Undoubtedly, the theory of rough sets, proposed by Pawlak [19,20], has become a well-established mathematical tool for the study of uncertainty in a wide variety of applications and intelligent systems characterized by inadequate and incomplete information where the equivalence classes created by the equivalence relation are used to define the lower and upper approximations to approximate an undefinable set. More over, in recent years, it has been widely used in a variety of fields, such as granular computing, graph theory, algebraic systems, partially ordered sets, medical diagnosis, data mining, conflict analysis (see, for example, [4,5,10,21,22,27]).

The generalization and extension of the rough set model is an important direction in the study of set theory. On the one hand, one such trend was introduced by Qian et al. [23,24], which has been called the multi-granulation rough set. It is defined by a family of equivalence relations, while Pawlak’s rough set is defined by only one equivalence relation, producing two types of multi-granulation rough sets called the optimistic multi-granulation rough set and the pessimistic multi-granulation rough set. The word "optimistic" is used in the lower and upper approximation to denote the idea that in multi independent granular structures, at least one granular structure must satisfy the inclusion relation between the equivalence class and the undefinable set, whereas the word "pessimistic" denotes the idea that each granular structure must satisfy the inclusion relation between the equivalence class and the undefinable set. Following that, a number of researchers looked into multi-granulation rough set models based on various types of relationships, and they came up with a number of interesting ideas (see, for example, [12,13,14,15,16,25])

On the other hand, one of these trends is to combine other theories that deal with uncertain knowledge, such as fuzzy set and rough set theory. While fuzzy set theory deals with potential uncertainties associated with inaccurate cases, perceptions, and preferences, we find that approximate sets, in turn, deal with uncertainty caused by ambiguity of information. Because the two types of uncertainty can occur in real-world problems, numerous approaches to combining fuzzy set theory with approximation set theory have been proposed. Rough fuzzy sets and fuzzy rough sets were presented by Dubois and Prade based on approximations of fuzzy sets by crisp approximation spaces and crisp sets by fuzzy approximation spaces, respectively [6,7]. In the same framework, the researchers presented an approach to enrich coarse fuzzy rough sets and rough fuzzy sets, (see for example, [2,9,11,17,18,28,29]).

As a result of the intuitionistic fuzzy sets given by Atanassov [3], which give the membership and nonmembership degrees to which an element belongs, dealing with incomplete and inaccurate information is more flexible and effective compared to Zadeh’s fuzzy sets [30].

Working under tthe name "intuitionistic" has sparked and doubts debate over the term’s applicability, particularly when dealing with of complete lattice L. Garcia and Rodabaugh [8] put an end to these doubts in 2005. They proved that in mathematics and applications, this word is inappropriate. They concluded that they work under the name "double".

The main contributions of the present paper are to further investigations into multi-granulation double fuzzy approximation spaces, mainly including double fuzzy upper and lower approximation operators with respect to multi-granulation double fuzzy approximation spaces. By using more than one pair of double fuzzy relations on U, two kinds of double fuzzy sets were introduced and the relationship between them was studied.

Throughout this paper, Let $U=\{x_1,x_2,...x_n\}$ be a nonempty and finite set of objects and $I=[0,1]$. A fuzzy set is a map from U to I. The set of all fuzzy sets on U is denoted by $I^U$. R is a fuzzy binary relation on U i.e. $R(x,y)\in [0,1]$ for any $x,y\in U$. The set of all fuzzy binary relation on U is denoted by $I^{U\times U}$.

Definition 1.1. [1] Let U and V be two arbitrary sets. A double fuzzy relation on $U\times V$ is a pair $(R,R^{*})$ of maps $R,R^{*}:U\times V\to I$ such that $R(x,y)\le 1-R^{*}(x,y)$ for all $(x,y)\in U\times V.$ If $R,R^{*}:U\times U\to I$, $(R,R^{*})$ is called a double fuzzy relation on U. $R(x,y)$ (resp. $R^{*}(x,y)$), referred to as the degree of relation (resp. non-relation ) between x and y.

