Results on generalized intuitionistic fuzzy hypergroupoids

Abstract

In this paper, we introduce and study the concept of generalized intuitionistic fuzzy hypergraph (shortly, g-if-hypergraph) and establish a relation between generalized if-hypergraph and if-hyperstructures. Further, we introduce partial if-hypergroupoid and extend the results for higher order if-hypergroupoids and study their properties.

1. Introduction

Graph theoretical studies, particularly, hypergraphs, are important for the development of modeling system architecture and to represent partition, covering and clustering in circuit design. Many mathematicians expanded graph models for the modeling of complex systems. Hypergraphs are the generalization of graphs to the set of multiary relations [1 , 13]. In 1970, Berge [6, 7] introduced the term hypergraph. In 1976, Berge introduced some more relevant concepts in [8, 9] for modeling systems with fuzzy binary and multiary relations between objects, which was a transition to fuzzy hypergraphs combining both fuzzy and graph models. In 1975, Rosenfield [23] introduced the concept of fuzzy graph on the basis of the idea introduced by Kauffmann [16] in 1973. Mordeson and Peng [21] introduced some operations on fuzzy graph. In [20], there is a very good presentation of fuzzy graph and fuzzy hypergraph theory. Recently, Parvati et al. [22] defined intuitionistic fuzzy hypergraphs. After that, many researchers in the field of hyperstructure theory established significant connections between hypergraphs and hyperstructures (see, for instance, [10 , 18]). In [24], Sen et al. introduced and studied fuzzy semihypergroups by using the concept of fuzzy hyperoperation.

Hypergraphs play a very significant role in circuit design and modeling system architectures. However, there are situations where some aspects of graph theoretic design may not be certain. In such cases, we must take help of fuzzy sets or fuzzy logic. But then again, the use of higher order fuzzy sets makes the solution procedure more complex. Intuitionistic fuzzy sets come to rescue and provides more flexibility in computation for such problems, which motivates the current study.

In this paper, we extend the concept of if-hypergraphs into generalized if-hypergraphs and establish some connection between generalized if-hypergraphs and if-hyperstructures. We also extend the results for higher order. We have discussed examples those motivate the necessity of studying the intuitionistic version of corresponding fuzzy graph model.

2. Preliminaries

First of all, we recall some essential notions and results of generalized intuitionistic fuzzy(in short “g-if") hypergraphs.

Definition 2.1. An intuitionistic fuzzzy set (in short “IFS") E in a non-empty set X is defined as E = {(x, μ_E (x) , ν_E (x)) : x ∈ X} where the functions μ_E : X → I and ν_E : X → I (where I is the unit interval $[0, 1] \subseteq ℝ$ ) define the degree of membership and non-membership of the element x ∈ X respectively and for every x ∈ X, 0 ≤ μ_E + ν_E ≤ 1.

The set of all fuzzy subsets of X will be denoted by I^X. For each μ₁, μ₂, ν₁, ν₂ ∈ I^X, say μ₂ ⊆ μ₁ and ν₂ ⊆ ν₁ if μ₂ (x) ≤ μ₁ (x) and ν₂ (x) ≤ ν₁ (x), for all x ∈ X. Suppose μ_α, ν_α ∈ I^X, in the index set $\bar{Λ}$ , we define (μ₁ ∪ μ₂) (x) = max {μ (x) , ν (x)}, (μ₁ ∩ μ₂) (x) = min {μ (x) , ν (x)} and (ν₁ ∪ ν₂) (x) = min {μ (x) , ν (x)}, (ν₁ ∩ ν₂) (x) = max {μ (x) , ν (x)},

$(μ_{1} ∖ μ_{2}) (x) = {\begin{matrix} μ_{1} (x) & if μ_{1} (x) > μ_{2} (x) \\ 0 & otherwise \end{matrix} and$ $(ν_{1} ∖ ν_{2}) (x) = {\begin{matrix} ν_{1} (x) & if ν_{1} (x) > ν_{2} (x), \\ 1 & otherwise \end{matrix}$ Moreover, we have $(⋃_{α \in \bar{Λ}} μ_{α}) (x) = ⋁_{α \in \bar{Λ}} {μ_{α} (x)},$ $where ⋁_{α \in \bar{Λ}} {μ_{α} (x)} = sup_{α \in \bar{Λ}} {μ_{α} (x)}$ $and (⋂_{α \in \bar{Λ}} μ_{α}) (x) = ⋀_{α \in \bar{Λ}} {ν_{α} (x)},$ $where ⋀_{α \in \bar{Λ}} {ν_{α} (x)} = inf_{α \in \bar{Λ}} {ν_{α} (x)} .$

If μ₁, μ₂, μ₃ and ν₁, ν₂, ν₃ are fuzzy subsets of X, then μ₂ ⊆ μ₁ and ν₂ ⊆ ν₁ satisfying that (μ₁ ∖ μ₂) ∪ μ₂ = μ₁ and (ν₁ ∖ ν₂) ∩ ν₁ = ν₁. Moreover μ₁ ∪ μ₂ = μ₁ ∪ μ₃ and ν₁ ∩ ν₂ = ν₁ ∩ μ₃ implies that μ₂ ∖ μ₃ ⊆ μ₁ and ν₂ ∖ ν₃ ⊆ ν₁.

Definition 2.2. If μ, ν ∈ I^X, then support of μ, ν, is defined by supp (μ, ν) = {x ∈ X | μ (x) >0, ν (x) >0}.

Definition 2.3. A fuzzy h-relation δ on X is a function from X × I^*^X to I, where I^*^X = I^X - {∅}. Let p ∈ I. The p-domain of δ is defined by Dom_p (δ) = {x ∈ X | δ (x, μ, ν) ≥ p, forsome μ ∈ I^*^X and μ + ν ≤ 1} and p-codomain of δ is defined by Cod_p (δ) = {μ, ν ∈ I^*^X | δ (x, μ, ν) ≥ p, forsome x ∈ X and μ + ν ≤ 1}. Also, for any x ∈ X, we define $x_{δ}^{p} = {μ, ν \in {I^{*}}^{X} | δ (x, μ, ν) \geq p}$ .

