A topological view on L -fuzzy soft sets: Connectedness degree

Abstract

In this study, we aim to introduce the notion of a parameterized connectedness degree of an L-fuzzy soft set in L-fuzzy (E, K)-soft topological spaces. We first define the notion of a parameterized separatedness degree of L-fuzzy soft sets in L-fuzzy (E, K)-soft topological spaces in terms of closure operators. Then we introduce the notion of a parameterized connectedness degree of L-fuzzy soft sets and generalize some connectedness properties, well known in general topology, to L-fuzzy (E, K)-soft topological spaces.

Keywords

Fuzzy soft set fuzzy soft topology separatedness degree connectedness degree

1 Introduction

In 1999, Molodtsov [19] proposed a completely new concept called soft set theory to model uncertainty, which associates a set with a set of parameters. Later, Maji et al. [17] introduced the concept of fuzzy soft sets which combines fuzzy sets and soft sets. Soft set and fuzzy soft set theories have a rich potential for applications in several directions [10–12 , 29]. So far, many scholars working in diverse areas have discussed the soft set and fuzzy soft set theories in different aspects. Topology of soft sets was defined by Shabir and Naz [22] and topology of fuzzy soft sets was defined by Tanay and Kandemir [26]. Parameterized soft topology and parameterized fuzzy soft topology was considered by Çetkin and Aygün [3] and Aygünoğlu et al. [2], respectively. Later the topological structures of lattice valued fuzzy soft sets were introduced and investigated with their fundamentals by Çetkin et al. [5 –8].

Connectivity is one of the most important notions in general topology (see [27]). It has been generalized to topology of L-fuzzy sets in terms of many forms. Sostak [25] first introduced a notion of connectedness degree by means of the level [0, 1]-topological spaces in [0, 1]-fuzzy topological spaces. Then Yue and Fang [28] presented connectivity in [0, 1]-fuzzy topological spaces which was defined for the whole space. Shi [24] defined the notions of separatedness degree and connectedness degree of L-fuzzy sets in L-fuzzy topological spaces. In addition, the notion of fuzzy soft connected sets take an important place in fuzzy soft topological spaces. In fuzzy soft setting, connectedness has been introduced by Karataş et al. [14]. Some types of separated fuzzy soft sets and fuzzy soft connected sets has been also introduced by Kandil et al. [13].

In the present paper, we intend to introduce the notions of separatedness degree and the connectedness degree of L-fuzzy soft sets with respect to parameters in the L-fuzzy (E, K)-soft topological spaces. So, we extend the results obtained by Shi [24] to the fuzzy soft universe. We first introduced the degree of separatedness of two L-fuzzy soft sets and examine its basic properties. Then we define the connectedness degree of an L-fuzzy soft set w.r.t a parameter and study its characteristics and elementary properties. We also give the relations between the proposed definitions and known definitions from fuzzy setting and fuzzy soft setting.

2 Preliminaries

Throughout this paper, X refers to an initial universe and L = (L, ∨ , ∧ , ′) denotes a completely distributive DeMorgan algebra with the smallest element 0_L and the largest element 1_L . Let a, b, c be elements in L . An element a in L is said to be coprime if a ≤ b ∨ c implies that a ≤ b or a ≤ c . The set of all coprime elements of L is denoted by c (L) . We say a is way below (wedge below) b, in symbols, a ⪡ b (a ⊲ b) or b ⪢ a (b ⊳ a) , if for every directed (arbitrary) subset D ⊆ L, ∨D ≥ b implies a ≤ d for some d ∈ D . Clearly if a ∈ L is coprime, then a ⪡ b if and only if a ⊲ b . A complete lattice L is completely distributive if and only if b = ⋁ {a ∈ L ∣ a ⊲ b} for each b ∈ L . For any b ∈ L, define β (b) = ⋁ {a ∈ L ∣ a ⊲ b} . The wedge below operation in a completely distributive lattice has an interpolation property, this means a ⊲ b implies there exists c ∈ L such that a ⊲ c ⊲ b . Some properties of the map β can be found in [9].

Let L^X denote the set of all L-fuzzy sets on X, and let E and K be the arbitrary nonempty sets viewed on the sets of parameters.

Definition 2.1. [21] f is called an L-fuzzy soft set on X, where f is a mapping from E into L^X, i.e., f_e : = f (e) is an L-fuzzy set on X, for each e ∈ E .

The family of all L-fuzzy soft sets on X is denoted by (L^X) ^E .

Definition 2.2. [20, 21] Let f and g be two L-fuzzy soft sets on X, then

we say that f is an L-fuzzy soft subset of g and write f ⊑ g if f_e ≤ g_e, for each e ∈ E. f and g are called equal if f ⊑ g and g ⊑ f.

the union of f and g is an L-fuzzy soft set h = f ⊔ g, where h_e = f_e ∨ g_e, for each e ∈ E.

the intersection of f and g is an L-fuzzy soft set h = f ⊓ g, where h_e = f_e ∧ g_e, for each e ∈ E.

the complement of an L-fuzzy soft set f is denoted by f′, where f′ : E → L^X is a mapping given by $f_{e}^{'} = (f_{e})^{'}$ , for each e ∈ E. Clearly (f′) ′ = f.

Definition 2.3. [21] (1) (Null L-fuzzy soft set) An L-fuzzy soft set f on X is called a null L-fuzzy soft set and denoted by $\tilde{0}$ , if f_e (x) =0_L, for each e ∈ E, x ∈ X.

(2) (Absolute L-fuzzy soft set) An L-fuzzy soft set f on X is called an absolute L-fuzzy soft set and denoted by $\tilde{1}$ , if f_e (x) =1_L, for each e ∈ E, x ∈ X.

Clearly $(\tilde{1})^{'} = \tilde{0}$ and $(\tilde{0})^{'} = \tilde{1}$ .

Proposition 2.1. [1] Let Δ be an index set, f, g, f_i, g_i ∈ (L^X) ^E, for all i ∈ Δ, then the following properties are satisfied:

f ⊓ (⊔ _i∈Δg_i) = ⊔ _i∈Δ (f ⊓ g_i) and f ⊔ (⊓ _i∈Δg_i) = ⊓ _i∈Δ (f ⊔ g_i)

${(⊓_{i \in Δ} f_{i})}^{'} = ⊔_{i \in Δ} f_{i}^{'}$ , ${(⊔_{i \in Δ} f_{i})}^{'} = ⊓_{i \in Δ} f_{i}^{'}$ .