Definition 1.2. [1] Let U be an arbitrary universal set and $(R,R^{*})$ a double fuzzy relation on U. Then for each fuzzy set $\lambda$ on $U,$ the pairs $({\underline{\mathcal{R}}}_R\lambda ,{\underline{\mathcal{R}}}_{R^{*}}^{*}\lambda ),({\overline{\mathcal{R}}}_R\lambda ,{\overline{\mathcal{R}}}_{R^{*}}^{*}\lambda )$ of maps ${\underline{\mathcal{R}}}_R\lambda ,{\underline{\mathcal{R}}}_{R^{*}}^{*}\lambda ,{\overline{\mathcal{R}}}_R\lambda ,{\overline{\mathcal{R}}}_{R^{*}}^{*}\lambda :U\to I$ are called double fuzzy lower approximation and double fuzzy upper approximation of a fuzzy set $\lambda ,$ respectively and are defined as follows: For all $x\in U,$ The quaternary $({\underline{\mathcal{R}}}_R\lambda ,{\underline{\mathcal{R}}}_{R^{*}}^{*}\lambda ,{\overline{\mathcal{R}}}_R\lambda ,{\overline{\mathcal{R}}}_{R^{*}}^{*}\lambda )$ is called double fuzzy rough set of $\lambda$. The pairs $({\underline{\mathcal{R}}}_R,{\underline{\mathcal{R}}}_{R^{*}}^{*}),({\overline{\mathcal{R}}}_R,{\overline{\mathcal{R}}}_{R^{*}}^{*})$ of operators ${\underline{\mathcal{R}}}_R,{\underline{\mathcal{R}}}_{R^{*}}^{*},{\overline{\mathcal{R}}}_R,{\overline{\mathcal{R}}}_{R^{*}}^{*}:I^U\to I^U$ are called double fuzzy lower approximation and double fuzzy upper approximation operators, respectively.

Definition 1.3. [1] For all $x,y\in U$ a double fuzzy relation $(R,R^{*})$ on U is called:

(1) Double fuzzy reflexive if $R(x,x)=1$ and $R^{*}(x,x)=0.$

(2) Double fuzzy transitive if $R(x,z)\ge {\bigvee }_{y\in U}(R(x,y)\wedge R(y,z))$ and $R^{*}(x,z)\le {\bigwedge }_{y\in U}(R^{*}(x,y)\vee R^{*}(y,z))\forall z\in U.$

(3) Double fuzzy symmetric if $R(x,y)=R(y,x)$ and $R^{*}(x,y)=R^{*}(y,x).$

Definition 2.1. Let U be an arbitrary universal set and $(R_1,R_1^{*})$ and $(R_2,R_2^{*})$ are double fuzzy relations on U. Then for each fuzzy set $\lambda$ on $U,$ the pairs $(\underline{{\mathcal{OR}}_{R_1+R_2}}\lambda ,\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda )$ and $(\overline{{\mathcal{OR}}_{R_1+R_2}}\lambda ,\overline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda )$ of maps $\underline{{\mathcal{OR}}_{R_1+R_2}}\lambda ,$$\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda ,$ $\overline{{\mathcal{OR}}_{R_1+R_2}}\lambda ,\overline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda :U\to I$ are called optimistic two-granulation double fuzzy lower approximation and optimistic two-granulation double fuzzy upper approximation of a fuzzy set $\lambda ,$ respectively and are defined as follows: For all $x\in U,$ The quaternary $(\underline{{\mathcal{OR}}_{R_1+R_2}}\lambda ,\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda ,\overline{{\mathcal{OR}}_{R_1+R_2}}\lambda ,\overline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda )$ is called optimistic two-granulation double fuzzy rough set of $\lambda$ (in short, OTGDFRS). The pairs $(\underline{{\mathcal{OR}}_{R_1+R_2}},\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}})$ and $(\overline{{\mathcal{OR}}_{R_1+R_2}},\overline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}})$ of operators $\underline{{\mathcal{OR}}_{R_1+R_2}},$$\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}},\overline{{\mathcal{OR}}_{R_1+R_2}},\overline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}:U\to I$ are called optimistic two-granulation double fuzzy lower approximation and optimistic two-granulation double fuzzy upper approximation operators, respectively.

The OTGDFRS approximations are defined by many separate pairs of double fuzzy relations, whereas the normal double fuzzy rough approximations are represented by those produced by only one pair of double fuzzy relation, as can be seen from the preceding definition. In fact, when $(R_1,R_1^{*})=(R_2,R_2^{*})$, the OTGDFRS degenerates into a double fuzzy rough set. To put it another way, a double fuzzy rough set model is a subset of the OTGDFRS.

Proposition 2.2. Let U be an arbitrary universal set and $(R_1,R_1^{*})$ and $(R_2,R_2^{*})$ be a double fuzzy relations on U. Then for each $\lambda \in I^U$ we obtain the following:

(1) $\underline{{\mathcal{OR}}_{R_1+R_2}}\lambda =\underline{{\mathcal{R}}_{R_1}}\lambda \vee \underline{{\mathcal{R}}_{R_2}}\lambda$ and $\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda =\underline{{\mathcal{R}}_{R_1^{*}}^{*}}\lambda \wedge \underline{{\mathcal{R}}_{R_2^{*}}^{*}}\lambda$.

(2) $\overline{{\mathcal{OR}}_{R_1+R_2}}\lambda =\overline{{\mathcal{R}}_{R_1}}\lambda \wedge \overline{{\mathcal{R}}_{R_2}}\lambda$ and $\overline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda =\overline{{\mathcal{R}}_{R_1^{*}}^{*}}\lambda \vee \overline{{\mathcal{R}}_{R_2^{*}}^{*}}\lambda$.

