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

The theory of rough sets, proposed by Pawlak [

Pawlak’s rough set theory is defined on the basis of an approximation space

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 [

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

The rest of the paper proceeds as follows: we briefly review in Section

A multigranulation approximation space [

Let

The multigranulation approximation space

Let

Let

We begin with an example, which shows that in some situations, the multigranulation approximation space is equivalent to a single-granulation approximation space.

Let

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 single-granulation approximation spaces are equivalent to each other.

Some preliminary results of Galois connections are briefly recalled below.

Let

Let

both

It was shown in [

Let

“

“

There are two cases to be considered below.

In this case, we have that

Then, we will further show that

Since

Let

The following proposition provides another necessary and sufficient condition for the equivalence between multigranulation approximation spaces and single-granulation approximation spaces.

Let

“

“

Similarly, we can prove the following result.

Let

Let

“

“

There are two cases to be considered below.

The following proposition can be shown in a similar way.

Let

In [

Let

Let

Let

“

Then, choose arbitrarily

“

Let

It can be proved in a similar way as that in Proposition

Main results in the present paper are summarized as follows.

Let

In this paper, we consider the equivalence between multigranulation rough sets and single-granulation rough sets from the lattice-theoretic viewpoint. The obtained results will help us gain more insights into the mathematical structure of multigranulation rough sets. Along this research line, some interesting topics are worthy of further research; for instance, what is the structure of multigranulation rough sets induced by general binary relations or fuzzy relations on the universe and what is the connection between multigranulation rough sets and knowledge reasoning for multiple agents? We will report them in forthcoming papers.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This project is supported by the National Nature Science Fund of China under Grant 61103133 and 61472471, the Natural Science Program for Basic Research of Shaanxi Province, China (no. 2014JQ1032), and the Innovation Foundation of Science and Technology for Young Scholars in Xi’an Shiyou University (no. 2012QN011).