On algebraic properties of DFS sets and its application in decision making problems

Abstract

In this paper, we apply the concept of double-framed soft set (briefly DFS-set) to non-associative and non-commutative structure called Abel Grassmann’s groupoid (briefly AG-groupoid). We define double-framed soft quasi-ideals (briefly DFS quasi-ideal) of AG-groupoids and discuss their properties in left regular AG-groupoids. We also characterize left regular AG-groupoids in terms of DFS quasi-ideals. Application of DFS sets in decision making situation is provided.

Keywords

Decision making

1 Introduction

In many real world problems, there frequently occur situations where uncertainty is involved. For example, if we consider the set of tall people of a particular area, then we are unable to clearly decide whether a particular person belongs to this set or not. To deal with such qualitative variables, Zadeh [45] introduced the concept of fuzzy set in 1965. Vagueness or the representation of imperfect knowledge had troubled the minds of many philosophers, logicians and mathematicians for a long time and recently it became a crucial issue for computer scientists, particularly in the area of artificial intelligence. To resolve this issue, Zadeh’s fuzzy set theory proved to be useful. To deal with uncertainities in different situations, several theories have been introduced from time to time (see for example [3 , 40]). But there are inherent difficulties associated with each of the proposed theories. Molodtsov [29] pointed out that all these techniques lack in the parameterization tool and hence conceptualized a framework known as soft set in 1999 and showed that a fuzzy set can be regarded as a soft set. Since then, the theory of soft set attracted many researchers and a number of articles published in recent years by reputable publishing platforms stands as witness to the fact that the soft set theory has gained a niche for itself in dealing with complex topics. For example, the soft set theory is applied in the field of optimization by Kovkov in [27], to decision making problems [5], data analysis [46], forecasting [42] etc. Soft set theory has also been applied to different algebraic structures. We refer the reader to the papers [1 , 28]. Some applications in decision making methods can be found in [33 –37] and [47 –59].

A left almost semigroup $(LA$ -semigroup) is a groupoid $S$ whose elements satisfy the following left invertive law (ab) c = (cb) a for all $a, b, c \in S$ . This concept was first given by Kazim and Naseeruddin in 1972 [19]. Several others authors that have contributed on left almost semigroups, can be found in references (cf. [6, 13] etc.). Stevanovic and Protic [38] called this structure an Abel-Grassmann groupoid (or simply AG-groupoid), which is the primary name under which this structure in known nowadays. There are many important applications of AG-groupoids in the theory of flocks [32]. For more study of AG-groupoids, the reader is invited to read [8–10 , 44].

Recently, Jun et al. extended the notions of intersectional and union soft sets into double-framed soft sets defining the double-framed soft subalgebra of a BCK/BCI-algebra and studied the related properties in [14]. Jun et al. also defined the concept of a double-framed soft ideal (briefly, DFS ideal) of a BCK/BCI-algebra and gave many valuable results. Khan et. al. [21] applied the idea of double framed soft sets to left almost semigroups.

The present paper is a continuation of the theory developed by Khan et al. in [18, 21]. The paper is divided into six sections. In Section 2, basic concepts in AG-groupoids are given. In Section 3, the concept of soft set and double-framed soft set and its operations are reviewed. In Section 4, we define double-framed soft quasi-ideals and discuss some properties of these ideals. In Section 5, we discuss properties of double-framed soft quasi-ideals in left regular AG-groupoids and characterize left regular AG-groupoids in terms of these ideals. Finally, in Section 6, we have presented an application of double-framed fuzzy soft expert-sets(briefly, DFFSE-sets) in a decision making problem by using the agree and disagree scores.

2 Preliminaries

A groupoid (S, ·) is called an AG-groupoid if it satisfies the left invertive law, that is, (ab) c = (cb) a for all a, b, c ∈ S

Every AG-groupoid S satisfies the medial law [19], that is, (ab) (cd) = (ac) (bd) for all a, b, c, d ∈ S .

It is a useful non-associative and non-commutative algebraic structure, midway between a groupoid and a commutative semigroup. It is important to mention here that if an AG-groupoid contains identity or even right identity, then it becomes a commutative monoid. An AG-groupoid may or may not contains left identity. If there exists left identity in an AG-groupoid, then it is unique [30]. Every AG-groupoid S with left identity satisfies the paramedial law [30], that is, (ab) (cd) = (db) (ca) for all a, b, c, d ∈ S .

In an AG-groupoid S with left identity, using the paramedial law, it is easy to prove that $(ab) (cd) = (dc) (ba) for all a, b, c, d \in S .$ (1) Moreover, in an AG-groupoid S with left identity, we have $a (bc) = b (ac) for all a, b, c \in S .$ (2)

Throughout this paper, S will represent an AG-groupoid unless otherwise stated. For non-empty subsets A and B of S we denote by AB : = {ab|a ∈ A and b ∈ B} . If A = {a} then we write aB instead of {a} B .

