On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

https://doi.org/10.1016/j.ijar.2022.01.012Get rights and content

Abstract

In rough set theory, upper and lower approximation operators are two crucial concepts. The study of these two approximation operators in the framework of lattice theory is an important generalization from the mathematical point of view. On the other hand, overlap and grouping functions, as two types of not necessarily associative binary aggregation functions different from the common binary aggregation functions triangular norms and triangular conorms, have not only rich theoretical results but also a wide range of practical applications. Therefore, based on overlap and grouping functions over complete lattices, this paper is devoted to proposing (O,G)-fuzzy rough sets as a further generalization of the notion of rough sets. Firstly, we define a pair of O-upper and G-lower L-fuzzy rough approximation operators and investigate basic properties of them. Then, the characterizations of (O,G)-fuzzy rough approximation operators are discussed by using different kinds of L-fuzzy relations. Meanwhile, we investigate the topological properties of (O,G)-fuzzy rough sets. Furthermore, we show a brief comparison of the (O,G)-fuzzy rough sets with other common rough set models. At the end of this paper, we further propose multigranulation (O,G)-fuzzy rough sets over complete lattices from the viewpoint of multigranulation structure.

Introduction

The generalization of rough approximation operators is an important research direction in rough set theory (RST, in short). This paper attempts to conduct a further study along this line. To be precise, we propose (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices. In the following, we present some related researches of overlap and grouping functions, as well as RST briefly.

Overlap and grouping functions, as two types of not necessarily associative binary aggregation functions on the unit interval [0,1] different from the common binary aggregation functions triangular norms (t-norms, in short) and triangular conorms (t-conorms, in short), were introduced by Bustince et al. respectively in [4], [5]. Over the decade, these two binary aggregation functions have obtained rich results both in real applications and theory.

In applications, overlap and grouping functions have been successfully applied in various practical fields, such as in image processing [24], decision making [5] and fuzzy community detection problems [19]. In theory, compared with t-norms and t-conorms, the classes of overlap and grouping functions are richer than their classes, so that overlap and grouping functions have a wider range of properties, such as migrativity [83], idempotency [11], homogeneity [55] and distributive laws [54]. In addition, overlap and grouping functions have been extended to other wider fields, as for instance, the n-dimensional cases [18], the interval-valued fuzzy sets [51] and the type-2 fuzzy sets [32]. In particular, Paiva et al. [38] introduced the notions of lattice-valued overlap and quasi-overlap functions, and showed some properties of (quasi-) overlap functions on bounded lattices. However, they did not give a concrete form of the continuity of overlap and grouping functions on bounded lattices. In recent years, Qiao [50] extended the concepts of overlap and grouping functions on the unit interval [0,1] to complete lattices and took the concrete form of the continuity of them into account, which establishes a more general framework for the study of overlap and grouping functions. In addition, Wang and Hu [63] further investigated the construction methods of overlap and grouping functions on complete lattices via complete homomorphisms and complete 0L, 1L-endomorphisms.

The notion of rough sets, as a new mathematical tool to deal with uncertainty, was proposed by Pawlak [41] in 1982. Ever since its introduction, RST has received extensive attention in both theoretical research and practical applications. In theory, various types of rough set models have been constructed and studied, such as covering-based rough sets [75], [82], dominance-based rough sets [12], [13], intuitionistic fuzzy rough sets [9], decision-theoretic rough sets [73], [74] and so on. Meanwhile, the topological and algebraic properties of rough sets have been discussed, see, e.g., [33], [54], [57], [72]. In applications, RST has been widely applied in many fields, such as attribute reduction [7], [8], conflict analysis [25], [26], multi-attribute decision-making [78], etc.

With the development of fuzzy set theory [76], RST has been generalized to fuzzy environment and different types of fuzzy rough set models have been proposed. Taking into account the unit interval [0,1] as the truth value table, diverse fuzzy extensions of rough approximation operators have been developed and studied by utilizing fuzzy binary relations and fuzzy logical connectives [10]. Dubois and Prade [14] first proposed the notion of fuzzy rough sets and defined a pair of rough approximations of a fuzzy set by means of the greatest t-norm (min), its dual t-conorm (max), and a fuzzy similarity relation. This work has influenced many authors to utilize different fuzzy logic connectives and fuzzy relations to construct fuzzy rough set models. For example, Radzikowska and Kerre [44] studied the fuzzy rough set model based on a t-norm and a fuzzy implication. Mi et al. [36] investigated the fuzzy rough set model by using a t-norm and a t-conorm. Mi and Zhang [37] utilized an implication and a coimplication to define a novel fuzzy rough set model. Recently, Qaio [49] utilized an overlap function and its residual implication separately to replace a t-norm and an implication to define the so-called (IO,O)-fuzzy rough set model.