Definition 2.4. The generalized intuitionistic fuzzy hypergraph (g-if hypergraph)is an ordered pair Γ = (X, E) where

X = {x₁, x₂ ⋯ x_n} is a finite set of vertices;

E = {E₁, E₂ ⋯ , E_n} is a family of intuitionistic fuzzy subsets of X;

E_j = {μ, ν ∈ I^*^X : δ (x_i, μ_j, ν_j) >0 forsome x ∈ X and μ_j (x_i) + ν_j (x_i) ≤1} for all i = 1, 2, ⋯ , n and j = 1, 2, ⋯ , m;

E_j ≠ φ, j = 1, 2, ⋯ , m;

⋃_jsupp (E_j) = X, j = 1, 2, ⋯ , m.

Here elements of X are called vertices and the IFS E_j = {μ, ν ∈ I^*^X : δ (x_i, μ_j, ν_j) >0 forsome x ∈ X and μ_j (x_i) + ν_j (x_i) ≤1} for all i = 1, 2, ⋯ , n and j = 1, 2, ⋯ , m are called IF hyperedges.

Example 2.5. Consider an g-if hypergraph Γ = (X, E) such that X = {x₁, x₂, x₃, x₄} and E = {E₁, E₂, E₃, E₄}, where E₁ = {(x₁, 0.1, 0.7) , (x₂, 0.2, 0.6)}, E₂ = {(x₂, 0.2, 0.6) , (x₃, 0.3, 0.7)}, E₃ = {(x₃, 0.3, 0.7) , (x₄, 0.6, 0.1)} and E₄ = {(x₄, 0.6, 0.1) , (x₁, 0.1, 0.7)}. Then the following figure represent the g-if hypergraph Γ:

Definition 2.6. A g-if Γ = (X, δ) is called v_p-linked if $x_{δ}^{p} \neq \emptyset$ , for all x ∈ X and it is called p-plenary if $\begin{matrix} ⋃_{μ, ν \in {Cod}_{p} (δ)} supp (μ, ν) = X . \end{matrix}$

A partial intuitionistic fuzzy hyperoperation on a non-empty set X mean the functions •, ⊲ from X × X to I^X. In other words, for any x, y ∈ X, x • y and x ⊲ y are intuitionistic fuzzy subset of X. Every mapping from X × X to I^*^X is called a if- hyperoperation. If μ₁, μ₂ ; ν₁, ν₂ ∈ I^*^X, then we define μ₁ • μ₂ = ⋃ {a • b | a ∈ supp (μ₁) , b ∈ supp (μ₂)} and ν₁ ⊲ ν₂ = ⋂ {a ⊲ b | a ∈ supp (ν₁) , b ∈ supp (ν₂)}, x • μ₂ = χ_{x} • μ₂ and μ₁ • y = μ₁ • χ_{y} and x • ν₂ = χ_{x} ⊲ ν₂ and ν₁ ⊲ y = ν₁ ⊲ χ_{y} where χ_X denotes the characteristic function of a given set X. If μ₁ =∅ or μ₂ =∅, then we define μ₁• μ₂ = ∅.

Definition 2.7. A (partial) intuitionistic fuzzy hypergroupoid is a pair (X, • , ⊲) where X is a non-empty set and •, ⊲ is a (partial) intuitionistic fuzzy hyperoperation. A if-hypergroupoid (X, • , ⊲) is called a intuitionistic fuzzy semihypergroup if the associative axiom is valid, i.e., x • (y • z) = (x • y) • z and x ⊲ (y ⊲ z) = (x ⊲ y) ⊲ z, for all x, y, z ∈ X and it is called reproductive if supp (x • χ_X) = supp (χ_X • x) = X and supp (x ⊲ χ_X) = supp (χ_X ⊲ x) = X, for all x ∈ X.

An intuitionistic fuzzy hypergroup is a reproductive intuitionistic fuzzy semihypergroup. The notion of H_v-structures was introduced by Vougiouklis [25]. A if-hypergroupoid (X, • , ⊲) is called if-H_v-semigroup if the weak associative axiom is valid, i.e., x• (y • z) ⋂ (x • y) • z ≠ ∅ and x⊲ (y ⊲ z) ⋂ (x ⊲ y) ⊲ z ≠ ∅, for all x, y, z ∈ X and it is called if-H_v-group if it is reproductive if-H_v-semigroup.

3. Main Results

Here we aim to establish some theorems and results on g-if hypergroupoids with the help of g-if hypergraph.

3.1. Partial (g-if)-hypergroupoids

Here we are going to introduce the concept of partial (g - if) ^p-hypergroupoid to each g-if hypergraph. We also establish some relations and conditions for separable intuitionistic fuzzy hypergroupoid and separable intuitionistic fuzzy semihypergroup.

Definition 3.1. Let Γ = (X, δ) be a (g-if)-hypergraph. Then $X_{Γ}^{p} = (X, •^{p}, ⊲^{p})$ is called the partial (g - if) ^p-hypergroupoid associated with Γ, if for p ∈ I, •^p and ⊲^p are defined by $\begin{matrix} x •^{p} y = {N_{•}}^{p} (x) ⋃ {N_{•}}^{p} (y) and \\ x ⊲^{p} y = {N_{⊲}}^{p} (x) ⋂ {N_{⊲}}^{p} (y) for all x, y \in X, \end{matrix}$ where

${N_{•}}^{p} (x) = ⋃_{δ (x, μ, ν) \geq p} μ$ and ${N_{⊲}}^{p} (x) = ⋂_{δ (x, μ, ν) \geq p} ν$ .

In the case that •^p, ⊲ ^p are intuitionistic fuzzy hyperoperations, $X_{Γ}^{p}$ is called the (g-if)^p-hypergroupoid associated with Γ.

Lemma 3.2. $X_{Γ}^{p}$ is called (g-if)^p-hypergroupoid if and only if Γ is v_p-linked.

Example 3.3. Let X = 1, 2, 3, 4 and p ∈ (0, 1]. Suppose that $(μ_{1}, ν_{1}) : \frac{1}{(0.2, 0.6)}, \frac{2}{(0.8, 0.1)}, \frac{3}{(0.3, 0.2)}, \frac{4}{(0.5, 0.1)}$ and $(μ_{2}, ν_{2}) : \frac{1}{(0.2, 0.1)}, \frac{2}{(0.3, 0.5)}, \frac{3}{(0, 0)}, \frac{4}{(0.7, 0.1)}$ are if-subsets of X.

