The structure of rough sets defined by reflexive relations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912385741094912 |
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| author | Järvinen, Jouni Radeleczki, Sándor |
| author_facet | Järvinen, Jouni Radeleczki, Sándor |
| contents | For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of rough set structures. Building upon D. Umadevi's work on describing joins and meets in DM(RS), we previously investigated pseudo-Kleene algebras defined on DM(RS) for reflexive relations. This paper delves deeper into the order-theoretic properties of DM(RS) in the context of reflexive relations. We describe the completely join-irreducible elements of DM(RS) and characterize when DM(RS) is a spatial completely distributive lattice. We show that even in the case of a non-transitive reflexive relation, DM(RS) can form a Nelson algebra, a property generally associated with quasiorders. We introduce a novel concept, the core of a relational neighborhood, and use it to provide a necessary and sufficient condition for DM(RS) to determine a Nelson algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The structure of rough sets defined by reflexive relations Järvinen, Jouni Radeleczki, Sándor Rings and Algebras Logic in Computer Science 68T37, 06B23 (Primary) 06D30, 03E20 (Secondary) For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of rough set structures. Building upon D. Umadevi's work on describing joins and meets in DM(RS), we previously investigated pseudo-Kleene algebras defined on DM(RS) for reflexive relations. This paper delves deeper into the order-theoretic properties of DM(RS) in the context of reflexive relations. We describe the completely join-irreducible elements of DM(RS) and characterize when DM(RS) is a spatial completely distributive lattice. We show that even in the case of a non-transitive reflexive relation, DM(RS) can form a Nelson algebra, a property generally associated with quasiorders. We introduce a novel concept, the core of a relational neighborhood, and use it to provide a necessary and sufficient condition for DM(RS) to determine a Nelson algebra. |
| title | The structure of rough sets defined by reflexive relations |
| topic | Rings and Algebras Logic in Computer Science 68T37, 06B23 (Primary) 06D30, 03E20 (Secondary) |
| url | https://arxiv.org/abs/2412.10863 |