-Stralka, and others. We undertake a detailed comparative study of the existing dualities for arbitrary bounded (non-distributive) lattices, including supplying the dual description of bounded lattice homomorphisms where it was lacking. This is achieved by working with relations instead of functions. As a result, we arrive at a landscape of categories that provide various generalizations of the category of Priestley spaces. We provide explicit descriptions of the functors yielding equivalences of these categories, together with explicit equivalences with the category of bounded lattices and bounded lattice homomorphisms. |
| title | Duality Theory for Bounded Lattices: A Comparative Study |
| topic | Logic 06B15, 06B35, 06A12, 06A15, 06E15, 06F30, 18F70 |
| url | https://arxiv.org/abs/2502.21307 |