Let δ be a if-h-relation on X as shown in the following figure-

Then from the above figure we have,

δ (1, μ₁, ν₁) =0.5, δ (1, μ₂, ν₂) =0; δ (2, μ₁, ν₁) =0.5, δ (2, μ₂, ν₂) =0.1; δ (3, μ₁, ν₁) =0.5, δ (3, μ₂, ν₂) =0.7; and δ (4, μ₁, ν₁) =0.5, δ (4, μ₂, ν₂) =0.

This implies, δ (x, μ₁, ν₁) =0.5 for all x ∈ X i.e. $x_{δ}^{0.5} \neq φ$ for all x ∈ X. Hence Γ = (X, δ) is v_0.5-linked.

Again, δ (2, μ₂, ν₂) =0.1 and δ (1, μ₂, ν₂) =0 implies that Γ = (X, δ) is not v_0.1-linked but it is 0.1-plenary.

Therefore, $X_{δ}^{0.5}$ is (g-if)^0.5-hypergroupoid associated with Γ.

Definition 3.4. A partial if-hypergroupoid (X, • , ⊲) is called separable if the following property holds: $\begin{matrix} x • y = x • x ⋃ y • y and \\ x ⊲ y = x ⊲ x ⋂ y ⊲ y, for all x, y \in X . \end{matrix}$

Remark 3.5. Let (X, • , ⊲) be a separable if- hypergroupoid and p ∈ (0, 1]. For each x ∈ X we define $δ (x, μ, ν) = {\begin{matrix} p & if μ = x • x and ν = x ⊲ x \\ 0 & otherwise . \end{matrix}$

Then, (X, • , ⊲) is the (g-if)^p-hypergroupoid associated with the v_p-linked (g-if)-hypergraph Γ = (X, δ). Therefore, every separable hypergroupoid can be considered as a (g-if)^p-hypergroupoid, where p ∈ (0, 1].

Lemma 3.6. Let (X, • ^p, ⊲ ^p) be a partial (g-if)^p-hypergroupoid. Then, for all x, y ∈ X and μ, ν ∈ I^*^X we have

x • ^py = y • ^px and x ⊲ ^py = y ⊲ ^px,

(x • ^px) • ^p (x • ^px) = ⋃ _{t∈supp(x•^px)}t • ^pt and (x ⊲ ^px) ⊲ ^p (x ⊲ ^px) = ⋂ _{t∈supp(x⊲^px)}t ⊲ ^pt,

(μ • ^pμ) ∘ ^p (μ • ^pμ) = ⋃ _{t∈supp(μ•^pμ)}t • ^pt and (ν ⊲ ^pν) ⊲ ^p (ν ⊲ ^pν) = ⋂ _{t∈supp(ν⊲^pν)}t ⊲ ^pt.

Lemma 3.7. Every separable if-hypergroupoid is a if-H_v-semigroup.

Proof. Let (X, • , ⊲) be a separable if-hypergroupoid. By Remark 3.1, (X, • , ⊲) can be considered as a (g-if)^p-hypergroupoid, for some p ∈ (0, 1]. Thus, for each x, y, z ∈ X, by using Lemma 3.6, we have

$\begin{matrix} (x • y) • z & = (x • x ⋃ y • y) • z \\ = (x • x) • z ⋃ (y • y) • z \end{matrix}$

$\begin{matrix} x • (y • z) & = (y • z) • x = (y • y ⋃ z • z) • x \\ = (y • y) • x ⋃ (z • z) • x . \end{matrix}$ And $\begin{matrix} (x ⊲ y) ⊲ z & = (x ⊲ x ⋂ y ⊲ y) ⊲ z \\ = (x ⊲ x) ⊲ z ⋂ (y ⊲ y) ⊲ z \end{matrix}$

$\begin{matrix} x ⊲ (y ⊲ z) & = (y ⊲ z) ⊲ x = (y ⊲ y ⋂ z ⊲ z) ⊲ x \\ = (y ⊲ y) ⊲ x ⋂ (z ⊲ z) ⊲ x . \end{matrix}$ Moreover, we observe that $\begin{matrix} (x • x) • z & = ⋃_{t \in supp (x • x)} t • z ∥ \\ = (⋃_{t \in supp (x • x)} t • t) ⋃ z • z ∥ \\ = [(x • x) • (x • x)] ⋃ z • z \end{matrix}$ and $\begin{matrix} (x ⊲ x) ⊲ z & = ⋂_{t \in supp (x ⊲ x)} t ⊲ z ∥ \\ = (⋂_{t \in supp (x ⊲ x)} t ⊲ t) ⋂ z ⊲ z ∥ \\ = [(x ⊲ x) ⊲ (x ⊲ x)] ⋂ z ⊲ z . \end{matrix}$ Therefore, we have $\begin{matrix} (x • y) • z = [(x • x) & • (x • x)] ⋃ z • z \\ ⋃ [(y • y) • (y • y)], \end{matrix}$

$\begin{matrix} x • (y • z) = [(y • y) & • (y • y)] ⋃ x • x \\ ⋃ [(z • z) • (z • z)] . \end{matrix}$ Also, we have $\begin{matrix} (x ⊲ y) ⊲ z = [(x ⊲ x) & ⊲ (x ⊲ x)] ⋂ z ⊲ z \\ ⋂ [(y ⊲ y) ⊲ (y ⊲ y)], \end{matrix}$

$\begin{matrix} x ⊲ (y ⊲ z) = [(y ⊲ y) & ⊲ (y ⊲ y)] ⋂ x ⊲ x \\ ⋂ [(z ⊲ z) ⊲ (z ⊲ z)] . \end{matrix}$

Hence, we obtain $\begin{matrix} \emptyset \neq (y • y) • (y • y) \subseteq (x • y) • z ⋂ x • (y • z), ∥ \\ \emptyset \neq (y ⊲ y) ⊲ (y ⊲ y) \subseteq (x ⊲ y) ⊲ z ⋂ x ⊲ (y ⊲ z) . \end{matrix}$ Therefore (X, • , ⊲) is if-H_v-semigroup.□

Corollary 3.8. Every (g-if)^p-hypergroupoid is a if-H_v-semigroup.

Corollary 3.9. A partial (g-if)^p-hypergroupoid $X_{Γ}^{p}$ is a if-H_v-semigroup if and only if Γ is v_p-linked.

Theorem 3.10. Let Γ = (X, δ) be a v_p-linked (g-if)-hypergraph. Then, the (g-if)^p-hypergroupoid $X_{Γ}^{p} = (X, •^{p}, ⊲^{p})$ is a if-H_v-group if and only if Γ is p-plenary.