A nonempty subset A of an AG-groupoid S is called sub AG-groupoid of S if A² ⊆ A.

A nonempty subset A of an AG-groupoid S is called left (resp. right) ideal of S if SA ⊆ A (resp. AS ⊆ A). If A is both a left and a right ideal of S, then it is called a two-sided ideal or simply an ideal of S.

A non-empty subset A of an AG-groupoid S is called generalized bi-ideal of S if (AS) A ⊆ A .

A non-empty subset A of an AG-groupoid S is called a bi-ideal of S if:

(i) A is a sub AG-groupoid of S,

(ii) A is a generalized bi-ideal of S.

A non-empty subset A of an AG-groupoid S is called interior ideal of S if (SA) S ⊆ A .

A non-empty subset A of an AG-groupoid S is called quasi-ideal of S if QS ∩ SQ ⊆ Q .

In an AG-groupoid with left identity e, the left ideal, bi-ideal and quasi-ideal of S generated by a are given by L [a] = a ∪ Sa, B [a] = a ∪ a² ∪ (aS) a and Q [a] = a ∪ (Sa ∩ aS) respectively. A subset A of an AG-groupoid S is called semiprime if for all a ∈ S, whenever a² ∈ A, then a ∈ A.

Lemma 2.1. In an AG-groupoid S, every one-sided ideal is quasi-ideal.

Proof. Straightforward.□

Lemma 2.2. ([31]) Let (S, ·) be an AG-groupoid with left identity and a ∈ S. Then the followings hold:

(i) Sa is a left ideal of S .

(ii) aS is a left ideal of S.

(iv) a²S, Sa² and (Sa²) S are ideals of S.

(v) Sa ∪ aS is an ideal of S.

(vi) a ∪ Sa ∪ aS is an ideal of S .

(vii) a² ∪ Sa² and a² ∪ a²S are ideals of S .

Corollary 2.3. In an AG-groupoid S with left identity, for all a ∈ S, Sa, aS, a²S, Sa², (Sa²) S are quasi-ideals of S.

Lemma 2.4. ([31]). If an AG-groupoid S contains left identity e, then S = S² and S = eS = Se .

Lemma 2.5. ([31]) If S is an AG-groupoid with left identity e, then (xy) ² = x²y² = y²x² for all x, y ∈ S.

Lemma 2.6. ([7]) If an AG-groupoid S contains left identity e, then Sa² = a²S = (Sa²) S for all a ∈ S .

3 Soft set (basic operations)

In [2], Atagun and Sezgin introduced some new operations on soft set theory and defined soft sets in the following way:

Let U be an initial universe, E a set of parameters, P (U) the power set of U and A ⊆ E. Then soft set f_A over U is a function defined by:

f_A : E ⟶ P (U) such that f_A (x) =∅ if x ∉ A.

Here f_A is called an approximate function. A soft set over U can be represented by the set of ordered pairs $f_{A} : = {(x, f_{A} (x)) : x \in E, f_{A} (x) \in P (U)} .$

It is clear that a soft set is a parameterized family of subsets of U. The set of all soft sets over U is denoted by S (U) .

Definition 3.1. Let f_A, f_B ∈ S (U) . Then f_A is a soft subset of f_B, denoted by $f_{A} \tilde{\subseteq} f_{B}$ if f_A (x) ⊆ f_B (x) for all x ∈ E. Two soft sets f_A, f_B are said to be equal soft sets if $f_{A} \tilde{\subseteq} f_{B}$ and $f_{B} \tilde{\subseteq} f_{A}$ and is denoted by $f_{A} \tilde{=} f_{B} .$

Definition 3.2. Let f_A, f_B ∈ S (U). Then the union of f_A and f_B, denoted by $f_{A} \tilde{\cup} f_{B}$ , is defined by $f_{A} \tilde{\cup} f_{B} = f_{A \cup B},$ where f_A∪B (x) = f_A (x) ∪ f_B (x) , for all x ∈ E.

Definition 3.3. Let f_A, f_B ∈ S (U). Then the intersection of f_A and f_B, denoted by $f_{A} \tilde{\cap} f_{B}$ , is defined by $f_{A} \tilde{\cap} f_{B} = f_{A \cap B},$ where f_A∩B (x) = f_A (x) ∩ f_B (x) , for all x ∈ E.

Definition 3.4. Let f_A, f_B ∈ S (U) . Then the soft product of f_A and f_B, denoted by $f_{A} \tilde{\circ} f_{B},$ is defined by $(f_{A} \tilde{\circ} f_{B}) (x) =$ ${\begin{matrix} ⋃_{x = yz} {f_{A} (y) \cap f_{B} (z)} if \exists y, z \in S such that x = yz \\ \emptyset otherwise . \end{matrix}$

Throughout this paper, let E = S, where S is an AG-groupoid and A, B, C, ⋯ are sub AG-groupoids, unless otherwise stated.