Definition 2.4. [15] Let φ : X₁ → X₂ and ψ : E₁ → E₂ be two functions, where E₁ and E₂ are parameter sets for the crisp sets X₁ and X₂, respectively. Then the pair φ_ψ is called an L-fuzzy soft mapping from X₁ to X₂.

Definition 2.5. [15] Let f and g be two L-fuzzy soft sets over X₁ and X₂, respectively and let φ_ψ be an L-fuzzy soft mapping from X₁ to X₂.

(1) The image of f under the L-fuzzy soft mapping φ_ψ, denoted by φ_ψ (f), is the L-fuzzy soft set on X₂ defined by for all k ∈ E₂, y ∈ X₂, $\begin{matrix} φ (f)_{k} (y) \\ = {\begin{matrix} ⋁_{φ (x) = y} ⋁_{ψ (a) = k} f_{a} (x), & if x \in φ^{- 1} (y), a \in ψ^{- 1} (k) \\ 0, & otherwise . \end{matrix} \end{matrix}$

(2) The pre-image of g under the L-fuzzy soft mapping φ_ψ, denoted by $φ_{ψ}^{- 1} (g)$ , is the L-fuzzy soft set on X₁ defined by for all e ∈ E₁, x ∈ X₁, $φ_{ψ}^{- 1} (g)_{e} (x) = g_{ψ (e)} (φ (x)) .$

If φ and ψ is injective (surjective), then φ_ψ is said to be injective (surjective).

Definition 2.6. [4] Let x ∈ X and λ : E → c (L) be a function. Then the L-fuzzy soft set defined as follows is called an L-fuzzy soft point, and denoted by x^λ. For all e ∈ E and y ∈ X, $x_{e}^{λ} (y) = {\begin{matrix} λ (e), & if y = x, \\ 0_{L}, & otherwise \end{matrix}$

An L-fuzzy soft point x^λ is said to be belonging to an L-fuzzy soft set f and denoted by x^λ ∈ f if λ (e) ≤ f_e (x), for each e ∈ E .

The set of all nonzero coprime elements of (L^X) ^E is denoted by c ((L^X) ^E). It is noted that c ((L^X) ^E) is exactly the set of all L-fuzzy soft points.

Definition 2.7. [2] A mapping τ : K → L^{(L^X)^E} is called an L-fuzzy (E, K)-soft topology on X if it satisfies the following conditions for each k ∈ K.

$τ_{k} (\tilde{0}) = τ_{k} (\tilde{1}) = 1_{L} .$

τ_k (f ⊓ g) ≥ τ_k (f) ∧ τ_k (g) , for all f, g ∈ (L^X) ^E .

τ_k (⊔ _i∈Δf_i) ≥ ⋀ _i∈Δτ_k (f_i) , for all f_i ∈ (L^X) ^E, i ∈ Δ .

Then the pair (X, τ) is called an L-fuzzy (E, K)-soft topological space. The value τ_k (f) is interpreted as the degree of openness of an L-fuzzy soft set f with respect to the parameter k ∈ K .

Let τ¹ and τ² be L-fuzzy (E, K)-soft topologies on X . We say that τ¹ is finer than τ² (τ² is coarser than τ¹), denoted by τ² ≤ τ¹, if $τ_{k}^{2} (f) \leq τ_{k}^{1} (f)$ for each k ∈ K and f ∈ (L^X) ^E .

Example 2.1. [7] Let L = {0, a, b, 1} be a diamond-type lattice with the order reversing involution ^′ : L → L defined by 0′ = 1, 1′ = 0, a′ = a and b′ = b. Then (L, ≤ , ′) is a completely distributive DeMorgan algebra. Let X = {x, y}, E = K = {1, 2} and f, g be two L-fuzzy soft sets defined as follows: f₁ (x) = f₂ (y) = g₁ (y) = g₂ (x) = a and f₁ (y) = f₂ (x) = g₁ (x) = g₂ (y) = b. Define a mapping τ : K → L^{(L^X)^E} as follows: $τ_{e} (h) = {\begin{matrix} 1, & if h \in {\tilde{0}, \tilde{1}}, \forall e \in E \\ a, & if h = f, e = 1 or h = g, e = 2 \\ b, & if h = g, e = 1 or h = f, e = 2 \\ 0, & otherwise . \end{matrix}$

Then it is easy to testify that the mapping τ is an L-fuzzy (E, K)-soft topology on X .

Definition 2.8. [2] Let (X₁, τ¹) be an L-fuzzy (E₁, K₁)-soft topological space and (X₂, τ²) be an L-fuzzy (E₂, K₂)-soft topological space. Let φ : X₁ → X₂, ψ : E₁ → E₂ and η : K₁ → K₂ be functions. Then φ_ψ,η : (X₁, τ¹) → (X₂, τ²) is called continuous if $τ_{k}^{1} (φ_{ψ}^{- 1} (g)) \geq τ_{η (k)}^{2} (g)$ for all g ∈ (L^{X
₂}) ^{E
₂}, k ∈ K₁ .

Definition 2.9. [4] An L-fuzzy (E, K)-soft quasi-coincident neighborhood (for short, q-nhood) system on X is a set $Q = {Q_{x^{λ}} ∣ x^{λ} \in c ((L^{X})^{E})}$ of maps $Q_{x^{λ}} : K \times (L^{X})^{E} \to L$ satisfying the following conditions for each k ∈ K and f, g ∈ (L^X) ^E,

$Q_{x^{λ}} (k, \tilde{1}) = 1_{L}, Q_{x^{λ}} (k, \tilde{0}) = 0_{L} .$

If $Q_{x^{λ}} (k, f) \neq 0_{L}$ , then x^λ ⋢ f′ .

$Q_{x^{λ}} (k, f ⊓ g) = Q_{x^{λ}} (k, f) \land Q_{x^{λ}} (k, g) .$

$Q_{x^{λ}} (k, f) = ⋁_{x^{λ} ⋢ g ⊒ f^{'}} ⋀_{y^{μ} ⋢ g} Q_{y^{μ}} (k, g^{'}) .$

Proposition 2.2. [4] Let τ be an L-fuzzy (E, K)-soft topology on X . Define the mapping $Q_{x^{λ}}^{τ} : K \times (L^{X})^{E} \to L$ as follows: for each k ∈ K, f ∈ (L^X) ^E and x^λ ∈ c ((L^X) ^E) , $Q_{x^{λ}}^{τ} (k, f) = ⋁_{x^{λ} ⋢ g ⊒ f^{'}} τ_{k} (g^{'}) .$

Then the set of $Q^{τ} = {Q_{x^{λ}}^{τ} ∣ x^{λ} \in c ((L^{X})^{E})}$ is an L-fuzzy (E, K)-soft q-nhood system on X, called induced L-fuzzy (E, K)-soft q-nhood system by τ .