Proof. Given that Γ = (X, δ) is a v_p-linked (g-if)-hypergraph. Suppose $X_{Γ}^{p}$ be a fuzzy H_v-group.

It suffices to show that X ⊆ ⋃ _{μ,ν∈Cod_p(δ)}supp (μ, ν).

Let x ∈ X be an arbitrary element. Since $X_{Γ}^{p}$ is a if-H_v-group, therefore we have, supp (x • ^pχ_X) = X and supp (x ⊲ ^pχ_X) = X. Therefore, there is y ∈ X such that $x \in supp (x •^{p} y) = supp ({N_{•}}^{p} (x)$ $⋃ {N_{•}}^{p} (y))$ and $x \in supp (x ⊲^{p} y) = supp ({N_{⊲}}^{p} (x) ⋃$ ${N_{⊲}}^{p} (y))$ . Thus, μ, ν ∈ Cod_p (δ) such that $\begin{matrix} x \in supp (μ, ν) \subseteq ⋃_{μ, ν \in {Cod}_{p} (δ)} supp (μ, ν), μ + ν \leq 1 . \end{matrix}$ Therefore, $\begin{matrix} X \subseteq ⋃_{μ, ν \in {Cod}_{p} (δ)} supp (μ, ν), 0 \leq μ + ν \leq 1 . \end{matrix}$

Conversely, let Γ be p-plenary. As Γ is a v_p-linked (g-if)-hypergraph, therefore by Lemma 3.2, $X_{Γ}^{p}$ is a (g-if)^p-hypergroupoid and so by Corollary 3.8, it is a if-H_v-semigroup. It is sufficient to show that

supp (x • ^pχ_X) = supp (χ_X • ^px) = X and supp (x ⊲ ^pχ_X) = supp (χ_X ⊲ ^px) = X, for each x ∈ X.

Trivially, supp (x • ^pχ_X) ⊆ X and supp (x ⊲ ^pχ_X) ⊆ X. We have to show that

X ⊆ supp (x • ^pχ_X) and X ⊆ supp (x ⊲ ^pχ_X). Let z ∈ X be an arbitrary element. Since Γ is p-plenary, there exists μ, ν ∈ Cod_p (δ) such that z ∈ supp (μ, ν). Since μ, ν ∈ Cod_p (δ), there is y ∈ X such that δ (y, μ, ν) ≥ p and so we have z ∈ supp (x • ^py) ⊆ supp (x • ^pχ_X) and z ∈ supp (x ⊲ ^py) ⊆ supp (x ⊲ ^pχ_X). This implies that X ⊆ supp (x • ^pχ_X) and X ⊆ supp (x ⊲ ^pχ_X). Therefore supp (x • ^pχ_X) = X and supp (x ⊲ ^pχ_X) = X. Similarly, we have supp (χ_X • ^px) = X and supp (χ_X ⊲ ^px) = X. Therefore $X_{Γ}^{p}$ is a if-H_v-group.

Corollary 3.11. $X_{Γ}^{p}$ is a reproductive (g-if)^p- hypergroupoid if and only if Γ is v_p-linked and p-plenary.

Theorem 3.12. Suppose that (X, • , ⊲) is a separable if-hypergroupoid. Then, •, ⊲ are associative if and only if the following conditions hold:

x • x ⊆ (x • x) • (x • x) and x ⊲ x ⊆ (x ⊲ x) ⊲ (x ⊲ x) , for all x ∈ X,

((x • x) • (x • x)) ∖ (x • x) ⊆ (y • y) • (y • y)and ((x ⊲ x) ⊲ (x ⊲ x)) ∖ (x ⊲ x) ⊆ (y ⊲ y) ⊲ (y ⊲ y) , forall x, y ∈ X.

Proof. Let •, ⊲ be associative and x, y be arbitrary elements of X. First, we prove condition (1). Suppose that supp (x • x) = {x₁, …, x_n} and supp (x ⊲ x) = {x₁, …, x_n}. Since $\begin{matrix} x • x_{i} = x • x ⋃ x_{i} • x_{i}, \end{matrix}$ and $\begin{matrix} x ⊲ x_{i} = x ⊲ x ⋂ x_{i} ⊲ x_{i}, \end{matrix}$ for each 1 ≤ i ≤ n, we have $\begin{matrix} (x • x) (x_{i}) \leq (x • x_{i}) (x_{i}) \leq (x • (x • x_{i})) (x_{i}) \\ = ((x • x) • x_{i}) (x_{i}) \\ = max {(x_{1} • x_{i}) (x_{i}), \dots, (x_{n} • x_{i}) (x_{i})} \\ = max {(x_{1} • x_{1}) (x_{i}), \dots, (x_{n} • x_{n}) (x_{i})} \\ = ((x • x) • (x • x)) (x_{i}) \end{matrix}$ and $\begin{matrix} (x ⊲ x) (x_{i}) \leq (x ⊲ x_{i}) (x_{i}) \leq (x ⊲ (x ⊲ x_{i})) (x_{i}) \\ = ((x ⊲ x) ⊲ x_{i}) (x_{i}) \\ = min {(x_{1} ⊲ x_{i}) (x_{i}), \dots, (x_{n} ⊲ x_{i}) (x_{i})} \\ = min {(x_{1} ⊲ x_{1}) (x_{i}), \dots, (x_{n} ⊲ x_{n}) (x_{i})} \\ = ((x ⊲ x) ⊲ (x ⊲ x)) (x_{i}) . \end{matrix}$ Thus (1) holds. Now, to prove condition (2) we have $\begin{matrix} \begin{matrix} (y • y) • x = ⋃_{t \in supp (y • y)} t • x = (⋃_{t \in supp (y • y)} t • t) \\ ⋃ x • x = [(y • y) • (y • y)] ⋃ x • x, \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} (y ⊲ y) ⊲ x = ⋂_{t \in supp (y ⊲ y)} t ⊲ x ∥ \\ = (⋂_{t \in supp (y ⊲ y)} t ⊲ t) ⋂ x ⊲ x = [(y ⊲ y) ⊲ (y ⊲ y)] ⋂ x ⊲ x, \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} y • (y • x) = ⋃_{t \in supp (y • x)} y • t = ⋃_{t \in supp (y • x)} (y • y ⋃ t • t) ∥ \\ = y • y ⋃ (⋃_{t \in supp (y • y)} t • t) ⋃ (⋃_{t \in supp (x • x)} t • t) \\ [(y • y) • (y • y)] ⋃ [(x • x) • (x • x)], \end{matrix} \end{matrix}$ also $\begin{matrix} \begin{matrix} y ⊲ (y ⊲ x) = ⋂_{t \in supp (y ⊲ x)} y ⊲ t = ⋂_{t \in supp (y ⊲ x)} (y ⊲ y ⋂ t ⊲ t) ∥ \\ = y ⊲ y ⋂ (⋂_{t \in supp (y ⊲ y)} t ⊲ t) ⋂ (⋂_{t \in supp (x ⊲ x)} t ⊲ t) ∥ \\ = [(y ⊲ y) ⊲ (y ⊲ y)] ⋂ [(x ⊲ x) ⊲ (x ⊲ x)] . \end{matrix} \end{matrix}$

