A Lattice-Theoretic Approach to Multigranulation Approximation Space

In this paper, we mainly investigate the equivalence between multigranulation approximation space and single-granulation approximation space from the lattice-theoretic viewpoint. It is proved that multigranulation approximation space is equivalent to single-granulation approximation space if and only if the pair of multigranulation rough approximation operators (Σi=1nRi¯,Σi=1nRi_) forms an order-preserving Galois connection, if and only if the collection of lower (resp., upper) definable sets forms an (resp., union) intersection structure, if and only if the collection of multigranulation upper (lower) definable sets forms a distributive lattice when n = 2, and if and only if ∀X⊆U,  Σi=1nRi_(X)=∩i=1nRi_(X). The obtained results help us gain more insights into the mathematical structure of multigranulation approximation spaces.


Introduction
The theory of rough sets, proposed by Pawlak [1,2], is a formal tool for the study of intelligent systems characterized by insufficient and incomplete information. After over thirty years of progress, it has become a well-established mechanism for uncertainty management in a wide variety of applications related to artificial intelligence (see [3,4]).
Pawlak's rough set theory is defined on the basis of an approximation space ( , ), where is a nonempty set, also called the universe of discourse, and is an equivalence relation on , representing the indiscernibility at the object level due to the lack of knowledge or information. Owing to the indiscernibility of objects, some subsets of the universe cannot be completely characterized with the available knowledge, thus forming a region of uncertainty. Pawlak's idea was to approximate those sets with two precise sets from below and above based on certainty and possibility, respectively. They are called lower and upper approximations and are defined, for any ⊆ , as Since in Pawlak's rough set theory, the concept is depicted by known knowledge induced from a single equivalence relation on the universe, in view of granular computing (see [5,6]), Pawlak's rough set theory was established through a single granulation, and therefore ( , ) is also called a singlegranulation approximation space. However, as illustrated in [7], in some cases it is more reasonable to describe the target concept through multiple relations on the universe according to user requirements or targets of problem solving. To more widely apply the rough set theory in practical applications, Qian et al. extended Pawlak's single-granulation rough set model to a multigranulation rough set model (see [7]). To date, the theory of multigranulation rough set progressed rapidly. Many interesting results have been reported in the literature ( [8][9][10][11][12][13][14][15]).
Superficially, the notion of multigranulation rough set differs significantly from that of Pawlak's single-granulation rough set, because the former is defined by using multiple equivalence relations on the universe whereas the latter is defined by employing a single one. However, in some situations (see Example 5), the collection of multigranulation rough sets generated by a multigranulation approximation space may coincide with that produced by a singlegranulation approximation space, such a phenomenon will 2 The Scientific World Journal be referred to as equivalence between these two kinds of approximation spaces. Since the lattice-theoretic properties of Pawlak's single-granulation rough sets have been extensively studied, then the equivalence between these two kinds of approximation spaces will help us gain more insights into the mathematical structure of multigranulation rough sets and hence deserves further study. Motivated by the above considerations, we attempt to investigate the conditions under which the multigranulation approximation space is equivalent to a single-granulation approximation space from the lattice-theoretic viewpoint.
The rest of the paper proceeds as follows: we briefly review in Section 2 multigranulation rough set theory and some of its basic properties. In Section 3, we present the main results and give their detailed proof. In Section 4, we complete this paper with some concluding remarks.

Preliminary
A multigranulation approximation space [7] is a pair is a nonempty set, also called the universe of discourse, and each (1 ≤ ≤ ) is an equivalence relation on , representing a particular kind of indiscernibility at the level of objects. For ⊆ , define Then we call Σ =1 ( ), Σ =1 ( ) the multigranulation lower approximation and the multigranulation upper approximation of , respectively. Note that, in [7], two kinds of multigranulation rough approximations were defined, they are optimistic multigranulation rough approximation and pessimistic rough multigranulation approximation. The above defined Σ =1 , Σ =1 are actually the optimistic rough approximation operators. However, since we are not concerned with the pessimistic one in the present paper, the term multigranulation approximation always means the optimistic multigranulation approximation. If Σ =1 ( ) = , then we call a multigranulation lower definable set, if Σ =1 ( ) = , then we call a multigranulation upper definable set and if Σ =1 ( ) = = Σ =1 ( ), then we call a multigranulation definable set. Let ∈ , if there exists an equivalence relation ∈ { 1 , . . . , } such that [ ] = ∩ =1 [ ] ; that is, [ ] is the smallest equivalence class containing ; then we say that has the SEC property.
Proposition 1 (see [1]). Let ( , ) be a single-granulation approximation space. Then the collection of definable sets in ( , ) forms a Boolean algebra under the usual set-theoretic operations.

Main Results
We begin with an example, which shows that in some situations, the multigranulation approximation space is equivalent to a single-granulation approximation space. Then one natural question arises: under what conditions the notion of multigranulation approximation space reduce to single-granulation space. Such a consideration leads us to investigate several necessary and sufficient conditions under which multigranulation approximation spaces and singlegranulation approximation spaces are equivalent to each other.
Some preliminary results of Galois connections are briefly recalled below.
The following example shows that does not form an intersection structure (see [17]) in the general case. The following proposition provides another necessary and sufficient condition for the equivalence between multigranulation approximation spaces and single-granulation approximation spaces. Proof . "⇒" It follows immediately from Proposition 1.
"⇐" Suppose, on the contrary, that the multigranulation approximation space is not equivalent to any singlegranulation approximation space; then, there exists an element of (say as ) such that does not have the SEC property. Since [ ] ∈ , 1 ≤ ≤ , we then have from the fact that forms an intersection structure that ∩ =1 [ ] ∈ . Clearly, ∈ ∩ =1 [ ] , and then according to (2) Similarly, we can prove the following result. Proof . "⇒" It follows immediately from Proposition 1.
"⇐" Suppose, on the contrary, that the multigranulation approximation space ( , { } 2 =1 ) is not equivalent to any single-granulation approximation space, then according to Proposition 1 . We assume, without any loss of generality, that where { , 1 , . . . , } = [ ] 1 ∩ [ ] 2 . There are two cases to be considered below.   In [7], some sufficient and necessary conditions for the equivalence between a multigranulation approximation space and a single-granulation approximation space are provided mainly from the viewpoint of topology, that is, as follows. , which shows that [ ] is a definable set in the single-granulation approximation space ( , ). Since the collection of definable sets in ( , ) forms a Boolean algebra and thus is closed under set-theoretic intersection, we conclude that ∩ =1 [ ] is also a definable set in ( , ). Moreover, since the collection of ∩ =1 [ ] is the set The Scientific World Journal 5 of equivalence classes produced by ∩ =1 , we further have that ∩ =1 is coarser than .