Soft Rough Approximation Operators on a Complete Atomic Boolean Lattice

Abstract

The concept of soft sets based on complete atomic Boolean lattice, which can be seen as a generalization of soft sets, is introduced. Some operations on these soft sets are discussed, and new types of soft sets such as full, keeping infimum, and keeping supremum are defined and supported by some illustrative examples. Two pairs of new soft rough approximation operators are proposed and the relationship among soft set is investigated, and their related properties are given. We show that Järvinen's approximations can be viewed as a special case of our approximation. If = ℘(), then our soft approximations coincide with crisp soft rough approximations (Feng et al. 2011).

On modified soft rough sets on a complete atomic Boolean lattice

“…In other words, a soft set over B is a parameterized family of elements of B. For each e ∈ A, f (e) is considered as e-approximate element of f A Definition 4 [14] Let B = (B, ≤) be a complete atomic Boolean lattice. Let A ⊆ E and let f A be a soft set over B.…”

“…Research on soft set theory is growing rapidly [9][10][11][12][13]. In [14], we presented soft sets on a complete atomic Boolean lattice as a generalization of soft sets and obtained the lattice structure of these soft sets.…”

“…In [23,24], the concept of bipolar soft rough set and bipolar soft rough relation are proposed In [25], Shabir introduced a new approach to soft rough sets called modified soft rough set (MSR-set) and studied some of their basic properties. In [14], we defined two pairs of soft rough approximation operators on a complete atomic Boolean lattice Band gave their properties. These operators suffer from unexpected properties such as soft upper approximation of non zero element might be equal zero and soft upper approximation of any element might not greater than this element.…”

“…Moreover, we study their properties and introduce a new rough set model, which is an improvement of Järvinen , s model [20]. We study the relations between soft rough approximations on B [14] and current approximations. Also, we introduce the deviation of some properties of the previous soft model [14] and our new model supported by counter examples.…”

“…We study the relations between soft rough approximations on B [14] and current approximations. Also, we introduce the deviation of some properties of the previous soft model [14] and our new model supported by counter examples. In "Relation between MSR sets and rough sets on a complete atomic Boolean lattice" section, we introduce the notion of Boolean lattice information system.…”