Now, associativity of •, ⊲ implies that $\begin{matrix} [(y • y) & • (y • y)] ⋃ x • x \\ = [(y • y) • (y • y)] ⋃ [(x • x) • (x • x)], \end{matrix}$

$\begin{matrix} [(y ⊲ y) & ⊲ (y ⊲ y)] ⋂ x ⊲ x \\ = [(y ⊲ y) ⊲ (y ⊲ y)] ⋂ [(x ⊲ x) ⊲ (x ⊲ x)] . \end{matrix}$

Thus, ((x • x) • (x • x)) ∖ (x • x) ⊆ (y • y) • (y • y) and ((x ⊲ x) ⊲ (x ⊲ x)) ∖ (x ⊲ x) ⊆ (y ⊲ y) ⊲ (y ⊲ y), for all x, y ∈ X. Therefore, (2) holds.

For the converse, suppose that x, y, z are arbitrary elements of X and conditions (1) and (2) hold. From the Lemma 3.1, we have

$\begin{matrix} (x • y) • z = [(x • x) & • (x • x)] ⋃ z • z \\ ⋃ [(y • y) • (y • y)], \end{matrix}$

$\begin{matrix} x • (y • z) = [(y • y) & • (y • y)] ⋃ x • x \\ ⋃ [(z • z) • (z • z)] . \end{matrix}$ Also, $\begin{matrix} (x ⊲ y) ⊲ z = [(x ⊲ x) & ⊲ (x ⊲ x)] ⋂ z ⊲ z \\ ⋂ [(y ⊲ y) ⊲ (y ⊲ y)], \end{matrix}$

$\begin{matrix} x ⊲ (y ⊲ z) = [(y ⊲ y) & ⊲ (y ⊲ y)] ⋂ x ⊲ x \\ ⋂ [(z ⊲ z) ⊲ (z ⊲ z)] . \end{matrix}$

Then, $\begin{matrix} μ_{1} = [(x • x) • (x • x)] ⋃ z • z \end{matrix}$ and $\begin{matrix} μ_{2} = [(z • z) • (z • z)] ⋃ x • x \end{matrix}$ . Similarly,

$\begin{matrix} ν_{1} = [(x ⊲ x) ⊲ (x ⊲ x)] ⋂ z ⊲ z ∥ \\ ν_{2} = [(z ⊲ z) ⊲ (z ⊲ z)] ⋂ x ⊲ x . \end{matrix}$ Now, $\begin{matrix} (x • y) • z = [(y • y) • (y • y)] ⋃ μ_{1} and \\ x • (y • z) = [(y • y) • (y • y)] ⋃ μ_{2}, ∥ \\ (x ⊲ y) ⊲ z = [(y ⊲ y) ⊲ (y ⊲ y)] ⋂ ν_{1} and \\ x ⊲ (y ⊲ z) = [(y ⊲ y) ⊲ (y ⊲ y)] ⋂ ν_{2} . \end{matrix}$ By using conditions (1) and (2) we have $\begin{matrix} μ_{1} = ([(x • x) • (x • x)] ∖ x • x) ⋃ x • x ⋃ z • z ∥ \\ \subseteq [(z • z) • (z • z)] ⋃ z • z ⋃ x • x ∥ \\ = [(z • z) • (z • z)] ⋃ x • x = μ_{2} . \end{matrix}$ and $\begin{matrix} μ_{1} = ([(x ⊲ x) ⊲ (x ⊲ x)] ∖ x ⊲ x) ⋂ x ⊲ x ⋂ z ⊲ z ∥ \\ \subseteq [(z ⊲ z) ⊲ (z ⊲ z)] ⋂ z ⊲ z ⋂ x ⊲ x ∥ \\ [(z ⊲ z) ⊲ (z ⊲ z)] ⋂ x • x = μ_{2} . \end{matrix}$ Similar, the inverse inclusion is proved and then •, ⊲ is associative.□

Theorem 3.13. Suppose that (X, • , ⊲) is a separable if-semihypergroup. Then, •, ⊲ is associative if and only if the following conditions hold:

μ • μ ⊆ (μ • μ) • (μ • μ) and ν ⊲ ν ⊆ (ν ⊲ ν) ⊲ (ν ⊲ ν) , forall μ, ν ∈ I^*^X,

((μ₁ • μ1) • (μ₁ • μ₁)) ∖ (μ₁ • μ₁) ⊆ (μ₂ • μ₂) • (μ₂ • μ₂) and ((μ₁ ⊲ μ1) ⊲ (μ₁ ⊲ μ₁)) ∖ (μ₁ ⊲ μ₁) ⊆ (μ₂ ⊲ μ₂) ⊲ (μ₂ ⊲ μ₂) , forall μ₁, μ₂, ν₁, ν₂ ∈ I^*^X.

