Couple fuzzy covering rough set models and their generalizations to CCD lattices
Introduction
Rough set theory and fuzzy set theory are the main tools being used to process uncertainty and incomplete information in information systems [5], [6], [11], [18], [28], [30], [31], and fuzzy rough set theory [4] is their combination. Covering rough set theory [31] as an extension of rough set theory has received great interest since 2001 [3], [12], [14], [15], [17], [19], [20], [21], [24], [29], [35], [36], [37]. Recently, intending to link the covering rough set theory with fuzzy rough set theory, investigators constructed some fuzzy rough set models based on fuzzy coverings [7], [10], [13], [25]. Especially in [16], based on new concepts of fuzzy β-covering and fuzzy β-neighborhood, two more general models called fuzzy β-covering rough set models were constructed. Afterwards, to generalize the results of [16], some improvements have been made by B. Yang and B.Q. Hu [26], [27] and by Y. Guan and F. Xu et al. [9]. In the soft rough set theory, J. Zhan and Q. Wang [32] defined some types of soft coverings based rough sets and studied their applications, and L. Zhang, J. Zhan et al. [33], [34] construct some fuzzy soft β-covering based fuzzy rough set models and apply them on decision-makings. Thus, the fuzzy β-covering rough set theory has been enriched and developed unprecedentedly.
For a fuzzy covering approximation space , and each , we divide into two parts and for some . Both the two parts and are important in describing x from positive aspect and negative aspects. So, in this paper, based on the two parts, we define couple fuzzy β-covering rough set models and couple fuzzy β-covering rough set models . These two types of couple fuzzy β-covering rough set models are both generalizations of the twin approximation operators which were defined in covering rough set theory [15]. So they can be viewed as bridges linking the covering rough set theory and the fuzzy rough set theory. They can also be used to analyze and solve practical problems from positive and negative aspects so as to make a crucial decision, so we then give some examples to show their important practical value. Moreover, the relationships between our models and some other models introduced in previous literature are investigated and the matrix representations for the new models are given to facilitate the calculations.
To generalize the couple fuzzy β-covering rough set models to the quasi-complemented lattice are of a bit complicated, because any two elements in the lattice can usually not be compared with each other. In fact, given a CCD lattice L and with , for a L-fuzzy β-covering of a universe U composed by some L-fuzzy sets from U to L, there are subfamilies and of . Both the two families are the descriptions of x just from positive and negative aspects, but the union of the and are generally not equal to , since that there may be an element satisfying neither nor . This is a essential difference between and a general lattice which make the work more complicated. After some effort we succeeded construct the ideal couple L-fuzzy β-covering rough set models on quasi-complemented lattice which are just the generalizations of the couple fuzzy β-covering rough set models, and we also obtain the lattice matrix representations of the models.
So, in this paper, we come true gradual generalizations from the concepts of twin approximation operators in covering rough set theory to the concepts of couple fuzzy β-covering rough set models in fuzzy rough set theory, and further to the concepts of couple L-fuzzy β-covering rough set models in L-fuzzy rough set theory.
The remainder of this paper is organized as follows. In Section 2, some preliminary definitions are reviewed. In Section 3, we introduce two concepts of fuzzy β-co-neighborhood and β-co-neighborhood, then we define two types of couple fuzzy covering rough set models. Properties of the new models are investigated and practical examples are provided. In Section 4, we thoroughly discuss the relationships between the couple fuzzy β-covering rough set models and some other models in the literature. The matrix representations of the new models are introduced in Section 5. We then generalize the two types of couple fuzzy β-covering rough set models and matrix representations to the quasi-complemented lattice to construct the ideal couple L-fuzzy β-covering rough set models in section 6. The paper is concluded in Section 7.
Section snippets
Preliminaries
We first review some concepts in covering rough set theory and fuzzy rough set theory which will be needed in this paper.
Two types of couple fuzzy β-covering rough set models
We use to denote the fuzzy complement set of a fuzzy set , i.e., for each .
Relationships with some known rough set models
If is a fuzzy β-covering of a universe U for some , we can define two fuzzy relations and on U as:
It follows from Proposition 3 that and have the following relationship.
Proposition 6 Let be a fuzzy β-covering approximation space, where is a fuzzy β-covering of U for some . Then for each , if and only if .
Note that, couple fuzzy β-covering rough set
Matrix representations of all couple fuzzy β-covering rough set models
The matrix representations we obtained in [16] can translate the calculations of the lower and upper approximations into matrix operations, which are algorithmic and can easily be implemented by the computer when the universe considered is finite and very big. In this section, we will present matrix representations of operators , and thus reveal relationships between matrix representations of two pairs of approximation operators in every couple fuzzy β-covering rough set
Couple L-fuzzy β-covering rough set models and their matrix representations
In this section, we will construct two types of couple L-fuzzy β-covering rough set models as generalizations of the couple fuzzy β-covering rough set models, as well as generalizations of the twin approximation operators.
Conclusions
We have achieved the generalizations from the concepts of twin approximation operators in covering rough set theory to the concepts of couple fuzzy β-covering rough set models in fuzzy rough set theory, and further to the concepts of couple L-fuzzy β-covering rough set models in L-fuzzy rough set theory. We have also shown that the matrix methods can be used to calculate the approximations of all these models more effectively.
Declaration of Competing Interest
The author declare that she have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was mainly completed during my visit to the Department of Mathematical Sciences at the University of Cincinnati. During my research, my tutor professor Dan Ralescu helped me a lot, so I am deeply grateful to him here.
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