Generalized textural rough sets: Rough set models over two universes
Introduction
Rough set models over two universes ensure a reasonable generalization of classical rough sets of Zdzislaw Pawlak [15]. They provide useful applications with respect to information systems (see e.g., [4], [12], [13], [14], [18], [19], [23]). The studies on the subject were first given by Wong et al. [22], and Yao et al. [24], [25]. An extensive study on two-universe rough sets can be also found in the paper of Pei and Xu [16]. On the other hand, recent studies on textures and rough sets are subjected to single universe rough set models [5], [6], [7], [8], [9]. In this work, we focus on the textural rough sets over two universes with the aim to exhibit the unnoticed connections between rough sets and textures. Note that the seriality of a relation is one of the conditions of a function while the inverse seriality corresponds to surjectivity. In a similar way, the textural seriality and textural inverse seriality given in this paper are in fact the conditions DF1 and SUR properties of a difunction, respectively [1]. Let (U, V, r) be a generalized approximation space. Then (U, V, r, R) is a generalized t-approximation space where and . This connection provides a systematic approach to the basic properties of rough set models over two universes as follows. Note that the compatibility notion is a necessity in defining the approximation operators for (U, V, r). Let us consider the mapping defined by for all v ∈ V. Then the mapping given by leads us to the upper approximation of Yao [24] (see Table 1). However, although the existence of the remarkable equalities
for all A ⊆ U and B ⊆ V, the researchers did not consider the approximations as the mappings and . Here, and are complementary duals of the mappings and respectively. Under the compatibility condition, Theorem 4 in [16] states that for all B ⊆ V,
(DL7)
(DH7) .
However, without any condition, we already have . Then under the seriality condition, we immediately conclude that . This is in fact one of the equivalent conditions in Theorem 7 in [16]. Further, we easily show that
The above inclusions are immediate results of Lemma 2.6 given in this paper. Note that the compatibility of r is not a necessary condition for the above inclusions (see Corollary 3.12). Hence, we see that a textural discussion not only makes Theorem 4 and the related results presented in [16] more understandable, but also removes the unnecessary conditions of the propositions. Another discussion may be given as follows. In [18], Proposition 8(1) asserts that for any generalized approximation space (U, V, r) and for any B ⊆ V, the inclusion r(r(B)) ⊆ B holds. The operation r is defined for the subsets of V and however, r(B) is a subset of U. Hence, the set r(r(B)) is not defined in [18]. Now let us look at the case from the textural point of view. If (r, R) is a t-inverse serial direlation from to then we obtain that
For any generalized approximation space (U, V, r), a result of the above inclusion may be given as
where r is an inverse serial relation (Corollary 3.20). Now the statement r(r(B)) ⊆ B should be the same as the first inclusion above. Further, the second inclusion is the proposition (DH7’) in Theorem 4 [16] given under the compatibility condition. Hence, the textural approach removes the deficiency occured in Proposition 8 [18] and Theorem 4 [16] giving a systematic approach for the inclusions from both sides as well. In addition to above inclusions, we also havewhere r is a serial relation (Corollaries 3.18 (ii) and 3.33 (i)). For the definability notion, it is worthy to note that if r is compatible and injective, then every subset A ⊆ U is successor definable. If is compatible and injective, then the dual case also holds, that is every subset B ⊆ V is predecessor definable. Moreover, for the corresponding conditions, the sets aprG(A) and and aprF(B) and are successor definable and predecessor definable sets, respectively. Another significant case in this work is the formulation given for the revised aproximations in textures aswhere (r, (U × V)∖r) is the direlation from to with . This gives all basic properties with respect to revised approximations through the textures. Here, we may also observe that for all A ⊆ U, AprG(A) and are successor definable. Similarly, for all B ⊆ V, and AprF(B) are predecessor definable. What is more, for any subset of B of V, the sets aprF(B) and are successor definable sets with respect to revised approximations on U. Moreover, in view of a textural result given in Theorem 6 in [9], the revised lower and upper approximations can be considered as natural generalizations of the subset approximations of Yao. Indeed, if we take then the revised approximation operators and correspond to the subset approximation operators given by Yao in [26]. For instance, if (U, r) is an approximation space, then for all A ⊆ U, we conclude that
In particular, for any two domain of discourses U and V with U ⊂ V, this argument allows us a comparison between the approximation operators of the spaces (U, r) and (U, V, r). Indeed, one of the approximations of subsets of (U, r) may change aseven if the relation r does not consider the objects of the set V∖U. Consequently, this paper reveals the useful aspects of textures for generalized approximation spaces in rough set theory.
This work is structured as follows. Section 2 is devoted to the basic concepts and results on textures. In Section 3, seriality and inverse seriality are discussed in textures and some basic formulations for approximations are presented. In particular, we present a comparison about the formulations with respect to mappings and approximations in the framework of textures and rough sets (see Table 1). In Section 4, the revised approximations given by Pei and Xu in [16] are considered in textures. It is known that the formal concept analysis [21] and rough set theory [15] are two complementary fields from the point view of data analysis. In the appendix, we also present a systematic discussion on the connections between the revised approximation operators and the object oriented formal concept lattices of Yao [28].
Section snippets
Textures
Definition 2.1 Let U be a set. Then is called a texturing of U, and is called a texture space, or in brief a texture, if is a complete lattice containing U and ∅, which has the property that arbitrary meets coincide with intersections, and finite joins coincide with unions, that is, for all index sets K,and for all finite index sets Kwhere . is completely distributive, that is, for all index set K, and for all k ∈ K, if Jk is an index set and if [3]
Generalized textural approximation spaces
It is known that a generalized rough set with respect to two universes can be defined using the notion of interval structure given in [22]. Note that this approach may be considered in view of the compatibility relation. If U and V are two domain of discourses and r be a binary relation from U to V, then r is said to be [17]
- (i)
serial, if for any u ∈ U, there exists a v ∈ V such that (u, v) ∈ r,
- (ii)
inverse serial, if for any v ∈ V, there exists a u ∈ U such that (u, v) ∈ r,
- (iii)
a compatibility relation if
Revised two-universe rough set models
Note that for a two-universe rough set model, an approximation of a subset of one of the domain of discourses is a subset of the other one (see, e.g., [16]). However, this is mostly not the case since the evaluations may have no meaning from the point of view of knowledge discovery for information systems. Here, we consider this case in the framework of textures using sections and presections of a direlation. Note that if (r, R) is a direlation from to then in view of the
Conclusion and appendix
The studies on rough sets and textures are mostly subjected to approximation spaces with a single universe. This paper is a first step for an extensive study on textures and two-universe approximation spaces. In view of the representation theorem of textures (Theorem 2.1 in [2]), this work also provides a base for the studies on textures, rough sets, fuzzy sets in the framework of concept lattices. For instance, recall that for any two domain of discourses U and V, and for all A ⊆ U, B ⊆ V, the
Declaration of Competing Interest
We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
Acknowledgement
The authors would like to express their sincere thanks to the anonymous reviewers for their valuable comments improving the presentation of this paper.
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