Proof. Let •, ⊲ be associative and μ₁, μ₂, ν₁, ν₂ be arbitrary non-empty fuzzy subsets of X. Then, by using Theorem 3.12 we have $\begin{matrix} μ • μ & = ⋃_{x \in supp (μ)} x • x \subseteq ⋃_{x \in supp (μ)} ((x • x) • (x • x)) \\ = ⋃_{x \in supp (μ)} (⋃_{t \in supp (x • x)} t • t) ∥ \\ = ⋃_{t \in supp (μ • μ)} t • t = (μ • μ) • (μ • μ) . \end{matrix}$ and $\begin{matrix} ν ⊲ ν & = ⋂_{x \in supp (ν)} x ⊲ x \subseteq ⋂_{x \in supp (ν)} ((x ⊲ x) ⊲ (x ⊲ x)) \\ = ⋂_{x \in supp (ν)} (⋂_{t \in supp (x ⊲ x)} t ⊲ t) ∥ \\ = ⋂_{t \in supp (ν • ν)} t ⊲ t = (ν ⊲ ν) ⊲ (ν ⊲ ν) . \end{matrix}$ Hence (1) is true. To prove condition (2), let y ∈ supp (μ₂) and y ∈ supp (ν₂) be an arbitrary element. Then, we have $\begin{matrix} ((μ_{1} • μ_{1}) • (μ_{1} • μ_{1})) ∖ (μ_{1} • μ_{1}) ∥ \\ \subseteq ⋃_{x \in supp (μ_{1})} (((x • x) • (x • x)) ∖ (x • x)) \\ \subseteq (y • y) • (y • y) . \end{matrix}$ and $\begin{matrix} ((ν_{1} ⊲ ν_{1}) ⊲ (ν_{1} ⊲ ν_{1})) ∖ (ν_{1} ⊲ ν_{1}) ∥ \\ \subseteq ⋂_{x \in supp (ν_{1})} (((x ⊲ x) ⊲ (x ⊲ x)) ∖ (x ⊲ x)) \\ \subseteq (y ⊲ y) ⊲ (y ⊲ y) . \end{matrix}$

Again, (y • y) • (y • y) ⊆ (μ₂ • μ₂) • (μ₂ • μ₂) and (y ⊲ y) ⊲ (y ⊲ y) ⊆ (ν₂ ⊲ ν₂) ⊲ (ν₂ ⊲ ν₂). Therefore, $\begin{matrix} ((μ_{1} • μ 1) • (μ_{1} • μ_{1})) ∖ (μ_{1} • μ_{1}) \\ \subseteq (μ_{2} • μ_{2}) • (μ_{2} • μ_{2}), ∥ \\ ((μ_{1} ⊲ μ 1) ⊲ (μ_{1} ⊲ μ_{1})) ∖ (μ_{1} ⊲ μ_{1}) \\ \subseteq (μ_{2} ⊲ μ_{2}) ⊲ (μ_{2} ⊲ μ_{2}) . \end{matrix}$ Hence, condition (2) holds.

For the converse, suppose that conditions (1) and (2) hold. Let x, y be arbitrary elements of X. By putting μ₁ = χ_{x}, μ₂ = χ_{y} and ν₁ = χ_{x}, ν₂ = χ_{y}, conditions (1) and (2) of Theorem 3.1 hold and therefore •, ⊲ is associative.□

Corollary 3.14. If a reproductive (g-if)^p-hypergroupoid $X_{Γ}^{p} = (X, •^{p}, ⊲^{p})$ satisfies anyone of the following conditions: $\begin{matrix} (x •^{p} x) •^{p} (x •^{p} x) = x •^{p} x and \\ (x ⊲^{p} x) ⊲^{p} (x ⊲^{p} x) = x ⊲^{p} x, for all x \in X, \\ (x •^{p} x) •^{p} (x •^{p} x) = ⋃_{t \in X} t •^{p} t and \\ (x ⊲^{p} x) ⊲^{p} (x ⊲^{p} x) = ⋂_{t \in X} t ⊲^{p} t, for all x \in X, \end{matrix}$ then it is a if-hypergroup.

3.2. Higher-order if-hypergroupoids

In this section we discuss some results and theorems of if-hypergroupoids for higher-order.

Let (X, • , ⊲) be a separable if-hypergroupoid. We construct a sequence of if-hypergroupoids X₀ = (X, • ₀) , X₁ = (X, • ₁) , X₂ = (X, • ₂) , … and X₀ = (X, ⊲ ₀) , X₁ = (X, ⊲ ₁) , X₂ = (X, ⊲ ₂) , … recursively as follows: for all x, y ∈ X we set x • ₀y = x • y, x • _k+1x = (x • _kx) • _k (x • _kx) and $\begin{matrix} x •_{k + 1} y = x •_{k + 1} x ⋃ y •_{k + 1} y, \end{matrix}$ and x ⊲ ₀y = x ⊲ y, x ⊲ _k+1x = (x ⊲ _kx) ⊲ _k (x ⊲ _kx) and $\begin{matrix} x ⊲_{k + 1} y = x ⊲_{k + 1} x ⋂ y ⊲_{k + 1} y, \end{matrix}$ where k ≥ 0. Suppose $N_{k} (x) = x •_{k} x$ and . We define $\begin{matrix} N_{k} (μ) = ⋃_{a \in supp (μ)} N_{k} (a) and \\ {N^{'}}_{k} (ν) = ⋂_{a \in supp (ν)} {N^{'}}_{k} (a), \end{matrix}$ where μ, ν are if-subset of X.

Lemma 3.15. Let μ, ν be two if-subsets of X. Then, $\begin{matrix} supp (N_{k} (μ)) = ⋃_{a \in supp (μ)} supp (N_{k} (a)) and \\ supp ({N^{'}}_{k} (ν)) = ⋂_{a \in supp (ν)} supp ({N^{'}}_{k} (a)) . \end{matrix}$

The following properties are holds for the higher-order where k ≥ 0:

$N_{k} (μ) = μ •_{k} μ$ and $N'_{k} (v) = v ⊲_{k} v$ , for all μ, ν ∈ I^*^X,

$N_{k + 1} (x) = N_{k} (N_{k} (x))$ and $N'_{k + 1} (x) = N'_{k} (N'_{k} (x))$ , for all x ∈ X,

$N_{k} (N_{k + 1} (x)) = N_{k + 1} (N_{k} (x))$ and $N' k (N_{k + 1} (x)) = N'_{k + 1} (N_{k} (x))$ , for all x ∈ X,

$N_{k + 1} (μ) = N_{k} (N_{k} (μ))$ and $N'_{k + 1} (v) = N_{k} (N'_{k} (v))$ , for all μ, ν ∈ I^*^X,

μ₁ ⊆ μ₂ implies that $N_{k} (μ_{1}) \subseteq N_{k} (μ_{2})$ and ν₁ ⊆ ν₂ implies that $N_{k} (ν_{1}) \subseteq N_{k} (ν_{2})$ , for all μ₁, μ₂, ν₁, ν₂ ∈ I^*^X,

$N_{k} (x) = N_{k + 1} (x)$ implies that $N_{k} (x) = N_{r} (x)$ and $N' k (x) = N'_{k + 1} (x)$ implies that $N' k (x) = N'_{r} (x)$ , for all r ≥ k.