Definition 2.10. [14] Two non-null fuzzy soft sets f and g are said to be fuzzy soft Q-separated in a topology of fuzzy soft sets $(X, T)$ if $Fcl (f) ⊓ g = f ⊓ Fcl (g) = \tilde{0}$ , where Fcl (f) denotes the closure of the fuzzy soft set f .

Definition 2.11. [13] In a topology of fuzzy soft sets $(X, T)$ , f is called an FSC_M-disconnected fuzzy soft set if there exist two non-null fuzzy soft Q-separated sets g, h over X such that f = g ⊔ h . Otherwise f is called an FSC_M-connected fuzzy soft set. If one takes $\tilde{1}$ instead of f, then the topology of fuzzy soft sets $(X, T)$ is called an FSC_M-connected space.

The details of the connectedness types of fuzzy soft sets and the relations between them can be found in the paper [13].

3 Separatedness degrees of L-fuzzy soft sets

In this section, we first propose the definition of a fuzzy soft closure operator. Then we introduce the separatedness degree of two L-fuzzy soft sets with respect to parameters by using the closure operator and discuss some of its fundamental properties.

Definition 3.1. An L-fuzzy (E, K)-soft closure operator on X is a mapping cl : K × (L^X) ^E → L^{c((L^X)^E)} which satisfies the following conditions: for each k ∈ K,

cl (k, f) (x^λ) = ⋀ _μ⊲λcl (k, f) (x^μ), for all x^λ ∈ c ((L^X) ^E) .

$cl (k, \tilde{0}) (x^{λ}) = 0_{L}$ for any x^λ ∈ c ((L^X) ^E) .

cl (k, f) (x^λ) =1_L for any x^λ ⊑ f .

cl (k, f ⊔ g) = cl (k, f) ∨ cl (k, g) .

σ_a (cl (k, ⊔ σ_a (cl (k, f)))) ⊆ σ_a (cl (k, f)) ,

where $σ_{a} (cl (k, f)) = {x^{λ} \in c ((L^{X})^{E}) ∣ cl (k, f) (x^{λ}) \geq a} .$

The quantity cl (k, f) (x^λ) is called the degree to which x^λ belongs to the parameterized closure of f .

Example 3.1. For a parameterized L-soft topological space (X, τ) in the sense of [3], the closure operator cl : K × (L^X) ^E → (L^X) ^E satisfies the following conditions:

$cl (k, \tilde{0}) = \tilde{0}$ , (2) f ⊑ cl (k, f), (3) cl (k, f ⊔ g) = cl (k, f) ⊔ cl (k, g), (c) cl (k, cl (k, f)) = cl (k, f) .

Now we consider such a mapping Cl (k, f) : c ((L^X) ^E) →2 defined by: $Cl (k, f) (x^{λ}) = {\begin{matrix} 1, & if x^{λ} ⊑ cl (k, f) \\ 0, & otherwise . \end{matrix}$

It is easy to check that the mapping defined as above satisfies the conditions of Definition 3.1.

Theorem 3.1. Let τ be an L-fuzzy (E, K)-soft topology on X and let $Q = {Q_{x^{λ}} ∣ x^{λ} \in c ((L^{X})^{E})}$ be the q-nhood system induced by τ. Define a mapping C : K × (L^X) ^E → L^{c((L^X)^E)} by $C (k, f) (x^{λ}) = (Q_{x^{λ}} (k, f^{'}))^{'} .$

Then the mapping C is an L-fuzzy (E, K)-soft closure operator on X, which is called the L-fuzzy (E, K)-soft closure operator induced by τ .

Proof. (C1)-(C3) are obvious and (C4) holds from (FSQ3). So it is sufficient to prove (C5).

(C5) Let x^λ ∉ σ_a (C (k, f)) for some a ∈ c (L) . Then by (FSQ4), we infer that $\begin{matrix} ⋁_{x^{λ} ⋢ g ⊒ f} ⋀_{y^{μ} ⋢ g} Q_{y^{μ}} (g^{'}) \\ = Q_{x^{λ}} (f^{'}) = (C (k, f) (x^{λ}))^{'} ≰ a^{'} . \end{matrix}$

Hence there exists g ∈ (L^X) ^E s.t. x^λ ⋢ g ⊒ f and ⋀_y^μ⋢g (C (k, g) (y^μ)) ′ = ⋀ _y^μ⋢gQ_{y
^μ} (k, g′) ≰ a′ . This implies that y^μ ∉ σ_a (C (k, g)) holds for any y^μ ⋢ g. By (C4), we have $x^{λ} ⋢ g ⊒ ⊔ σ_{a} (C (k, g)) ⊒ ⊔ σ_{a} (C (k, f)) .$

Therefore, $\begin{matrix} {(C (k, ⊔ σ_{a} (C (k, f))) (x^{λ}))}^{'} \\ = Q_{x^{λ}} (k, (⊔ σ_{a} (C (k, f)))^{'}) \\ = ⋁_{x^{λ} ⋢ g ⊒ ⊔ σ_{a} (C (k, f))} ⋀_{y^{μ} ⋢ g} Q_{y^{μ}} (k, g^{'}) ≰ a^{'} . \end{matrix}$

This shows that x^λ ∉ σ_a (C (k, ⊔ σ_a (C (k, f)))) which completes the proof.

The next corollary gives a representation of the L-fuzzy (E, K)-soft closure operator by means of an L-fuzzy (E, K)-soft topology.

Corollary 3.1. Let τ be an L-fuzzy (E, K)-soft topology on X and let the mapping C : K × (L^X) ^E → L^{c((L^X)^E)} be the L-fuzzy (E, K)-soft closure operator induced by τ . Then for each x^λ ∈ c ((L^X) ^E) , f ∈ (L^X) ^E and k ∈ K, the following equality is satisfied. $\begin{matrix} C (k, f) (x^{λ}) \\ = ⋀ {(τ_{k} (g^{'}))^{'} ∣ g \in (L^{X})^{E}, x^{λ} ⋢ g ⊒ f} \\ = ⋀_{x^{λ} ⋢ g ⊒ f} (τ_{k} (g^{'}))^{'} . \end{matrix}$

Proof. It is clear from Proposition 2.2 and Theorem 3.1.

