Constructive and axiomatic approaches of fuzzy approximation operators

Covering soft rough sets and its topological properties with application

In this article, we introduce the notion of the complementary soft neighborhood and present three kinds of covering soft rough set ($$\mathcal {CSR}$$ CSR ) models. The basic properties of these models are investigated. The relationships among these models are also discussed. Moreover, we establish the topological approach to $$\mathcal {CSR}$$ CSR say, $$\varDelta $$ Δ -topological spaces (i.e., $$\varDelta $$ Δ -TS). Hence, the topological properties for $$\varDelta $$ Δ -TS models such as $$\varDelta $$ Δ -open sets, $$\varDelta $$ Δ -closed sets, $$\varDelta $$ Δ -interior, $$\varDelta $$ Δ -closure, $$\varDelta $$ Δ -boundary, $$\varDelta $$ Δ -neighborhood and $$\varDelta $$ Δ -limit point are studied and the relationships between them are given. Finally, we make use of an algorithm for these proposed models to deal with uncertainties for solving the MGDM problems using the constructed topologies.

Fuzzy preorder based fuzzy rough sets using multiplication and division

By a fuzzy preorder, it means a (0, 1]-binary relation on a nonempty set which is self-reflective and transitive. For a fuzzy preorder on a universe, this paper constructs a kind of fuzzy rough set model based on the multiplication and division of real numbers. The definable sets and the related fuzzy topology are studied.

Transformations and information granularity of knowledge structures in set-based granular computing

Granular computing is a relatively new platform for constructing, describing and processing information or knowledge. For crisp information granulation, the universe is decomposed into granules by binary relations on the universe, say, preorder, tolerance and equivalence relations. A knowledge structure is composed of all information granules induced by a relation that corresponds to the granulation. This paper establishes a novel theoretical framework for the measurement of information granularity of knowledge structures. First, two new relations between knowledge structures are introduced through the use of their respective Boolean relation matrices, where the granular equality relation is defined based on an orthogonal transformation with the transformation matrix being a permutation matrix, and the granularly finer relation is presented by combining the classical finer relation and the orthogonal transformation. Then, it is demonstrated that the simplified knowledge structure base with the granularly finer relation is a partially ordered set, which can be represented by a Hasse diagram. Subsequently, an axiomatic definition of information granularity is proposed to satisfy the constraints regarding these two relations. Moreover, a general form of the information granularity is given, and some existing measures are proved to be its special cases. Finally, as an application of the proposed measure, the attribute significance measure is developed based on the information granularity.

Certain types of fuzzy soft β-covering based fuzzy rough sets with application to decision-making

The aim of this paper is to introduce and study different kinds of fuzzy soft β-neighborhoods called fuzzy soft β-adhesion neighborhoods and to analyze some of their properties. Further, the concepts of soft β-adhesion neighborhoods are investigated and the related properties are studied. Then, we present new kinds of lower and upper approximations by means of different fuzzy soft β-neighborhoods. The relationships among our models (i.e., Definitions 3.9, 3.12, 3.15 and 3.18) and Zhang models [48] are also discussed. Finally, we construct an algorithm based on Definition 3.12, when k = 1 to solve the decision-making problems and illustrate its applicability through a numerical example.

Multi-Granulation Picture Hesitant Fuzzy Rough Sets

We lay the theoretical foundations of a novel model, termed picture hesitant fuzzy rough sets, based on picture hesitant fuzzy relations. We also combine this notion with the ideas of multi-granulation rough sets. As a consequence, a new multi-granulation rough set model on two universes, termed a multi-granulation picture hesitant fuzzy rough set, is developed. When the universes coincide or play a symmetric role, the concept assumes the standard format. In this context, we put forward two new classes of multi-granulation picture hesitant fuzzy rough sets, namely, the optimistic and pessimistic multi-granulation picture hesitant fuzzy rough sets. Further, we also investigate the relationships among these two concepts and picture hesitant fuzzy rough sets.

On Categories of L-Fuzzifying Approximation Spaces, L-Fuzzifying Pretopological Spaces and L-Fuzzifying Closure Spaces

This paper investigates the essential connections among several categories with a weaker structure than that of L-fuzzifying topology, namely category of L-fuzzifying approximation spaces based on reflexive L-fuzzy relations, category of L-fuzzifying pretopological spaces and category of L-fuzzifying interior (closure) spaces. The interrelations among these structures are established in categorical setup.