Rough quotient in topological rough sets

Abstract In this paper, we introduce a rough quotient. Also, we present conditions ensuring that G/H are partitions of G. The rough projection map is also presented. We discuss first, second and third rough isomorphism theorems and other related results. At the end, an orbit and a stabilizer in topological rough groups are considered.


Introduction
The rough set theory was introduced by Pawlak in [1]. Since then, many authors worked on rough set theory. For more details, see [2][3][4][5][6]. The classical rough set theory is based on the equivalence relations.
In , Bagirmaz et al. introduced the notion of topological rough groups. They extended the notion of a topological group to include algebraic structures of rough groups in [7]. For more detailed de nitions about rough groups, rough subgroups, rough normal subgroups and rough homomorphisms and kernel, see the recent paper [8].
The main purpose of this paper is to initiate rough quotient groups. For instance, we present conditions that we need to ensure that G/H are partitions of G. We also de ne the rough projection maps. Moreover, the rst, the second and the third rough isomorphism theorems are given with other important results. Moreover, the concepts of an orbit and stabilizer in topological rough groups are de ned. For the details of topological group theory, we follow [9]. This paper is produced from the PhD thesis of Ms. Nof Alharbi registered in King Abdulaziz University.

Preliminaries
First, we give the de nition of rough groups introduced by Biswas and Nanda in .
Let (U, R) be an approximation space such that U is any set and R is an equivalence relation on U. For a subset X ⊂ U, then the set X = (X, X) is called a rough set of U.
Suppose that (*) is a binary operation de ned on U. We will use xy instead of x * y for all composition of elements x, y ∈ U, as well as, for composition of subsets XY, where X, Y ⊆ U.
De nition 2.1. [7] Let G = (G, G) be a rough set in the approximation space (U, R). Then G = (G, G) is called a rough group if the following conditions are satis ed: 1. ∀x, y ∈ G, xy ∈ G (closed); 2. (xy)z = x(yz), ∀x, y, z ∈ G (associative law); 3. ∀x ∈ G, ∃e ∈ G such that xe = ex = x (e is the rough identity element); 4. ∀x ∈ G, ∃y ∈ G such that xy = yx = e (y is the rough inverse element of x. It is denoted as x − ).

De nition 2.2. [7] A non-empty rough subset H = (H, H) of a rough group G = (G, G) is called a rough subgroup if it is a rough group itself.
The rough set G = (G, G) is a trivial rough subgroup of itself. Also the rough set e = (e, e) is a trivial rough subgroup of the rough group G if e ∈ G. Also, a rough normal subgroup can be de ned. Let N be a rough subgroup of the rough group G, then N is called a rough normal subgroup of G if for all x ∈ G, xN = Nx De nition 2.3. [5] Let (U , R ) and (U , R ) be two approximation spaces and *, * be two binary operations on U and U , respectively. Suppose that G ⊆ U , G ⊆ U are rough groups. If the mapping φ : G → G satis es φ(x * y) = φ(x) * φ(y) for all x, y ∈ G , then φ is called a rough homomorphism.
Here, we present a topological rough group, which is an ordinary topology on a rough group, i.e., a topology τ on G induced a subspace topology τ G on G. Suppose that (U, R) is an approximation space with a binary operation * on U. Let G be a rough group in U.

De nition 2.4. [7]
A topological rough group is a rough group G with a topology τ G on G satisfying the following conditions: 1. the product mapping f : G × G → G de ned by f (x, y) = xy is continuous with respect to a product topology on G × G and the topology τ on G induced by τ G ; 2. the inverse mapping ι : G → G de ned by ι(x) = x − is continuous with respect to the topology τ on G induced by τ G .
Elements in the topological rough group G are elements in the original rough set G with ignoring elements in approximations.
Then G is a rough group with addition. It is also a topological rough group with the usual topology on R.

Example 2.2. [7] Consider U = S the set of all permutations of four objects. Let (*) be the multiplication operation of permutations. Let
The two conditions in De nition . are satis ed, hence G is a topological rough group.

Rough quotient
Let G be a rough group such that G is a group. Let H be a rough subgroup of G where both H and H are not subgroups in G. Then G/H and G/H do not divide G to cosets (partitions of G). The following example con rms our argument. Here, we need that G is a rough group such that G is a group and H is a rough subgroup of G. We also need that H is a subgroup in G (or H is a subgroup in G). Consequently, by group theory,  If H is a normal subgroup in G, then G/H is a group and the projection map π is a homomorphism.