Definition 3.2. Let (X, τ) be an L-fuzzy (E, K)-soft topological space. Define a map S : K × (L^X) ^E × (L^X) ^E → L for all k ∈ K and f, g ∈ (L^X) ^E as follows: $\begin{matrix} S (k, f, g) & = & (⋀ {(C (k, g) (x^{λ}))^{'} ∣ x^{λ} ⊑ f}) \\ \land (⋀ {(C (k, f) (y^{μ}))^{'} ∣ y^{μ} ⊑ g}) \\ S (k, f, g) & = & (⋀_{x^{λ} ⊑ f} (C (k, g) (x^{λ}))^{'}) \\ \land (⋀_{y^{μ} ⊑ g} (C (k, f) (y^{μ}))^{'}) \end{matrix}$

Then the value S (k, f, g) is said to be the separatedness degree of f and g with respect to parameter k.

Remark 3.1. If the parameter sets E and K are singletons, then Definition 3.2 coincides with Definition 3.1 in [24].

Proposition 3.1. Let $T = {T_{k} : (L^{X})^{E} \to 2 ∣ k \in K}$ be a topology of fuzzy soft sets on X in the sense of Çetkin et al. [3], and f, g ∈ (L^X) ^E. Then S (k, f, g) =1_L if and only if f, g are fuzzy soft Q-separated in $(X, T_{k})$ , for each k ∈ K .

Lemma 3.1. Let (X, τ) be an L-fuzzy (E, K)-soft topological space and f, g ∈ (L^X) ^E . If $f ⊓ g \neq \tilde{0}$ , then the parameterized separatedness degree S (k, f, g) =0_L, for all k ∈ K .

Proof. From $f ⊓ g \neq \tilde{0}$ , we can take x^λ ∈ c ((L^X) ^E) such that x^λ ⊑ f ⊓ g. Then the following is true for each k ∈ K, $\begin{matrix} S (k, f, g) \\ = (⋀_{x^{λ} ⊑ f} (C (k, g) (x^{λ}))^{'}) \land (⋀_{x^{λ} ⊑ g} (C (k, f) (x^{λ}))^{'}) \\ \leq (C (k, g) (x^{λ}))^{'} \land (C (k, f) (x^{λ}))^{'} \\ = 1_{L}^{'} \land 1_{L}^{'} = 0_{L} . \end{matrix}$

Lemma 3.2. Let (X, τ) be an L-fuzzy (E, K)-soft topological space and f, g, u, v ∈ (L^X) ^E . If f ⊑ g and u ⊑ v, then S (k, g, v) ≤ S (k, f, u) for each k ∈ K .

Proof. If f ⊑ g and u ⊑ v, then C (k, f) ≤ C (k, g) and C (k, u) ≤ C (k, v) , for all k ∈ K . Hence the following inequality is verified as: for each k ∈ K, $\begin{matrix} S (k, g, v) \\ = (⋀_{x^{λ} ⊑ g} (C (k, v) (x^{λ}))^{'}) \land (⋀_{y^{μ} ⊑ v} (C (k, g) (y^{μ}))^{'}) \\ \leq (⋀_{x^{λ} ⊑ g} (C (k, u) (x^{λ}))^{'}) \land (⋀_{y^{μ} ⊑ v} (C (k, f) (y^{μ}))^{'}) \\ \leq (⋀_{x^{λ} ⊑ f} (C (k, u) (x^{λ}))^{'}) \land (⋀_{y^{μ} ⊑ u} (C (k, f) (y^{μ}))^{'}) \\ = S (k, f, u) \end{matrix}$

Lemma 3.3. Let (X, τ) be an L-fuzzy (E, K)-soft topological space and f, g ∈ (L^X) ^E, k ∈ K and a ∈ c (L). Then the following is satisfied

(S (k, f, g)) ′≱a ⇔ ∃ u, v ∈ (L^X) ^E s.t. $u ⊒ f, v ⊒ g, u ⊓ g = v ⊓ f = \tilde{0}$ and (τ_k (u′)) ′ ∨ (τ_k (v′)) ′≱a .

Proof. Suppose that (S (k, f, g)) ′≱a . Then (S (k, f, g)) ′≱b for some b ∈ β^* (a) . This implies $⋁_{x^{λ} ⊑ f} (C (k, g) (x^{λ})) \lor ⋁_{y^{μ} ⊑ g} (C (k, f) (y^{μ})) ≱ b$

Furthermore, we have $⋁_{x^{λ} ⊑ f} ⋀_{x^{λ} ⋢ v ⊒ g} (τ_{k} (v^{'}))^{'} \lor ⋁_{y^{μ} ⊑ g} ⋀_{y^{μ} ⋢ u ⊒ f} (τ_{k} (u^{'}))^{'} ≱ b .$

Hence for any x^λ ⊑ f and for any y^μ ⊑ g, there are u_{y
^μ}, v_{x
^λ} ∈ (L^X) ^E such that x^λ ⋢ v_{x
^λ} ⊒ g, y^μ ⋢ u_{y
^μ} ⊒ f and $(τ_{k} (u_{y^{μ}}^{'}))^{'} \lor (τ_{k} (v_{x^{λ}}^{'}))^{'} ≱ b .$

Let v = ⊓ _x^λ⊑fv_{x
^λ} and u = ⊓ _y^μ⊑gu_{y
^μ} . Then obviously, we have that $u ⊒ f, v ⊒ g, u ⊓ g = v ⊓ f = \tilde{0}$ and $\begin{matrix} (τ_{k} (u^{'}))^{'} \lor (τ_{k} (v^{'}))^{'} \\ = {(τ_{k} (⊔_{y^{μ} ⊑ g} u_{y^{μ}}^{'}))}^{'} \lor {(τ_{k} (⊔_{x^{λ} ⊑ f} v_{x^{λ}}^{'}))}^{'} \\ \leq ⋁_{y^{μ} ⊑ g} (τ_{k} (u_{y^{μ}}^{'}))^{'} \lor ⋁_{x^{λ} ⊑ f} (τ_{k} (v_{x^{λ}}^{'}))^{'} ≱ a . \end{matrix}$

Conversely, if there exist u, v ∈ (L^X) ^E such that $u ⊒ f, v ⊒ g, u ⊓ g = v ⊓ f = \tilde{0}$ and (τ_k (u′)) ′ ∨ (τ_k (v′)) ′≱a .

