Pawlak Algebra and Approximate Structure on Fuzzy Lattice
The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties.
The theory of rough sets was originally proposed by Pawlak  in 1982 as a mathematical approach to handle imprecision, vagueness, and uncertainty in data analysis, which has been applied successfully to many areas such as knowledge representation, data mining, pattern recognition, and decision making (Sun et al. , Wang et al. ). This theory takes into consideration the indiscernibility between objects. The indiscernibility is typically characterized by an equivalence relation. Rough sets are the results of approximating crisp sets using equivalence classes. However, the requirement of an equivalence relation seems to be a very restrictive condition which may limit the applications of rough set theory since this requirement can deal only with complete information systems. Therefore, some interesting and meaningful extensions of Pawlak’s rough set models have been proposed in the literature. For example, some interesting extensions to equivalence relations have been proposed, such as tolerance relations or similarity relations, general binary relations on the discourse, partitions and general binary relations on the neighborhood system from topological space, and general approximation spaces (examples of this approach can be found in Chen et al. , Wang and Hu , and Yin et al. [6–8]); some general notions of rough sets such as rough fuzzy sets, fuzzy rough sets, and soft rough sets have been proposed and discussed (examples of this approach can be found in Ali et al. , Feng et al. , Li and Yin , Yao et al. , and Zhang et al. ); and rough set models on two universes of discourse which can be interpreted by the notions of interval structure and generalized approximation space have been extensively studied by Li and Zhang , Ma and Sun , and Yao and Lin .
The relationships between lattice theory and rough sets are another topic receiving much attention in recent years. Cattaneo and Ciucci  focus on the study on lattices with interior and closure operators and abstract approximation spaces, in which the nonequational notion of abstract approximation space for roughness theory is introduced, and its relationship with the equational definition of lattice with Tarski interior and closure operations is studied. Their categorical isomorphism is also proved, and the role of the Tarski interior and closure with an algebraic semantic of a S4-like model of modal logic is widely investigated. Järvinen  investigates lattice-theoretical foundations of rough set theory, in which closure operators in a more general setting of ordered sets, fixpoints of Galois connections, rough set approximations and definable sets, and the lattice structures of the ordered set of all rough sets determined by different kinds of indiscernibility relations are studied in detail. The purpose of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. The relationships between the Pawlak approximation structures and these mathematic structures are established.
The remaining part of the paper is organized as follows. Section 2 introduces the relevant definitions which will be used throughout the paper. Section 3 investigates the Pawlak algebra and weak Pawlak algebra on fuzzy lattice. Section 4 discusses the relationships between Pawlak algebra and auxiliary ordering. Section 5 focuses on the study of molecular Pawlak algebra. Section 6 investigates the properties of Pawlak algebra based on binary relation. Finally, Section 7 concludes the paper and suggests some future research topics.
Definition 1. A partially ordered set is said to be a lattice if and , denoted by and , respectively, exist, for all , . A lattice is said to be complete if, for every , and exist.
Definition 2. Let be a complete lattice with the maximum element 1 and minimum element 0 and “” a binary relation on . If the following conditions hold: for all , , , , ,(i),(ii),(iii), ,(iv), ,then the relation “” is called an auxiliary ordering on . If the relation “” satisfies conditions (i), (ii), and (iv), it is called a weak auxiliary ordering on . The weak auxiliary ordering “” is called completely approximate if Furthermore, if the relation “” satisfies conditions (i), (ii), (iv), and(iii)′,then “” is called a strong auxiliary ordering on .
Definition 3. Let be a completely distributive lattice. If the mapping satisfies the following conditions:(i)reverse law: , , if , then ,(ii)recovery law: , ,then is called a fuzzy lattice.
Remark 4. The notion of fuzzy lattice just given is first introduced in Wang  and is also called de Morgan completely distributive lattice in the literature. To keep consistency, we adopt the term “fuzzy lattice” in this paper.
Definition 5. Let be a fuzzy lattice and . If the subset satisfies the following conditions:(i),(ii), , ,(iii), , , , ,(iv) for all ,(v)if and is linear order subset, then ,then is called a molecular set, and the element of is called a molecule. The fuzzy lattice with molecule is called a molecular lattice.
Definition 6. Let be a fuzzy lattice and . If the subset satisfies the following conditions:(1), ,(2),(3), ,then is called a lattice topology space.
3. The Pawlak Algebra and Weak Pawlak Algebra on Fuzzy Lattices
In this section, we investigate the structural properties of Pawlak rough approximations on fuzzy lattices. Let us begin with introducing the following concepts.
Definition 7 (see [20). Let be a fuzzy lattice. If the dual mappings and satisfy the following conditions: (P1) , (P2) , (P3) , (P4) ,then is called a Pawlak algebra. (resp., , pair () ) is called upper approximation operator (resp., lower approximation operator, dual approximation operator) on . If , then is called a definable element. If , then is called a rough element.