Theorem 3.3. Let G be a topological rough group such that G is a group. If H is a closed rough subgroup such that H is a subgroup of G, then G/H is a T -space.
Proof. We know that Lg is a homeomorphism, so Lg(H) = gH = π(g). Thus for every g ∈ G, π(g) is closed in G/H. It follows that G/H is a T -space.

Theorem 3.4. (The First Rough Isomorphism Theorem) Let f : G → H be a topological rough group homomorphism from G into H such that G is a group. Let K = kerf , then ϕ : G/K → H, de ned by ϕ(xK) = f (x), is a continuous rough isomorphism. If f is open, then ϕ is a rough homeomorphism.
Proof. Consider the projection map π : G → G/K. It is clear that f = ϕ • π. We know that π is continuous and open. Thus, ϕ is continuous. Since K is a normal subgroup in G, then π is a homomorphism. This implies that ϕ is a homomorphism. Let x ∈ G such that ϕ(xK) = e H . Then f (x) = ϕ(π(x)) = e H . If x ∈ K, then xK = K. Thus ϕ is a rough isomorphism. Now assume that f is open. Since π is onto, we have that ϕ is open (let W be an open set in G/K, then the image ϕ(W) = f (π − (W)) is open in H) which implies that ϕ − is continuous. Hence, ϕ is a rough homeomorphism.
Before stating the second rough isomorphism theorem, we need the following proposition.

Proposition 3.1. Let G and H be topological rough groups such that G and H are groups. Let f : G → H be a rough homeomorphism. Let G be a rough subgroup of G and be a normal subgroup in G. Take H = f (G ), then ϕ : G/G → H/H is a topological rough group homeomorphism.
Proof. Consider the projection maps π : G → G/G and π : H → H/H . We have π • f = ϕ • π, then ϕ is a continuous open homomorphism due to the fact that f , π and π are open continuous homomorphisms. Let xG ∈ G/G and set y = f (x). If ϕ(xG ) = H , then π (y) = H . Therefore, y ∈ H and x ∈ G . Thus, the kernel of ϕ is trivial. By the rst rough isomorphism theorem, ϕ is a topological rough group homeomorphism. Let (U, R) be an approximation space with a binary operation * de ned on U. Let G be a topological rough group in U, and X be a topological space induced by topological space X, where X is a rough set in U. Suppose that G acts on X from left (right). We can de ne an equivalence relation on X by setting x ∼ x if there is an element g ∈ G such that x = gx (x = xg).
Consider the quotient space X/G and the projection map π : X → X/G. Each element in X/G is It is called an orbit of x by G. Each orbit is an equivalence class of ∼ . Now, let G be a topological rough group such that G is a group. Given a rough action of G on X. For an element x of X, consider the set Gx = {g ∈ G : gx = x} (or Gx = {g ∈ G : xg = x}). Then Gx is called the stabilizer of x.
Theorem 3.7. The stabilizer of x is a subgroup of G.

Proof.
1. For g , g ∈ Gx, we have g x = x and g x = x. Then g g x = g x = x. Thus, g g ∈ Gx . 2. The identity element e ∈ Gx (since ex = x). 3. Let g ∈ Gx . Then g − x = g − (gx) = (g − g)x = ex = x. Hence, for every g ∈ Gx , we have g − ∈ Gx . From (1), (2) and (3), we conclude that Gx is a subgroup of G. Remark 3.1. In de nition of the stabilizer, we always need G to be a group. Without this condition, we cannot con rm that Gx is closed under multiplication. Now, for x ∈ X, we de ne the map µx : G → X by µx(g) = gx (or µx(g) = xg), where µx is continuous.

Remark 3.2.
Let G be a topological rough group such that G is a group acting on X. Then (i) G acts transitively on X ⇐⇒ µx is surjective; (ii) G acts e ectively on X ⇐⇒ ∩ x∈X Gx = {e}. Theorem 3.8. Let X be a topologically rough homogeneous space of G.

If µx is an open map, then hx is a homeomorphism.
Proof.

Conclusion
In this paper, we investigated cosets and quotients in topological rough groups giving conditions ensuring that G/H or G/H are partitions of G and G/H or G/H is a group. We also have discussed rough isomorphism theorems and other related results in this theory. So far, we have many applications of rough sets into making decision theory, but we could not see more applications of rough algebraic structures. In future, we want to nd applications of these rough algebraic topological structures with into making decision theory.