Then by the following $\begin{matrix} (S (k, f, g))^{'} \\ = ⋁_{x^{λ} ⊑ f} C (k, g) (x^{λ}) \lor ⋁_{y^{μ} ⊑ g} C (k, f) (y^{μ}) \\ = ⋁_{x^{λ} ⊑ f} ⋀_{x^{λ} ⋢ w ⊒ g} (τ_{k} (w^{'}))^{'} \lor ⋁_{y^{μ} ⊑ g} ⋀_{y^{μ} ⋢ h ⊒ f} (τ_{k} (h^{'}))^{'} \\ \leq (τ_{k} (u^{'}))^{'} \lor (τ_{k} (v^{'}))^{'}, \end{matrix}$

we obtain the desired claim that $(S (k, f, g))^{'} ≱ a .$

4 Connectedness degree of L-fuzzy soft sets

In this section, we introduce the connectedness degree of an L-fuzzy soft set with respect to parameters. Then we generalize some well known connectedness properties to the fuzzy soft universe.

Definition 4.1. Let (X, τ) be an L-fuzzy (E, K)-soft topological space, h ∈ (L^X) ^E and k ∈ K. Define a mapping Con : K × (L^X) ^E → L as follows: $\begin{matrix} Con (k, h) \\ = ⋀ {(S (k, f, g))^{'} ∣ f, g \in (L^{X})^{E} \ {\tilde{0}}, \\ h = f ⊔ g} . \\ Con (k, h) \\ = ⋀ {⋁_{x^{λ} ⊑ f} C (k, g) (x^{λ}) \lor ⋁_{y^{μ} ⊑ g} C (k, f) (y^{μ}) ∣ \\ f, g \in (L^{X})^{E} \ {\tilde{0}}, h = f ⊔ g} . \end{matrix}$

Then the value Con (k, h) is said to be the connectedness degree of an L-fuzzy soft set h with respect to k.

Example 4.1. Let X = {x, y} , L = [0, 1] and E = {e} , K = {k₁, k₂}. Define L-fuzzy soft sets f and g by f_e (x) =0.5, f_e (y) =0, g_e (x) =0, g_e (y) =0.5, respectively. Let the mapping τ : K → L^{(L^X)^E} be given as follows: $\begin{matrix} τ_{k_{1}} (h) = {\begin{matrix} 1, & if h \in {\tilde{0}, \tilde{1}, \tilde{0.5}}, \\ 0.5, & if h \in {f^{'}, g^{'}}, \\ 0, & otherwise \end{matrix} \\ τ_{k_{2}} (h) = {\begin{matrix} 1, & if h \in {\tilde{0}, \tilde{1}}, \\ 0, & otherwise . \end{matrix} \end{matrix}$

Then τ is an L-fuzzy (E, K)-soft topology on X . It is easy to verify that $Con (k_{1}, \tilde{α}) = 0.5$ for any α ∈ (0, 0.5] and $Con (k_{1}, \tilde{α}) = 1,$ for any α ∈ (0.5, 1]. (Here $\tilde{α}$ denotes the constant fuzzy soft set.) Also, Con (k₂, h) =1 for any h ∈ (L^X) ^E .

Remark 4.1. If the parameter sets E and K are singletons, then Definition 4.1 coincides with Definition 4.1 in [24].

Proposition 4.1. Let $T = {T_{k} : (L^{X})^{E} \to 2 ∣ k \in K}$ be a topology of fuzzy soft sets on X in the sense of Çetkin et al. [3], and f ∈ (L^X) ^E. Then Con (k, f) =1 if and only if f is FSC_M-connected fuzzy soft set in $(X, T_{k}),$ for each k ∈ K .

Theorem 4.1. Let (X, τ) be an L-fuzzy (E, K)-soft topological space, g ∈ (L^X) ^E and k ∈ K. Then the following equality is valid:

$Con (k, g) = ⋀ {(τ_{k} (u^{'}))^{'} \lor (τ_{k} (v^{'}))^{'} ∣ g ⊓ u \neq \tilde{0}, g ⊓ v \neq \tilde{0}, g ⊓ u ⊓ v = \tilde{0} and g ⊑ u ⊔ v}$ .

Proof. On one hand we have the following; $\begin{matrix} Con (k, g) \\ = ⋀ {⋁_{x^{λ} ⊑ u} C (k, v) (x^{λ}) \lor ⋁_{y^{μ} ⊑ v} C (k, u) (y^{μ}) \\ ∣ u, v \in (L^{X})^{E} \ {\tilde{0}}, g = u ⊔ v} \\ = ⋀ {⋁_{x^{λ} ⊑ u} ⋀_{x^{λ} ⋢ f ⊒ v} (τ_{k} (f^{'}))^{'} \lor ⋁_{y^{μ} ⊑ v} ⋀_{y^{μ} ⋢ h ⊒ u} (τ_{k} (h^{'}))^{'} \\ ∣ u, v \in (L^{X})^{E} \ {\tilde{0}}, g = u ⊔ v} \\ = ⋀ {⋁_{x^{λ} ⊑ g ⊓ u} ⋀_{x^{λ} ⋢ f ⊒ g ⊓ v} (τ_{k} (f^{'}))^{'} \\ \lor ⋁_{y^{μ} ⊑ g ⊓ v} ⋀_{y^{μ} ⋢ h ⊒ g ⊓ u} (τ_{k} (h^{'}))^{'} ∣ g ⊓ u \neq \tilde{0}, \\ g ⊓ v \neq \tilde{0} and g ⊑ u ⊔ v} \\ \leq ⋀ {⋁_{x^{λ} ⊑ g ⊓ u} (τ_{k} (v^{'}))^{'} \lor ⋁_{y^{μ} ⊑ g ⊓ v} (τ_{k} (u^{'}))^{'} \\ ∣ g ⊓ u \neq \tilde{0}, g ⊓ v \neq \tilde{0}, g ⊓ u ⊓ v = \tilde{0}, g ⊑ u ⊔ v} \\ = ⋀ {(τ_{k} (v^{'}))^{'} \lor (τ_{k} (u^{'}))^{'} ∣ g ⊓ u \neq \tilde{0}, g ⊓ v \neq \tilde{0}, \\ g ⊓ u ⊓ v = \tilde{0} and g ⊑ u ⊔ v} . \end{matrix}$

On the other hand, in order to prove the inverse implication, we suppose that Con (k, g) ≱a where a ∈ c (L). Then there exist $u, v \in (L^{X})^{E} \ {\tilde{0}}$ such that g ⊑ u ⊔ v and (S (k, u, v)) ′≱a . By Lemma 3.3, we know that there exist f, h ∈ (L^X) ^E such that $\begin{matrix} f ⊒ u, h ⊒ v, f ⊓ v = h ⊓ u = \tilde{0} and \\ (τ_{k} (f^{'}))^{'} \lor (τ_{k} (h^{'}))^{'} ≱ a . \end{matrix}$

Obviously, $g ⊓ f \neq \tilde{0}, g ⊓ h \neq \tilde{0}, g ⊓ f ⊓ h = \tilde{0} and g ⊑ f ⊔ h .$

Hence we have $\begin{matrix} ⋀ {(τ_{k} (v^{'}))^{'} \lor (τ_{k} (u^{'}))^{'} ∣ g ⊓ u \neq \tilde{0}, g ⊓ v \neq \tilde{0}, \\ g ⊓ u ⊓ v = \tilde{0}, g ⊑ u ⊔ v} ≱ a . \end{matrix}$

Therefore, $\begin{matrix} Con (k, g) \geq ⋀ {(τ_{k} (v^{'}))^{'} \lor (τ_{k} (u^{'}))^{'} ∣ g ⊓ u \neq \tilde{0}, \\ g ⊓ v \neq \tilde{0}, g ⊓ u ⊓ u = \tilde{0} and g ⊓ u ⊔ v} . \end{matrix}$

This completes the proof.