Nevertheless, Goguen [17] pointed out that the degree of membership may not be always represented by elements in the unit interval [0,1]. Moreover, the unit interval [0,1] is a chain, which seems to be a very restriction limiting the applications of fuzzy rough sets and their extensions. Therefore, some lattice structures are put forward to replace the unit interval [0,1] as the truth values set of membership degrees. In light of this fact, it is meaningful to investigate rough set models in the framework of lattice theory from the mathematical viewpoint. For example, some scholars selected a residuated lattice as a lattice background to define L-fuzzy rough sets [1], [45], [57], in which the operators & and → are separately used to define the upper and lower approximation operators. Qiao and Hu [52] discussed the (,&)-fuzzy rough sets in the framework of complete residuated and co-residuated lattices, where two L-fuzzy approximation operators are determined by binary operators ⊙ and &, respectively. In addition, some scholars chose a fuzzy lattice as a lattice background to define L-fuzzy rough sets [29], [39], [40], in which the operators ∧ and ∨ are separately used to define the upper and lower approximation operators.

In light of the perspective of granular computing [76], Pawlak's rough set model and most of its generalization forms are established by using only one granular structure, which is induced by a binary relation (or a fuzzy binary relation). Thus, these models can be referred to as single-grained rough sets. However, a single granulation structure limits the applications of these models in some fields. To address this limitation, Qian et al. [47] proposed the concept of multigranulation rough sets (MGRSs, in short), where the two approximation sets are defined by means of multiple equivalence relations on the universe. In the MGRS model, two basic models, namely, optimistic MGRSs and pessimistic MGRSs are defined. In recent years, MGRSs have received wide attention due to its merits in information fusion and analysis, and various models of MGRSs have been introduced from different viewpoints, for instance, neighborhood-based MGRSs [27], multigranulation fuzzy rough sets [69], fuzzy covering-based MGRSs [77] and double-quantitative decision-theoretic MGRSs [68].

As we have stated, taking the unit interval [0,1] as a basic structure, Pawlak's rough sets [41], [42] have been generalized to fuzzy environment. However, the unit interval [0,1] can not be supplied as a truth value table any more in the partial ordering world. For example, in the approximation space of a high-dimensional information system, by using the unit interval [0,1], some useful information will be lost more or less. People also may think that [0,1] is a very special lattice with lots of nice properties. Sometimes, we do not need all these properties. Hence, some types of non-linearly partial ordered and non-completely distributive lattices can be considered as more general truth value tables for fuzzy rough set theory, see, e.g., [33], [40], [45], [57], [61].

In fact, the crisp power set of a universe is an atomic Boolean lattice, and in the Pawlak's rough sets, equivalence classes are the elements of this power set. From the aspect of the order relation on a lattice, the lower approximation of a subset X of the universe is some element of its power set lattice which is not larger than X, and the upper approximation of the subset X is some element not smaller than X. In addition, the fuzzy power set of a universe is a molecular lattice, as known as a complete completely distributive lattice, in which every non-zero element is a supremum of some moleculars. Therefore, in order to further research the different kinds of generalizations of rough set models and bring them into a unified framework, lattices become a wider mathematical foundation applied to construct rough set models. Meanwhile, the extended rough sets based on a binary relation also play an important part in rough set theory. So in this paper, we adopt a complete lattice L with a negation “N” as the lattice background and select an arbitrary L-fuzzy binary relation to define L-fuzzy rough sets, which can be seen as a unified framework for the research of rough sets based on ordinary binary relations in [56], [70], [71], [72], fuzzy rough sets in [14], [28], [43], [67] and L-rough fuzzy sets in [29], [39], [40].

On the other side, it should be noticed that overlap and grouping functions can be regarded as the new extension of the intersection and union operations of classical logic on the unit interval that differ from the usual fuzzy logical connectives t-norms and t-conorms, respectively, although they are closely related to continuous t-norms and t-conorms with no non-trivial zero divisors. Therefore, overlap functions can be used to replace the classical conjunction operator to define the upper fuzzy rough approximation operators in fuzzy rough sets, and grouping functions can be used to replace the classical disjunction operator to define the lower fuzzy rough approximation operators in fuzzy rough sets. Based on this idea, Qiao [49] used overlap functions to replace the classical conjunction operator to define the upper fuzzy rough approximation operators in fuzzy rough sets, correspondingly, he utilized the residual implications induced by overlap functions to define the lower fuzzy rough approximation operators. However, he only discussed two approximation operators on a special lattice structure of the unit interval [0,1]. Thus, it is necessary to construct more general fuzzy rough set models based on overlap and grouping functions over a wider lattice structure. Recently, scholars have begun to explore the construction methods and properties of overlap and grouping functions on complete lattices, and have carried out in-depth research. For instance, Qiao [50] firstly introduced the notions of overlap and grouping functions on complete lattices, and gave two construction methods of them. Meanwhile, Wang and Hu [63] further investigated the construction methods of overlap and grouping functions on complete lattices via complete homomorphisms and complete 0L, 1L-endomorphisms. In addition, Wang and Hu [64] discussed the ordinal sums of overlap functions and grouping on complete lattices.