Proof. The proofs follow directly from Definition 1.2 and Definition 2.1.

Theorem 2.3. Let U be an arbitrary universal set and $(R_1,R_1^{*})$ and $(R_2,R_2^{*})$ be a double fuzzy relations on U. Then for each $\lambda \in I^U$ we obtain the following:

Proof. (1) For each $x\in U,\lambda \in I^U$ we have Hence, $\overline{{\mathcal{OR}}_{R_1+R_2}}\lambda \le \tilde{1}-\overline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda$. Similarly, $\underline{{\mathcal{OR}}_{R_1+R_2}}\lambda \ge \tilde{1}-\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda$.

(2) Since, for each $x\in U,\tilde{1}\left(x\right)=1,$ we obtain and Therefore, we obtain $\underline{{\mathcal{OR}}_{R_1+R_2}}\tilde{1}=\tilde{1}$ and $\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\tilde{1}=\tilde{0}.$

(3) It is similar to the proof of (2).

(4) For each $x\in U$, we have Thus, we obtain $\overline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}(\tilde{1}-\lambda )=\tilde{1}-\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda$. Similarly, we can prove $\overline{{\mathcal{OR}}_{R_1+R_2}}(\tilde{1}-\lambda )=\tilde{1}-\underline{{\mathcal{OR}}_{R_1+R_2}}\lambda$

(5) Similarly to that of (4).

Theorem 2.4. Let U be an arbitrary universal set and $(R_1,R_1^{*})$ and $(R_2,R_2^{*})$ be a double fuzzy relations on U. Then for each $\lambda ,\mu \in I^U$:

Proof. (1) For each $x\in U$ and $\lambda ,\mu \in I^U,$ we have

Also, for each $x\in U,$ we have

(2) Similar to (1).

(3) If $\lambda \le \mu ,$ then for all $y\in U,\lambda \left(y\right)\le \mu \left(y\right)$ we have and Form Equations (2.1) and (2.2) we have Therefore, $\underline{{\mathcal{OR}}_{R_1+R_2}}\lambda \le \underline{{\mathcal{OR}}_{R_1+R_2}}\mu$. Also, and Form Equations (2.3) and (2.4) we have Hence $\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda \ge \underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\mu$.

(4) Similar to (3).

(5) Since $\lambda \le \lambda \vee \mu$ and $\mu \le \lambda \vee \mu$, by (3) we have Therefore $\underline{{\mathcal{OR}}_{R_1+R_2}}\lambda \vee \underline{{\mathcal{OR}}_{R_1+R_2}}\mu \le \underline{{\mathcal{OR}}_{R_1+R_2}}(\lambda \vee \mu )$. Also, we have This implies that $\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda \wedge \underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\mu \ge \underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}(\lambda \vee \mu )$.

(6) Similar to (5).

Example 2.5. Let $U=\{x,y,z\}.$ Define $R_1,R_1^{*},R_2,R_2^{*}:U\times U\to I$ as follows:

Define $\lambda ,\mu \in I^U$ as follows:

Then,

Therefore, $\underline{{\mathcal{OR}}_{R_1+R_2}}(\lambda \wedge \mu )\ne \underline{{\mathcal{OR}}_{R_1+R_2}}\lambda \wedge \underline{{\mathcal{OR}}_{R_1+R_2}}\mu .$

Therefore, $\underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}(\lambda \wedge \mu )\ne \underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\lambda \vee \underline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}\mu$.

Theorem 2.6. Let $(R_1,R_1^{*})$ and $(R_2,R_2^{*})$ be a double fuzzy relations on an universal set U. Then the following statements are equivalent:

Proof.$\left(1\right)\Rightarrow \left(2\right)$ Let $(R_1,R_1^{*})$ and $(R_2,R_2^{*})$ are a double fuzzy reflexive relations. Then $R_i(x,x)=1$ and $R_i^{*}(x,x)=0$ for all $i\in \{1,2\}$ and $x\in U.$ Therefore, and $\left(2\right)\Rightarrow \left(1\right)$ Suppose that there exists some $x\in U$ such that $R_i(x,x)=a_i\ne 1$ and $R_i^{*}(x,x)=b_i\ne 0$ for all $i\in \{1,2\}$, then we can define fuzzy set ${\delta }_x:U⟶I$ as: Then and Therefore ${\delta }_x\nleqq \overline{{\mathcal{OR}}_{R_1+R_2}}{\delta }_x$ and $\tilde{1}-{\delta }_x\ngeqq \overline{{\mathcal{OR}}_{R_1^{*}+R_2^{*}}^{*}}{\delta }_x$. This is a contradiction. Hence, $R_i(x,x)=1$ and $R_i^{*}(x,x)=0$ for all $i\in \{1,2\}$ and $x\in U.$