Definition 3.5. [14] A double-framed soft pair 〈 (α, β) ; A〉 is called a double-framed soft set of S over U (briefly, DFS-set of A), where α and β are mappings from A to P (U).

The set of all DFS-sets of S over U will be denoted by DFS (U) .

For a DFS-set 〈 (α, β) ; A〉 of A and two subsets γ and δ of U, the γ-inclusive set and the δ-exclusive set of 〈 (α, β) ; A〉, denoted by i_A (α ; γ) and e_A (β ; δ), respectively, are defined as follows: $i_{A} (α; γ) : = {x \in A | α (x) \supseteq γ}$ and $e_{A} (β; δ) : = {x \in A | β (x) \subseteq δ}$ respectively. The set ${DF}_{A} {(α, β)}_{(γ, δ)} : = {x \in A | α (x) \supseteq γ, β (x) \subseteq δ}$ is called a double framed soft including set [14] of 〈 (α, β) ; A 〉 . It is clear that DF_A (α, β) _(γ,δ) : = i_A (α ; γ) ∩ e_A (β ; δ) .

Let 〈 (α, β) ; A〉 and 〈 (f, g) ; B〉 be two double-framed soft sets of S over U. Then the int-uni soft product is denoted by 〈 (α, β) ; A〉 ◇ 〈 (f, g) ; B 〉 and is defined as a double framed soft set $〈 (α \tilde{\circ} f, β \tilde{\circ} g); S 〉$ defined to be a double-framed soft set over U, in which $α \tilde{\circ} f,$ and $β \tilde{\circ} g$ are soft mappings from S to P (U), given as follows: $α \tilde{\circ} f : S ⟶ P (U)$ $x ⟼ {\begin{matrix} ⋃_{x = yz} {α (y) \cap f (z)} if \exists y, z \in S such that x = yz, \\ \emptyset otherwise, \end{matrix}$ $β \tilde{\circ} g : S ⟶ P (U)$ $x ⟼ {\begin{matrix} ⋂_{x = yz} {β (y) \cup g (z)} if \exists y, z \in S such that x = yz, \\ U otherwise . \end{matrix}$

One can easily see that the operation “◇”is well-defined.

Let 〈 (α, β) ; A〉 and 〈 (f, g) ; B〉 be two double-framed soft sets of S over a common universe U. Then 〈 (α, β) ; A〉 is called a double-framed soft subset (briefly, DFS subset) of 〈 (f, g) ; B〉, denoted by 〈 (α, β) ; A〉 ⊑ 〈 (f, g) ; B 〉, if

(i) A ⊆ B,

(ii) (∀ e ∈ A) $(\begin{matrix} α (e) \subseteq f (e) \\ β (e) \supseteq g (e) \end{matrix}) .$

Two DFS-sets 〈 (α, β) ; A〉 and 〈 (f, g) ; B〉 are DFS equal if 〈 (α, β) ; A〉 ⊑ 〈 (f, g) ; B 〉 and 〈 (f, g) ; B〉 ⊑ 〈 (α, β) ; A 〉 denoted by $〈 (α, β); A 〉 \tilde{=} 〈 (f, g); B 〉 .$

For any two DFS sets 〈 (α, β) ; A〉 and 〈 (f, g) ; A〉 of S over U, the DFS int-uni set [14] of 〈 (α, β) ; A〉 and 〈 (f, g) ; A 〉 , is defined to be a DFS set $〈 (α \tilde{\cap} f, β \tilde{\cup} g); A 〉$ where $α \tilde{\cap} f,$ and $β \tilde{\cup} g$ are mappings given by $α \tilde{\cap} f : A \to P (U), x \to α (x) \cap f (x)$ and $β \tilde{\cup} g : A \to P (U), x \to β (x) \cup g (x)$ .

It is denoted by $〈 (α, β); A 〉 ⊓ 〈 (f, g); A 〉 = 〈 (α \tilde{\cap} f, β \tilde{\cup} g); A 〉$ .

For a non-empty subset A of S, the DFS set $X_{A} = (χ_{A}, χ_{A}^{c}; A)$ is called the double framed characteristic soft set where

$χ_{A} : S \to P (U), x \to {\begin{matrix} U if x \in A \\ \emptyset if x \notin A \end{matrix}$ and $χ_{A}^{c} : S \to P (U), x \to {\begin{matrix} \emptyset if x \in A \\ U if x \notin A \end{matrix} .$

We have the following lemmas.

Lemma 3.6. (cf. [21]) If S is an AG-groupoid, then the set (DFS (U) , ◇) is an AG-groupoid.

Lemma 3.7. (cf. [21]) If S is an AG-Groupoid, then the medial law holds in DFS (U).