These works triggered the present research and motivated us to explore L-fuzzy rough sets based on overlap and grouping functions over complete lattices. To be more specific, based on the constructive approach, we first propose the definition of O-upper and G-lower L-fuzzy rough approximation operators by means of overlap and grouping functions over complete lattices, where we use an overlap function O on complete lattices to define an O-upper L-fuzzy rough approximation operator, and use a grouping function G and a negation N on complete lattices to define a G-lower L-fuzzy rough approximation operator. The connections between special L-fuzzy relations and the proposed new L-fuzzy rough approximation operators are examined. Moreover, the topological properties of the (O,G)-fuzzy rough sets are also investigated. The proposed (O,G)-fuzzy rough sets can be regarded as the extension of the classical case in [56], [70], [71], [72] and the fuzzy case in [14], [29], [39], [40], [43], [67] of rough set models from the theoretical point of view. All the conclusions in [14], [29], [39], [40], [43], [56], [67], [70], [71], [72] can be included as special cases of this paper.

In addition, exploring rough sets in the viewpoint of multigranulation is becoming one of desirable directions in rough set theory, in which lower and upper approximations are approximated by granular structures induced by multiple binary (fuzzy) relations. Through combining this idea as well as the complement of the theoretical aspect of multigranulation rough sets, the another objective of this study is to develop a new multigranulation rough set model, called the multigranulation (O,G)-fuzzy rough sets. More specifically, based on overlap and grouping functions, and a family of arbitrary L-fuzzy binary relations over complete lattices, we define two types of multigranulation (O,G)-fuzzy rough sets, i.e., the optimistic multigranulation (O,G)-fuzzy rough sets and the pessimistic multigranulation (O,G)-fuzzy rough sets. Furthermore, we discuss some basic properties of these two types of multigranulation (O,G)-fuzzy rough sets. The theoretical analysis shows that the proposed multigranulation (O,G)-fuzzy rough sets are generalized versions of the (O,G)-fuzzy rough sets and the multigranulation fuzzy rough sets in [59], [69].

The remainder of this paper is set out as follows. In Section 2, we concisely recall several preliminary concepts and results, which are utilized throughout the paper. In Section 3, we give the definition of (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices and study their essential properties, including the connections between the new proposed L-fuzzy rough approximation operators and different types of L-fuzzy relations, and the topological properties of (O,G)-fuzzy rough sets. In Section 4, we study two kinds of multigranulation (O,G)-fuzzy rough sets over complete lattices. Section 5 concludes our work.

Section snippets

Preliminaries

This section presents a concise retrospect to several fundamental notions and related results. Notice that, these fundamental notions and related results are essential for the sequel, especially for the following Sections 3 and 4.

(O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

In this section, we introduce (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices and study some basic properties of the model. Assume that U and V are two nonempty universes and R is an L-fuzzy relation from U to V. Then, the triple (U,V,R) is called an L-fuzzy approximation space. Particularly, if U=V and R is an L-fuzzy relation on U, we call the pair (U,R) an L-fuzzy approximation space [39], [54].

Multigranulation (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

In this section, according to the (O,G)-fuzzy rough sets and the theory of MGRSs [47], we introduce two types of multigranulation L-fuzzy rough sets based on overlap and grouping functions over complete lattices. In the following, we first give the definition of multigranulation L-fuzzy approximation space on complete lattices.

Definition 4.1

Assume that U, V are two nonempty universes and R={Ri:RiLU×V,i=1,2,,l} is a family of L-fuzzy relations from U to V. Then, the triple (U,V,R) is referred to as a

Conclusions

From the theoretical viewpoint, this paper investigates (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices, which can be regarded as a further study on the topic of the extension models for rough sets. The characterizations of (O,G)-fuzzy rough approximation operators by using different kinds of L-fuzzy relations and the topological properties of (O,G)-fuzzy rough sets are discussed in detail. Moreover, from the viewpoint of multigranulation structure, we

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant nos. 11971365 and 11571010).

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      P, if the proof of conclusion in [7] is incorrect. R, if the conclusion in [7] includes some redundant condition(s). S, if the conclusion in [7] is a special case of the conclusion in our paper.

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