Corollary 4.1. Let (X, τ) be an L-fuzzy (E, K)-soft topological space. Then the following is satisfied for each k ∈ K; $\begin{matrix} Con (k, \tilde{1}) \\ = ⋀ {(τ_{k} (u^{'}))^{'} \lor (τ_{k} (v^{'}))^{'} ∣ u \neq \tilde{0}, v \neq \tilde{0}, \\ u ⊓ v = \tilde{0} and \tilde{1} = u ⊔ v} \\ = ⋀ {(τ_{k} (u))^{'} \lor (τ_{k} (v))^{'} ∣ u \neq \tilde{0}, v \neq \tilde{0}, \\ u ⊓ v = \tilde{0} and \tilde{1} = u ⊔ v} . \end{matrix}$

Theorem 4.2. Con (k, x^λ) =1_L, for each k ∈ K and x^λ ∈ c ((L^X) ^E).

Proof. From Theorem 4.1, the following is true $\begin{matrix} Con (k, x^{λ}) \\ = ⋀ {(τ_{k} (u^{'}))^{'} \lor (τ_{k} (v^{'}))^{'} ∣ x^{λ} ⊓ u \neq \tilde{0}, \\ x^{λ} ⊓ v \neq \tilde{0}, x^{λ} ⊓ u ⊓ v = \tilde{0} and x^{λ} ⊑ u ⊔ v} \\ = ⋀ \emptyset = 1_{L} . \end{matrix}$

Theorem 4.3. Let (X, τ) be an L-fuzzy (E, K)-soft topological space. Then for any k ∈ K and g ∈ (L^X) ^E, the following inequality is valid: $⋁_{r \in c (L)} Con (k, ⊔ σ_{r} (C (k, g))) \geq Con (k, g) .$

Proof. Let Con (k, g) ≥ a for any a ∈ c (L) . Now we prove $⋁_{r \in c (L)} Con (k, ⊔ σ_{r} (C (k, g))) \geq a .$

Suppose that $⋁_{r \in c (L)} Con (k, ⊔ σ_{r} (C (k, g))) ≱ a .$

Then $Con (k, ⊔ σ_{a} (C (k, g))) ≱ a .$

By Theorem 4.1, we know that there exist u, v ∈ (L^X) ^E such that $\begin{matrix} (⋁ σ_{a} (C (k, g))) ⊓ u \neq \tilde{0}, (⋁ σ_{a} (C (k, g))) \\ ⊓ v \neq \tilde{0}, (⋁ σ_{a} (C (k, g))) ⊓ u ⊓ v = \tilde{0} \end{matrix}$ and $(τ_{k} (u^{'}))^{'} \lor (τ_{k} (v^{'}))^{'} ≱ a .$

By $(⋁ σ_{a} (C (k, g))) ⊓ u \neq \tilde{0},$ we know that there exists x^λ ⊑ u such that C (k, g) (x^λ) ≥ a .

Furthermore by $(⋁ σ_{a} (C (k, g))) ⊓ u ⊓ v = \tilde{0},$ we obtain x^λ ⋢ v .

Now we prove $g ⊓ u \neq \tilde{0} .$ In fact, if $g ⊓ u = \tilde{0},$ then by g ⊑ ⋁ σ_a (C (k, g)) ⊑ u ⊔ v we have g ⊑ v, hence it follows that $a \leq C (k, g) (x^{λ}) = ⋀_{x^{λ} ⋢ h ⊒ g} (τ_{k} (h^{'}))^{'} \leq (τ_{k} (v^{'}))^{'} .$

This contradicts with a ≰ (τ_k (v′)) ′ .

Analogously, we can prove $g ⊓ v \neq \tilde{0} .$ Thus by $\begin{matrix} g ⊓ u \neq \tilde{0}, g ⊓ v \neq \tilde{0}, g ⊓ u ⊓ v = \tilde{0}, \\ g ⊑ u ⊔ v and τ_{k} (u^{'}))^{'} \lor (τ_{k} (v^{'}))^{'} ≱ a, \end{matrix}$ and by Theorem 4.1, we know that Con (k, g) ≱a which contradicts Con (k, g) ≥ a . It is proved that the desired claim as $⋁_{r \in c (L)} Con (k, ⊔ σ_{r} (C (k, g))) \geq Con (k, g) .$

Theorem 4.4. Let (X, τ) be an L-fuzzy (E, K)-soft topological space, g, h ∈ (L^X) ^E and k ∈ K, Then the following implication for the degree of connectedness of the union g ⊔ h with respect to k is satisfied:

Con (k, g ⊔ h) ≥ (S (k, g, h)) ′ ∧ Con (k, g) ∧ Con (k, h) .

Proof. Let a ∈ c (L) and $a \leq (S (k, g, h))^{'} \land Con (k, g) \land Con (k, h) .$

Now we prove Con (k, g ⊔ h) ≥ a . Suppose that for some k ∈ K, Con (k, g ⊔ h) ≱a .

Then by Theorem 4.1, we know that there exist u, v ∈ (L^X) ^E such that $\begin{matrix} (g ⊔ h) ⊓ u \neq \tilde{0}, (g ⊔ h) ⊓ v \neq \tilde{0}, \\ (g ⊔ h) ⊓ u ⊓ v = \tilde{0}, g ⊔ h ⊑ u ⊔ v \\ and (τ_{k} (u^{'}))^{'} \lor (τ_{k} (v^{'}))^{'} ≱ a . \end{matrix}$

By $(g ⊔ h) ⊓ u \neq \tilde{0},$ we know that one of $g ⊓ u \neq \tilde{0}$ or $h ⊓ u \neq \tilde{0}$ must be true.