By Theorem 3.12, X_k is a semihypergroup if and only if the following conditions hold:

Lemma 3.16. The if-hyperoperation •_k, ⊲ _k defined as above has the following properties:

μ • _k+1μ = (μ • _kμ) • _k (μ • _kμ) and ν ⊲ _k+1ν = (ν ⊲ _kν) ⊲ _k (ν ⊲ _kν), for all μ, ν ∈ I^*^X,

x • _k+2x = ((x • _k+1x) • _k (x • _k+1x)) • _k ((x • _k+1x) • _k (x • _k+1x)) and x ⊲ _k+2x = ((x ⊲ _k+1x) ⊲ _k (x ⊲ _k+1x)) ⊲ _k ((x ⊲ _k+1x) ⊲ _k (x ⊲ _k+1x)), for all x ∈ X.

Proof. (1) Let μ, ν be a non-empty if-subsets of X. Then, $\begin{matrix} μ •_{k + 1} μ & = N_{k + 1} (μ) = N_{k} (N_{k} (μ)) \\ = N_{k} (μ) •_{k} N_{k} (μ) = (μ •_{k} μ) •_{k} (μ •_{k} μ), \end{matrix}$ and $ν ⊲_{k + 1} ν = N_{k + 1}^{'} + 1 (ν) = N_{k}^{'} (N_{k}^{'} (ν)) = N_{k}^{'} (ν) ⊲_{k} N_{k}^{'} (ν) = (ν ⊲_{k} ν) ⊲_{k} (ν ⊲_{k} ν) .$ (2) The result follows from part (1) and the definition of •_k+2, ⊲ _k+2.□

Theorem 3.17. Let (X, • , ⊲) be a separable if- hypergroupoid. Then, the following assertions hold:

If X_k = (X, • _k, ⊲ _k) satisfies condition (α) for some k ≥ 0, then $N_{r} (x) \subseteq N_{r + 1} (x)$ and $N'_{r} (x) \subseteq N'_{r + 1} (x)$ , for all x ∈ X and r ≥ k.

If X_k = (X, • _k, ⊲ _k) satisfies condition (β) for some k ≥ 0, then $N_{r + 1} (x) \subseteq N_{r} (x)$ and $N'_{r + 1} (x) \subseteq N'_{r} (x)$ , for all x ∈ X and r > k.

The procedure of the proof of above theorem 3.17 is similar as the proof of Theorem 3.3 in [11].

Corollary 3.18. If (X, • _k, ⊲ _k) is a separable if-semihypergroup, then $N_{r} (x) = N_{r + 1} (x)$ and $N'_{r} (x) = N'_{r + 1} (x)$ , for all x ∈ X and r > k.

Corollary 3.19. If (X, • _k, ⊲ _k) is a separable if-semihypergroup, then $N_{r} (μ) = N_{r + 1} (μ)$ and $N'_{r} (v) = N'_{r + 1} (v)$ , for all μ, ν ∈ I^*^X and r > k.

Next proposition is a direct consequence of Theorem 3.12.

Proposition 3.20. If there exists a natural number k such that $N_{k} (x) = N_{k + 1} (x)$ and $N'_{k} (x) = N'_{k + 1} (x)$ , for all x ∈ X, then

X_k = (X, • _k, ⊲ _k) is a separable if- semihypergroup,

X_r = X_k, for all r ≥ k.

4. Application

Some applications in connection with our present study is discussed in this section.

First we discuss an application of intuitionistic fuzzy hypergraph model for radio transmission that can be used to determine station programming or marketing strategies or to establish an emergency broadcast network.

Application 4.1. Let U denote a finite set of radio receivers those can be considered as vertices in a graph. This vertices may represent locations at the centroid of a geographic region. For each of the n radio transmitters, an intuitionistic fuzzy set ”listening area of station j” may be defined, where A_i (x) = (μ_i (x) , ν_i (x)) represents the "quality of reception of station i at location x”.

Membership and nonmembership values near 1 and 0 denote “very clear reception on a very poor radio", respectively, whereas membership and nonmembership values near 0 and 1, could respectively, denote “very poor reception on even a very sensitive radio". As the signal strength gets affected by the location, each listening area may be considered as an intuitionistic fuzzy set. Further, for a fixed radio, the reception quality will be different between different stations. Such a model, therefore, gives rise to an intuitionistic fuzzy hypergraph.

A natural question arises that if there is a minimal subset of stations that reaches every radio with a given least signal strength. This could be an interesting investigation in future.

Next we show that an intuitionistic fuzzy quotient space model having the hypergroupoid structure can be utilized for the improvement of efficiency of fuzzy reasoning.

Application 4.2. Consider an equivalence relation $\hat{G} (a_{1}, a_{2})$ on $W$ satisfying:

$\forall a_{1} \in W$ , $\hat{G} (a_{1}, a_{1}) = 1$ ,

$\forall a_{1}, a_{2} \in W$ , $\hat{G} (a_{1}, a_{2}) = \hat{R} (a_{2}, a_{1})$ ,

$\forall a_{1}, a_{2}, a_{3} \in W$ , $\hat{G} (a_{1}, a_{3}) \geq sup_{a_{2}} (min (\hat{G} (a_{1}, a_{2}), \hat{G} (a_{2}, a_{3})))$ ,

then

\hat{G}

is called a fuzzy equivalence relation on

W

. An equivalence class [a₁] in

W / G

is defined as

[a_{1}] = {a_{2} \in W : a_{2} = a_{1} + g, g \in G}

. Collection of all equivalence classes in

W / G

is denoted by

[W]

. Construct a function

d : [W] \times [W] \to ℝ

such that

d ([a_{1}], [a_{2}]) = {\begin{matrix} 0, & if [a_{1}] = [a_{2}] \\ 1 - \frac{\hat{G} (a_{1}, a_{2})}{\hat{G} (a_{1}, a_{2}) + 1}, & otherwise . \end{matrix}

Then d is a distance function on

[W]

. Thus, for a given fuzzy equivalence relation

\tilde{G}

W

, we can establish a distance function (metric) on

[W]

, i.e., a fuzzy equivalence relation can be grasped under quotient space structure.

