32,666 research outputs found
Covering rough sets based on neighborhoods: An approach without using neighborhoods
Rough set theory, a mathematical tool to deal with inexact or uncertain
knowledge in information systems, has originally described the indiscernibility
of elements by equivalence relations. Covering rough sets are a natural
extension of classical rough sets by relaxing the partitions arising from
equivalence relations to coverings. Recently, some topological concepts such as
neighborhood have been applied to covering rough sets. In this paper, we
further investigate the covering rough sets based on neighborhoods by
approximation operations. We show that the upper approximation based on
neighborhoods can be defined equivalently without using neighborhoods. To
analyze the coverings themselves, we introduce unary and composition operations
on coverings. A notion of homomorphismis provided to relate two covering
approximation spaces. We also examine the properties of approximations
preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin
Covering matroid
In this paper, we propose a new type of matroids, namely covering matroids,
and investigate the connections with the second type of covering-based rough
sets and some existing special matroids. Firstly, as an extension of
partitions, coverings are more natural combinatorial objects and can sometimes
be more efficient to deal with problems in the real world. Through extending
partitions to coverings, we propose a new type of matroids called covering
matroids and prove them to be an extension of partition matroids. Secondly,
since some researchers have successfully applied partition matroids to
classical rough sets, we study the relationships between covering matroids and
covering-based rough sets which are an extension of classical rough sets.
Thirdly, in matroid theory, there are many special matroids, such as
transversal matroids, partition matroids, 2-circuit matroid and
partition-circuit matroids. The relationships among several special matroids
and covering matroids are studied.Comment: 15 page
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
widely used in diverse fields, especially search algorithm design which plays
important role in covering reductions. In this paper, we construct four
geometric lattice structures of covering-based rough sets through matroids, and
compare their relationships. First, a geometric lattice structure of
covering-based rough sets is established through the transversal matroid
induced by the covering, and its characteristics including 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, sufficient and necessary
conditions for three types of covering upper approximation operators to be
closure operators of matroids are presented. We exhibit three types of matroids
through closure axioms, and then obtain three geometric lattice structures of
covering-based rough sets. Third, these four geometric lattice structures are
compared. 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
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