If (P3) is replaced by the following condition (P3)*: (P3)*,then (resp., , pair ()) is called a weak upper approximation operator (resp., weak lower approximation operator, weak dual approximation operator) on , and the system is called a weak Pawlak algebra.
Let be a fuzzy lattice. Denote by (resp., ) the set of all dual approximation operators (resp., dual weak approximation operators) on , respectively, and by the set of all definable elements in the Pawlak algebra .
Definition 8. Let be a fuzzy lattice and , . Then the dual weak approximation operator is called rougher than , denoted by , if the following inequalities hold:
Proposition 9. Let be a Pawlak algebra. Then .
Proof. Let . For any with , we have , and so Therefore, , implying that condition (P3)* holds. Hence , as required.
Define two dual approximation operators and as follows: Then it is easy to see that , and for any .
It is possible to characterize according to the following result.
Theorem 10. Let be a fuzzy lattice. Then (, ) is a complete distributive lattice with maximum element and minimum element .
Proof. Suppose that , where is an index set. Define by
And , , and can be similarly defined. Then the following assertions hold. (1) Consider , . In fact, for any , by Definitions 6 and 7, we have (P1) = = = = , (P2) , (P3) if , then , , implying that , (P4) since for all . (2) Consider . The proof is analogous to that of (1). (3) Consider and = . Clearly, ≺ is true for all . In fact, let be such that for all . Then for all and by definition. It follows that
On the other hand, it is obvious that for all . Hence, . In a similar way by duality, we have .
Summing up the above analysis, () is a complete lattice. It is obvious that is the maximum element and is the minimum element and that () is distributive. This completes the proof.
Proposition 11. Let be a fuzzy lattice and a lattice topology space. Define the operators and by respectively, for all . Then the system is a Pawlak algebra.
Proof. It is straightforward and omitted.
Proposition 12. Let be a Pawlak algebra. Then is a zero-dimensional lattice topology space.
Proof. It is evident that . And we conclude that the following assertions hold.(a) is union-closed. Suppose that , where is an index set. Now, for any , we have , and so , implying that . Since ≥ , we have by conditions (P1) and (P4), as required.(b) is intersection-closed. Suppose that , where is an index set. For any , we have , and so , implying that . By condition (P4), we have , as required.(c) is complement-closed. For any ∩, by condition (P1), we have = = , implying that ∩. Hence is complement-closed.
Summing up the above analysis, is a zero-dimensional lattice topology space.
Now, let us turn our attention to the study of the transitive closure of approximation operators.
Let be a fuzzy lattice. Define the operator “” on as follows: That is, where . For any positive integer , define . Denote , which is called the transitive closure of . Then, we obtain the following result.
Lemma 13. Let be a fuzzy lattice. Then(1) for all ;(2);(3) is a dual approximation operator.
Proof. It is straightforward and omitted.
It suffices to prove that satisfies conditions (P1)–(P4) in Definition 7. In fact, for any , we have (P1) , (P2) , (P3) = = = ∧ = ∧ ,(P4), .
From the proof of Theorem 10, we have And it is easy to prove by mathematical induction that Therefore, It follows that satisfies condition (P3). Similarly, we can prove that also satisfies conditions (P1), (P2), and (P4). Hence, is a dual approximation operator.
Theorem 14 (see ). If is a Pawlak algebra, then the subset is a lattice topology on .
Theorem 15. Let be a fuzzy lattice and . Then for any , one has
Proof. It suffices to prove (13). Equation (14) can be proved by the duality of approximation operators. By the infinite-union-closing in , for any , we have implying that . Conversely, for any , It follows that , and so . Hence, (13) holds.
As a consequence of Theorem 15, we obtain the following result.
Corollary 16. If is a Pawlak algebra, then is a closure operator on .
4. The Relationships between Pawlak Algebra and Auxiliary Ordering
Gierz  introduces the concept of auxiliary ordering for the study of continuous lattice. In fact, the auxiliary ordering is an order relation which is rougher than the initial ordering and can be regarded as the approximation of initial ordering. Inspired by this idea, we can use approximation operator to describe order approximately.
The following results present the relationships between Pawlak algebra and strong auxiliary ordering.
Theorem 17. Let be a fuzzy lattice and “” a strong auxiliary ordering on . Define two operators and : as follows: then is a Pawlak algebra.
Proof. It is obvious that condition (P1) holds. In what follows, we prove that conditions (P2)–(P4) are satisfied.(1)The strong auxiliary ordering implies that is true, and so we have
implying that condition (P2) holds.(2)Let , , be such that . Then and . By condition (ii) in Definition 1, we have , and so
It follows that and , and hence
On the other hand, by condition (iii) in Definition 3, we have It follows from that Hence implying that Thus , and so . Hence condition (P3) holds.(3)By the duality of approximation operators, it suffices to prove that . And this is clearly true by the definition of .
Summing up the above statements, is a Pawlak algebra.
Theorem 18. Let be a Pawlak algebra. Define a binary relation “” as follows: Then “” is an auxiliary ordering on .