5. Conclusion

Recently, Ma et al. [19] have conducted a very interesting survey of decision making methods based on hybrid soft set models, which can further be explained and extended using intuitionistic fuzzy hypergroupoid structure. Zhang et al. [30] used fuzzy rough set models and applied them to multi-criteria fuzzy group decision making. Some very recent and significant applications of intuitionistic fuzzy hypergraphs in decision making with soft set/rough set models were established by Akram and his coauthors [3 –5].

In this paper we have studied the concept of g-if-hypergraph and partial-if-hypergroupoid. We also established some important results and extended them to higher order. The concept of graph and hypergraph structures are highly utilized in computer science. Intuitionistic fuzzy models give more precision and compatibility to the system as compared to classical and fuzzy models. Therefore, intuitionistic fuzzy hypergraphs are more flexible than fuzzy hypergraphs. Extension and application of our results in decision making using the models of Zhan and coauthors [15 , 26–31] will be very important topics for future study.

Footnotes

Acknowledgments

The authors are immensely thankful to the reviewers and the Associate Editor for their constructive feedback towards overall improvement of the paper.

References

Akram ,

Chen and

Davvaz , On N-hypergraphs, J Intell Fuzzy Syst 26 (2014), 2937–2944.

Akram and

W.A.

Dudek , Intuitionistic fuzzy hypergraphs with applications, Inform Sci 218 (2013), 182–193.

Akram and

Luqman , A new decision-making method based on bipolar neutrosophic directed hypergraphs, J Appl Math Comput 57(120) (2018), 547–575.

Akram and

Sarwar , Transversals of m-polar fuzzy hypergraphs with applications, J Intell Fuzzy Syst 33(1) (2017), 351–364.

Akram and

Shahzadi , Hypergraphs in m-polar fuzzy environment, Mathematics 6(2) (2018), 28. doi: 10.3390/math6020028

Berge , Hypergraphs generalizing bipartite graphs. Integer and Nonlinear Programming, North-Holland, Amsterdam, 1970, pp. 507–509.

Berge , Graphcs et Hypcrgraphcs. Dunod, Pum” 1972.

Berge , The multicolorings of graphs and hypergraphs- theory and applications of graphs, Proc Internat Conf, Western Mich Univ, Kalamazoo, Mich, 1976, Lecture Notes in Math, 642, Springer, Berlin, 1978, pp. 23–36.

Berge , Hypergraphs- Combinatorics of finite sets, Translated from the French, North-Holland Mathematical Library, 45. North-Holland Publishing Co., Amsterdam” 1989.

10.

Corsini , Hypergraphs and hypergroups, Algebra Universalis 35 (1996), 548–555.

11.

Farshi and

Davvaz , Generalized fuzzy hypergraph and hypergroupoids, Filomat 30(9) (2016), 2375–2387.

12.

Farshi ,

Davvaz and

Mirvakili , Degree hypergroupoids associated with hypergraphs, Filomat 28(1) (2014), 119–129.

13.

Feng , p-fuzzy hypergroupoids associated to the product of p-fuzzy Ital, J Pure Appl Math 28 (2011), 153–162.

14.

M.N.

Iradmusa and

Iranmanesh , The combinatorial and algebraic structure of the hypergroup associated to a hypergraph, J Mult-Valued Logic Soft Comput 11 (2005), 127–136.

15.

Jiang ,

Zhan and

Chen , Covering based variable precision (I,T)-fuzzy rough sets with applications to multi-attribute decision-making, IEEE Transactions on Fuzzy Systems (2018). DOI: 10.1109/TFUZZ.2018.2883023

16.

Kauffman , Introduction a la theorie des sous-emsembles flous, Masson et Cie 1 (1973).

17.

Leoreanu and

Leoreanu , Hypergroups associated with hypergraphs, Ital J Pure Appl Math 4 (1998), 119–126.

18.

Leoreanu-Fotea ,

Corsini and

Iranmanesh , On the sequence of hypergroups and membership functions determined by a hypergraph, J Mult-Valued Logic Soft Comput 14 (1998), 119–126.

19.

Ma ,

Zhan ,

M.I.

Ali and

Mehmood , A survey of decision making methods based on two classes of hybrid soft set models, Artif Intell Rev 49(4) (2018), 511–529.

20.

J.N.

Mordeson and

Fuzzy Graphs and Fuzzy Hypergraphs, New York, Physica-Verlag, 2000.

21.

J.N.

Mordeson and

C.S.

Peng , Operations on fuzzy graphs, Inform Sci 79 (1994), 159–170.

22.

Parvathi ,

M.G.

Karunambigai and

Thilagavathi , Intuitionistic fuzzy hypergraphs, Cybernet Inform Technol 9 (2009), 46–48.

23.

Rosenfeld , Fuzzy graphs in:

L.A.

Zadeh ,

K.S.

Fu ,

Shimura , (Eds). Fuzzy sets and their applications, Academic Press, New York, 1975, pp. 77–95.

24.

M.K.

Sen ,

Ameri and

Chowdhury , Fuzzy hypersemigroups, Soft Comput 9 (2008), 891–900.

25.

Vougiouklis , Hyperstructures and their representations, Hadronic Press Inc., 1994.

26.

Zhan ,

Sun and

J.C.R.

Alcantud , Covering based multigranulation (i,t)-fuzzy rough set models and applications in multi-attribute group decision-making, Inform Sci 476 (2019), 290–318.

27.

Zhan and

Wang , Certain types of soft coverings based rough sets with applications, Int J Mach Learn Cybern (2018). 10.1007/s13042-018-0785-x

28.

Zhan and

Xu , Two types of coverings based multigranulation rough fuzzy sets and applications to decision making, Artif Intell Rev (2018). 10.1007/s10462-018-9649-8

29.

Zhang and

Zhan , Fuzzy soft-covering based fuzzy rough sets and corresponding decision-making applications, Int J Mach Learn Cybern (2018). 10.1007/s13042-018-0828-3.

30.

Zhang and

Zhan , Novel classes of fuzzy softcoverings-based fuzzy rough sets with applications to multicriteria fuzzy group decision making, Soft Comput (2018). doi.org/10.1007/s00500-018-3470-9

31.

Zhang ,

Zhan and

Z.X.

Xu , Covering-based generalized IF rough sets with applications to multi-attribute decisionmaking, Inf Sci 478 (2019), 275–302.