Suppose that $g ⊓ u \neq \tilde{0}$ (the case of $h ⊓ u \neq \tilde{0}$ is analogous). Then we must have $g ⊓ v = \tilde{0}$ , otherwise if $g ⊓ v \neq \tilde{0},$ then by $\begin{matrix} g ⊓ u \neq \tilde{0}, g ⊓ v \neq \tilde{0}, g ⊓ u ⊓ v = \tilde{0}, g ⊑ u ⊔ v \\ and (τ_{k} (u^{'}))^{'} \lor (τ_{k} (v^{'}))^{'} ≱ a, \end{matrix}$ we know that Con (k, g) ≱a .

This contradicts the fact that Con (k, g) ≥ a . In this case by $(g ⊔ h) ⊓ v \neq \tilde{0},$ we know that $h ⊓ v \neq \tilde{0} .$ Similarly, we can prove $h ⊓ u = \tilde{0} .$ Then by g ⊔ h ⊑ u ⊔ v we can obtain that g ⊑ u and h ⊑ v. Hence by $\begin{matrix} g ⊑ u, h ⊑ v, g ⊓ v = h ⊓ u = \tilde{0} and \\ (τ_{k} (u^{'}))^{'} \lor (τ_{k} (v^{'}))^{'} ≱ a, \end{matrix}$ and by Lemma 3.3, we know that (S (k, g, h)) ′ ≥ a . This shows that Con (k, g ⊔ h) ≥ a . So, the claim is proved as follows, $\begin{matrix} Con (k, g ⊔ h) \geq (S (k, g, h))^{'} \land Con (k, g) \\ \land Con (k, h), for all k \in K . \end{matrix}$

Corollary 4.2. Let (X, τ) be an L-fuzzy (E, K)-soft topological space and g, h ∈ (L^X) ^E. If $g ⊓ h \neq \tilde{0}$ , then for each k ∈ K, $Con (k, g ⊔ h) \geq Con (k, g) \land Con (k, h) .$

Proof. The proof is clear from Lemma 3.1 and Theorem 4.4.

Theorem 4.5. Let (X, τ) be an L-fuzzy (E, K)-soft topological space and g ∈ (L^X) ^E. Then the following equality is satisfied for each k ∈ K, $\begin{matrix} Con (k, g) \\ = ⋀_{x^{λ}, y^{μ} ⊑ g} ⋁ {Con (k, f_{λ, μ}) ∣ x^{λ}, y^{μ} ⊑ f_{λ, μ} ⊑ g} \end{matrix}$

Proof. It is obvious that $\begin{matrix} Con (k, g) \\ \leq ⋀_{x^{λ}, y^{μ} ⊑ g} ⋁ {Con (k, f_{λ, μ}) ∣ x^{λ}, y^{μ} ⊑ f_{λ, μ} ⊑ g} . \end{matrix}$

Now we need to prove the converse inequality. Suppose that for some a ∈ c (L) , $⋀_{x^{λ}, y^{μ} ⊑ g} ⋁ {Con (k, f_{λ, μ}) ∣ x^{λ}, y^{μ} ⊑ f_{λ, μ} ⊑ g} \geq a$

Take a fixed x^λ ⊑ g . Then for any y^μ ⊑ g, there exists a fuzzy soft set f_λ,μ ∈ (L^X) ^E such that x^λ, y^μ ⊑ f_λ,μ ⊑ g and Con (k, f_λ,μ) ≥ a . Let f_λ = ⊔ _y^μ⊑gf_λ,μ . Obviously, f_λ = g and $⊓_{y^{μ} ⊑ g} f_{λ, μ} \neq \tilde{0} .$

By Corollary 4.2, we easily obtain $Con (k, g) = Con (k, f_{λ}) \geq ⋀_{y^{μ} ⊑ g} Con (k, f_{λ, μ}) \geq a .$

This shows that, $\begin{matrix} Con (k, g) \\ \geq ⋀_{x^{λ}, y^{μ} ⊑ g} ⋁ {Con (k, f_{λ, μ}) ∣ x^{λ}, y^{μ} ⊑ f_{λ, μ} ⊑ g} . \end{matrix}$

Theorem 4.6. Let (X₁, τ¹) and (X₂, τ²) be an L-fuzzy (E₁, K₁)-soft topological space and L-fuzzy (E₂, K₂)-soft topological space, respectively. If the function φ_ψ,η : (X₁, τ¹) → (X₂, τ²) is continuous, then for all k ∈ K₁ and g ∈ (L^{X
₁}) ^{E
₁}, $Con (η (k), φ_{ψ} (g)) \geq Con (k, g) .$

Proof. Let k ∈ K₁ and g ∈ (L^{X
₁}) ^{E
₁} be given. Then we have the following inequality $\begin{matrix} Con (η (k), φ_{ψ} (g)) \\ = ⋀ {(τ_{η (k)}^{2} (u^{'}))^{'} \lor (τ_{η (k)}^{2} (v^{'}))^{'} ∣ φ_{ψ} (g) ⊓ u \neq \tilde{0}, \\ φ_{ψ} (g) ⊓ v \neq \tilde{0}, φ_{ψ} (g) ⊓ u ⊓ v = \tilde{0}, \\ φ_{ψ} (g) ⊑ u ⊔ v} \\ \geq ⋀ {τ_{η (k)}^{2} (u^{'}))^{'} \lor (τ_{η (k)}^{2} (v^{'}))^{'} ∣ g ⊓ φ_{ψ}^{- 1} (u) \neq \tilde{0}, \\ g ⊓ φ_{ψ}^{- 1} (v) \neq \tilde{0}, g ⊓ φ_{ψ}^{- 1} (u) ⊓ φ_{ψ}^{- 1} (v) = \tilde{0}, \\ g ⊑ φ_{ψ}^{- 1} (u) ⊔ φ_{ψ}^{- 1} (v)} \\ \geq ⋀ {(τ_{k}^{1} (φ_{ψ}^{- 1} (u^{'})))^{'} \lor (τ_{k}^{1} (φ_{ψ}^{- 1} (v^{'})))^{'} \\ ∣ g ⊓ φ_{ψ}^{- 1} (u) \neq \tilde{0}, g ⊓ φ_{ψ}^{- 1} (v) \neq \tilde{0}, and \\ g ⊓ φ_{ψ}^{- 1} (u) ⊓ φ_{ψ}^{- 1} (v) = \tilde{0}, \\ g ⊑ φ_{ψ}^{- 1} (u) ⊔ φ_{ψ}^{- 1} (v)} \\ \geq ⋀ {(τ_{k}^{1} (f^{'}))^{'} \lor (τ_{k}^{1} (h^{'}))^{'} ∣ g ⊓ h \neq \tilde{0}, \\ g ⊓ f \neq \tilde{0}, g ⊓ f ⊓ h = \tilde{0}, g ⊑ f ⊔ h} \\ = Con (k, g) . \end{matrix}$

Hence the result.

