On axiomatic characterizations of three pairs of covering based approximation operators

Four matroidal structures of covering and their relationships with rough sets

“…As a useful tool to deal with covering data, covering-based rough sets have been attracting more and more research interest [1,5,19,41], and in their returns these lead to many interesting and significant problems. For example, from the theoretical point of view, covering approximation models have been constructed [37,42], covering axiomatic systems have been established [17,39], covering reduction problems have been defined [10,21], and covering decision systems have been proposed [11,29]. From the viewpoint of the application, covering-based rough sets have been used in knowledge reduction [3,13].…”

On minimization of axiom sets characterizing covering-based approximation operators

“…Since axiom systems of rough sets are significant for using rough set theory in logical reasoning systems, the independence of the axiom sets is also a very important issue in the study of the axiomatization of rough sets. The independence of the axiom sets of generalized crisp approximation operators [26], covering-based approximation operators [30,32,35], rough fuzzy approximation operators and fuzzy rough approximation operators [27], and generalized fuzzy rough approximation operators [8] has been examined.…”

“…Some properties of the first covering-based lower approximation operator or covering-based upper approximation operator were discussed in [12,23,24,28,33], and several axiom sets of this pair of approximation operators were presented in [30]. Some properties of the second pair of covering-based approximation operators were studied by Qin et al [13], and the upper approximation operator was also introduced by Chen et al [1].…”

“…Some properties of the second pair of covering-based approximation operators were studied by Qin et al [13], and the upper approximation operator was also introduced by Chen et al [1]. Moreover, axiomatic characterizations for this pair of approximation operators were studied by Zhang et al [30]. The third covering-based lower approximation operator was introduced by Zakowski [28], and an axiom set of the lower approximation operator was FHðXÞ ¼ [fKjK 2 C; K \ X -;g; (1)-(5) were shown in [12,28,30], hence we just prove (6).…”

“…Moreover, axiomatic characterizations for this pair of approximation operators were studied by Zhang et al [30]. The third covering-based lower approximation operator was introduced by Zakowski [28], and an axiom set of the lower approximation operator was FHðXÞ ¼ [fKjK 2 C; K \ X -;g; (1)-(5) were shown in [12,28,30], hence we just prove (6). For any X; Y # U; FHðXÞ \ Y -; () there is K 2 C such that Y \ K -; and X \ K -; () X \ FHðYÞ -;.…”