Proof. (1) Let , be such that . Then since and .
(2) Let , , , be such that . Then and . It follows that ; that is, .
(3) Let , , be such that and . Then and . Thus, implying that .(4)It follows from that for all .
Summing up the above statements, “” is an auxiliary ordering on .
5. Molecular Pawlak Algebra and Rough Topology
In this section, we focus on the approximate structure on fuzzy lattices. This structure can be regarded as abstract system of rough set.
Definition 19. Let be a molecular Pawlak algebra, a directed set, and a molecular net and . If there exists such that for any , then is called a rough limit of , denoted by . The set of all rough limits of is denoted by .
In the sequel, we provide two examples of molecular Pawlak algebra in topology space.
Example 20. Let be a metric space, , and , where denotes the set of all subsets of . Define where is the closed ball. Then is a molecular Pawlak algebra. Let be the -limit of the molecular sequence (i.e., ), which means that there exists such that for . Obviously, .
Example 21. Let be a nonempty set, an equivalent relation on , fuzzy power set on , , and . For any , define two fuzzy sets and on as
Then we have, for all ,(1),(2),(3),(4),(5).Moreover, for any , define and as
for any . Then is a molecular Pawlak algebra.
In the system , the rough neighborhood of molecule is represented as if and only if there exists such that and for , where and is poly-point set.
If , namely, if and only if , then we have for fuzzy point and it is know that if and only if there exists such that for .
The following is now straightforward.
Proposition 23. The -fuzzy-rough-convergent classes satisfy the Moore-Smith conditions can induce a topology, called -rough fuzzy topology, which is a nullity topology with square members.
Theorem 24. Let be a molecular lattice. Then any mapping satisfying can at least induce a molecular Pawlak algebra.
Proof. Set , and define Then it is evident that is a molecular Pawlak algebra.
6. The Application of Approximation Operators to Rough Sets
Yao and Lin  introduce the concept of rough sets with general binary relation. They defined -neighborhood of from a universe by the binary relation on , and defined general dual approximation operators and as, for all ,
In this section, we further investigate the properties of approximation operators induced by a binary relation.
The following results recall some basic properties of approximation operators induced by a binary relation.
Proposition 25 (see ). Let be a similar relation on . Then the following assertions hold: for all and , ,(R1), ;(R2), ;(R3), ;(R4), ;(R5);(R6).
Proposition 25 indicates that the system is a Pawlak algebra, where denotes the set of all subsets of .
Proposition 26 (see ). Let be a binary relation on . Then(1) is reflexive if and only if ;(2) is symmetric if and only if is equivalent to ;(3)the reflexive relation is transitive if and only if is a closure operator.
Proposition 27. Let , where is a closure operator. Then
Proof. It is straightforward and omitted.
Theorem 28. Let ( be a Pawlak algebra, where is a closure operator that satisfies (P6) for all . Then we have the following:(1)the binary relation defined as is an equivalent relation on ;(2). Moreover, if is a finite universe or the cover of is finite, then .
Proof. (1) It follows from Proposition 29 and condition (P6) that
Then it is easy to see that is an equivalent relation, where for all .
(2) For any , we have On the other hand, implies that ; hence . Thus, we have .
Suppose that universe is finite and let . Then we have Analogous to the above proof, we have . It follows that .
Now suppose that the cover of is finite; that is, there exist , such that is a cover of . Analogous to the above proof, to prove that , it suffices to prove that, for all , . In fact, it follows from being a cover of that . Hence as required.
In the sequel, denote by the set of all similar relations on . Then it is evident that is infinite-intersection-closed. And the following result holds.
Proposition 29. Let be the set of all similar relations on . Then we have(1) for , where is an index set;(2), , .
Proof. (1) Consider = = = = .
(2) Let be such that . Then for any , we have . Now, let . If , then , implying that ; that is, . Therefore, and by the duality. It follows that .
Conversely, suppose that and . It follows that . Since for any by the assumption, we know that , and hence , implying that . Therefore, .
Theorem 30. Let be a finite set and a similar relation on . Then there exists an equivalent relation such that and that is the transitive closure of .
Proof. It follows from the proof of Theorem 10 that and that is a closure operator satisfying (R6). Now, we define a binary relation on as follows:
Then, it is evident that is a similar relation on and by Theorem 10. In the sequel, we prove that is the transitive closure of , which also implies that is an equivalent relation on .
Suppose that is an equivalent relation such that . By Proposition 29, we have and is a closure operator, and hence . Thus, by Proposition 29, .
In this paper, we have investigated the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. The relationships between the Pawlak approximation structures and these mathematic structures are established, and some related properties are presented. These works would provide a new direction for the study of rough set theory and information systems. As for future research, it will be interesting to continue the study of molecular Pawlak algebra and general partial approximation spaces.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the National Natural Science Foundation of China (nos. 61305057 and 71301022) and the Natural Science Research Foundation of Kunming University of Science and Technology (no. 14118760).
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