5 Conclusion

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness, in other words being an “all one piece”, is one of the principal topological properties that are used to distinguish topological spaces. The topological study of connectedness is heavily geometric (or visual). Thus connectedness-like properties play an important role in most topological characterization theorems, as well as the study of constructions to the extension of functions. The notion of connectedness also takes an important place in manifolds, Lie groups, graphs and categories. So, many researchers extended and studied the notion of connectedness to fuzzy setting and also fuzzy soft setting in different forms. Therefore, we found it reasonable to study connectedness degree in the parameterized fuzzy soft topological spaces. Also there is a strong linkage between rough sets and soft sets in application (see [16 , 30–33]). For further research, this connection may be applied to investigations of topological structures.

Footnotes

Acknowledgments

The authors wish to thank the associate editor Prof. Jianming Zhan and the referees for their valuable suggestions.

References

Ahmad

and Kharal

, On fuzzy soft sets, Advances in Fuzzy Systems 2009 (2009), Article ID 586507.

Aygünoğlu

, Çetkin

and Aygün

, An Introduction to fuzzy soft topological spaces, Hacettepe Journal of Mathematics and Statistics 43(2) (2014), 197–208.

Çetkin

and Aygün

, A Note on fuzzy soft topological spaces, The 8th Congerence of the European Society for Fuzzy Logic and Technology, Atlantis Press Proceedings, EUSFLAT 2013, Milan- Italy, 2013, pp. 56–60. doi: 10.2991/eusflat.2013.8

Çetkin

and Aygün

, On convergence of fuzzy soft filters, 3rd International Eurasian Conference on Mathematical Sciences and Applications, Vienna-Austria, 2014.

Çetkin

and Aygün

, On fuzzy soft topogenous structure, Journal of Intelligent and Fuzzy Systems 27 (2014), 247–255.

Çetkin

, Aygünoğlu

and Aygün

, CATs of soft topological spaces, Journal of Intelligent and Fuzzy Systems 30 (2016), 1903–1913.

Çetkin

, Aygünoğlu

and Aygün

, A topological view on application of L-fuzzy soft sets: Compactness, Journal of Intelligent and Fuzzy Systems 32 (2017), 781–790.

Çetkin

and Aygün

, Hutton uniformity in the context of fuzzy soft sets, European Journal of Pure and Applied Mathematics 9(4) (2016), 419–433.

Gierz

, et al, A compendium of continuous lattices, Springer-Verlag, 1980.

10.

Ìrfan Ali

, Feng

, Liu

, Min

W.K.

and Shabir

, On some new operations in soft set theory, Computers and Mathematics with Applications 57 (2009), 1547–1553.

11.

Ìrfan Ali

, Mahmood

, Muti Ur Rehman

and Fahim Aslam

, On lattice ordered soft sets, Applied Soft Computing 36 (2015), 499–505.

12.

Ìrfan Ali

, Shabir

and Gulistan

, Applicatiob of L-fuzzy soft sets to semirings, Journal of Intelligent and Fuzzy Systems 27(4) (2014), 1731–1742.

13.

Kandil

, El-Tantawy

O.A.

, El-Shiekh

S.A.

and El-Sayed

S.S.S.

, Fuzzy soft connected sets in fuzzy soft topological spaces I, Journal of Advances in Mathematics 8(12) (2016), 6473–6488.

14.

Karataş

, Kihc

and Tellioğlu

, On fuzzy soft connected topological spaces, Journal of Linear and Topological Algebra 3(4) (2015), 229–240.

15.

Kharal

and Ahmad

, Mappings on fuzzy soft classes, Advances in Fuzzy Systems 2009 (2009), Article ID 407890.

16.

, Liu

and Zhan

, A survey of decision making methods based on certain hybrid soft set models, Artifical Intelligence Review 47 (2017), 507–530.

17.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001), 589–602.

18.

Maji

P.K.

, Roy

A.R.

and Biswas

, An application of soft sets in a decision making problem, Computers and Mathematics with Applications 44(8-9) (2002), 1077–1083.

19.

Molodtsov

, Soft set theory-First results, Computers Mathematics with Applications 37(4/5) (1999), 19–31.

20.

Pei

and Miao

, From soft sets to information systems, Granular Computing, 2005 IEEE International Conference on (2) (2005), pp. 617–621.

21.

Roy

A.R.

and Maji

P.K.

, A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics 203 (2007), 412–418.

22.

Shabir

and Naz

, On soft topological spaces, Computers and Mathematics with Applications 61 (2011), 1786–1799.

23.

Shi

F.-G.

, L-fuzzy interiors and L-fuzzy closures, Fuzzy Sets and Systems 160 (2009), 1218–1232.

24.

Shi

F.-G.

, Connectedness degrees in L-fuzzy topological spaces, International Journal of Mathematics and Mathematical Sciences 2009 (2009), Article ID 892825.

25.

Sostak

A.P.

, The degree of connectivity of fuzzy sets in fuzzy topological spaces, Matematicki Vesnik 40(2) (1988), 159–171.

26.

Tanay

and Kandemir

M.B.

, Topological structure of fuzzy soft sets, Computers and Mathematics with Applications 61 (2011), 412–418.

27.

Willard

, General Topology, Dover Publications, Inc., Mineola, New-York, 2004.

28.

Yue

and Fang

, Generalized connectivity, Proyecciones Journal of Mathematics 25(2) (2006), 191–203.

29.

Yin

, Zhan

, Xu

and Wang

, The L-fuzzy hypermodules, Compuetrs and Mathematics with Applications 59(2) (2010), 953–963.

30.

Zhan

, Liu

and Herawan

, A novel soft rough set: Soft rough hemirimgs and its multicriteria group decision making, Applied Soft Computing 54 (2017), 393–402.

31.

Zhan

, Ali

M.Ý.

and Mehmood

, On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods, Applied Soft Computing 56 (2017), 446–457.

32.

Zhan

and Zhu

, A novel soft rough fuzzy set: Z-soft rough fuzzy ideals of hemirings and corresponding decision making, Soft Computing 21 (2017), 1923–1936.

33.

Zhan

, Liu

and Zhu

, Another approach to rough soft hemirings and corresponding decision making, Soft Computing 21 (2017), 3769–3780.