Geometric Lattice Structure of Covering-Based Rough Sets through Matroids
Covering-based rough set theory is a useful tool to deal with inexact, uncertain, or vague knowledge in information systems. Geometric lattice has been widely used in diverse fields, especially search algorithm design, which plays an important role in covering reductions. In this paper, we construct...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/236307 |
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| author | Aiping Huang William Zhu |
| author_facet | Aiping Huang William Zhu |
| author_sort | Aiping Huang |
| collection | DOAJ |
| description | Covering-based rough set theory is a useful tool to deal with inexact, uncertain, or vague knowledge in information systems. Geometric lattice has been widely used in diverse fields, especially search algorithm design, which plays an important role in covering reductions. In this paper, we construct three geometric lattice structures of covering-based rough sets through matroids and study the
relationship among them. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by a covering. Then its characteristics, such as atoms, modular elements, and modular pairs, are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, we present three sufficient and necessary conditions for two types of covering upper approximation operators to be closure operators of matroids. We also represent two types of matroids through closure axioms and then obtain two geometric lattice structures of covering-based rough sets. Third, we study the relationship among these three geometric lattice structures. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely, geometric lattice, to study covering-based rough sets. |
| format | Article |
| id | doaj-art-25f6d74ffe864f5e9a64cd01b8f7f1f8 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-25f6d74ffe864f5e9a64cd01b8f7f1f82025-02-03T06:11:31ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/236307236307Geometric Lattice Structure of Covering-Based Rough Sets through MatroidsAiping Huang0William Zhu1Lab of Granular Computing, Zhangzhou Normal University, Zhangzhou 363000, ChinaLab of Granular Computing, Zhangzhou Normal University, Zhangzhou 363000, ChinaCovering-based rough set theory is a useful tool to deal with inexact, uncertain, or vague knowledge in information systems. Geometric lattice has been widely used in diverse fields, especially search algorithm design, which plays an important role in covering reductions. In this paper, we construct three geometric lattice structures of covering-based rough sets through matroids and study the relationship among them. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by a covering. Then its characteristics, such as atoms, modular elements, and modular pairs, are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, we present three sufficient and necessary conditions for two types of covering upper approximation operators to be closure operators of matroids. We also represent two types of matroids through closure axioms and then obtain two geometric lattice structures of covering-based rough sets. Third, we study the relationship among these three geometric lattice structures. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely, geometric lattice, to study covering-based rough sets.http://dx.doi.org/10.1155/2012/236307 |
| spellingShingle | Aiping Huang William Zhu Geometric Lattice Structure of Covering-Based Rough Sets through Matroids Journal of Applied Mathematics |
| title | Geometric Lattice Structure of Covering-Based Rough Sets through Matroids |
| title_full | Geometric Lattice Structure of Covering-Based Rough Sets through Matroids |
| title_fullStr | Geometric Lattice Structure of Covering-Based Rough Sets through Matroids |
| title_full_unstemmed | Geometric Lattice Structure of Covering-Based Rough Sets through Matroids |
| title_short | Geometric Lattice Structure of Covering-Based Rough Sets through Matroids |
| title_sort | geometric lattice structure of covering based rough sets through matroids |
| url | http://dx.doi.org/10.1155/2012/236307 |
| work_keys_str_mv | AT aipinghuang geometriclatticestructureofcoveringbasedroughsetsthroughmatroids AT williamzhu geometriclatticestructureofcoveringbasedroughsetsthroughmatroids |