On lower and upper intension order relations by different cover concepts

Ghanim, M. H.; Mustafa, Heba I.; Aziz, Sehar

doi:10.1016/j.ins.2011.04.041

Cited by 4 publications

(2 citation statements)

References 37 publications

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“…It is a new mathematical tool to handle inexact, uncertain or vague knowledge and has been successfully applied to many fields such as machine learning, pattern recognition and data mining [6,7,13,17,20]. In order to meet various practical applications, rough set theory has been connected with other theories, such as fuzzy set theory [18,22], boolean algebra [21,23], topology [15,26,30], lattice theory [2,3,4,5,9,10] and so on.…”

Section: Introductionmentioning

confidence: 99%

Closed-set lattice of regular sets based on a serial and transitive relation through matroids

Zhu

2013

Int. J. Mach. Learn. & Cyber.

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Abstract. Rough sets are efficient for data pre-processing in data mining. Matroids are based on linear algebra and graph theory, and have a variety of applications in many fields. Both rough sets and matroids are closely related to lattices. For a serial and transitive relation on a universe, the collection of all the regular sets of the generalized rough set is a lattice. In this paper, we use the lattice to construct a matroid and then study relationships between the lattice and the closed-set lattice of the matroid. First, the collection of all the regular sets based on a serial and transitive relation is proved to be a semimodular lattice. Then, a matroid is constructed through the height function of the semimodular lattice. Finally, we propose an approach to obtain all the closed sets of the matroid from the semimodular lattice. Borrowing from matroids, results show that lattice theory provides an interesting view to investigate rough sets.

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Section: Introductionmentioning

confidence: 99%

Closed-set lattice of regular sets based on a serial and transitive relation through matroids

Zhu

2013

Int. J. Mach. Learn. & Cyber.

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“…All these works help us comprehend of rough sets on lattice and have greatly enriched rough set theory and its applications. Moreover, M.H.Ghanim et al [10] defined two relations by lower intension and upper intension, respectively. They pointed that two equivalence classes of these two relations are two partially ordered sets, respectively.…”

Section: Introductionmentioning

confidence: 99%

Lattice structures of fixed points of the lower approximations of two types of covering-based rough sets

Li,

Zhu

2012

Preprint

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Covering is a common type of data structure and covering-based rough set theory is an efficient tool to process this data. Lattice is an important algebraic structure and used extensively in investigating some types of generalized rough sets. In this paper, we propose two family of sets and study the conditions that these two sets become some lattice structures. These two sets are consisted by the fixed point of the lower approximations of the first type and the sixth type of covering-based rough sets, respectively. These two sets are called the fixed point set of neighborhoods and the fixed point set of covering, respectively. First, for any covering, the fixed point set of neighborhoods is a complete and distributive lattice, at the same time, it is also a double p−algebra. Especially, when the neighborhood forms a partition of the universe, the fixed point set of neighborhoods is both a boolean lattice and a double Stone algebra. Second, for any covering, the fixed point set of covering is a complete lattice.When the covering is unary, the fixed point set of covering becomes a distributive lattice and a double p−algebra. a distributive lattice and a double p−algebra when the covering is unary. Especially, when the reduction of the covering forms a partition of the universe, the fixed point set of covering is both a boolean lattice and a double Stone algebra.

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Covering Rough Sets and Formal Topology – A Uniform Approach Through Intensional and Extensional Constructors

Pagliani¹

2016

Transactions on Rough Sets XX

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On lower and upper intension order relations by different cover concepts

Cited by 4 publications

References 37 publications

Closed-set lattice of regular sets based on a serial and transitive relation through matroids

Closed-set lattice of regular sets based on a serial and transitive relation through matroids

Lattice structures of fixed points of the lower approximations of two types of covering-based rough sets

Covering Rough Sets and Formal Topology – A Uniform Approach Through Intensional and Extensional Constructors

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