That is, for 〈 (α, β) ; S 〉 , 〈 (f, g) ; S 〉 , 〈 (h, k) ; S〉 and 〈 (p, q) ; S 〉 ∈ DFS (U), we have $(α \tilde{\circ} f) \tilde{\circ} (h \tilde{\circ} p) = (α \tilde{\circ} h) \tilde{\circ} (f \tilde{\circ} p)$ and $(β \tilde{\circ} g) \tilde{\circ} (k \tilde{\circ} q) = (β \tilde{\circ} k) \tilde{\circ} (g \tilde{\circ} q) .$

Lemma 3.8. (cf. [21]) If S is an AG-groupoid with left identity, then the paramedial law holds in DFS (U) .

That is, for all 〈 (α, β) ; S 〉 , 〈 (f, g) ; S 〉 , 〈 (h, k) ; S〉 and 〈 (p, q) ; S 〉 ∈ DFS (U) ,

$(α \tilde{\circ} f) \tilde{\circ} (h \tilde{\circ} p) = (p \tilde{\circ} f) \tilde{\circ} (h \tilde{\circ} α)$ and $(β \tilde{\circ} g) \tilde{\circ} (k \tilde{\circ} q) = (q \tilde{\circ} g) \tilde{\circ} (k \tilde{\circ} β)$ .

Lemma 3.9. (cf. [21]) If S is an AG-groupoid with left identity then for all 〈 (f, g) ; S 〉 , 〈 (h, k) ; S〉 and 〈 (p, q) ; S 〉 ∈ DFS (U) ,

$f \tilde{\circ} (h \tilde{\circ} p) = h \tilde{\circ} (f \tilde{\circ} p)$ and $g \tilde{\circ} (k \tilde{\circ} q) = k \tilde{\circ} (g \tilde{\circ} q) .$

Lemma 3.10. Let 〈 (α, β) ; S〉 , 〈 (f, g) ; S 〉 , 〈 (h, k) ; S 〉 and 〈 (p, q) ; S 〉 ∈ DFS (U). Then,

i) 〈 (α, β) ; S 〉 ◇ (〈 (f, g) ; S 〉 ⊓ 〈 (h, k) ; S 〉) = (〈 (α, β) ; S 〉 ◇ 〈 (f, g) ; S 〉) ⊓ (〈 (α, β) ; S 〉 ◇ 〈 (h, k) ; S 〉) .

ii) If 〈 (f, g) ; S〉 ⊑ 〈 (h, k) ; S 〉, then 〈 (α, β) ; S 〉 ◇ 〈 (f, g) ; S 〉 ⊑ 〈 (α, β) ; S 〉 ◇ 〈 (h, k) ; S 〉 .

iii) If 〈 (α, β) ; S〉 ⊑ 〈 (f, g) ; S 〉 and 〈 (h, k) ; S〉 ⊑ 〈 (p, q) ; S 〉, then 〈 (α, β) ; S 〉 ◇ 〈 (h, k) ; S 〉 ⊑ 〈 (f, g) ; S 〉 ◇ 〈 (p, q) ; S 〉 .

Proof. Straightforward.□

Lemma 3.11. Let A and B be two non empty subsets of an AG-groupoid S. Then the following properties hold:

i) If A ⊆ B then X_A ⊑ X _B .

ii) X_A ⊓ X_B = X_A∩B .

iii) X_A ◇ X_B = X_AB .

Proof. Straightforward.□ The following definitions are available in [24].

Let S be an AG-groupoid and 〈 (α, β) ; A〉 be a DFS-set of S over U. Then:

(i) A DFS-set 〈 (α, β) ; A〉 is called a double-framed soft AG-groupoid (briefly, DFS AG-groupoid) of S over U if α (xy) ⊇ α (x) ∩ α (y) and β (xy) ⊆ β (x) ∪ β (y) for all x, y ∈ A.

(ii) A DFS-set 〈 (α, β) ; A〉 is called a double-framed soft left (resp. right) ideal (briefly, DFS left (resp. right)) ideal of S over U if α (ab) ⊇ α (b) (resp . α (ab) ⊇ α (a)) and β (ab) ⊆ β (b) (resp . β (ab) ⊆ β (a)) for all a, b ∈ A.

A DFS-set 〈 (α, β) ; A〉 of A over U is called a double-framed soft two sided ideal (briefly, DFS two-sided ideal) of S over U if it is both a DFS left and a DFS right ideal of A over U.

(iii) A DFS-set 〈 (α, β) ; A〉 is called a double-framed soft semiprime if α (a) ⊇ α (a²) and β (a) ⊆ β (a²) for all a ∈ A . (iv) A DFS-set 〈 (α, β) ; A〉 of A over U is called a double-framed soft generalized bi-ideal (briefly, DFS generalized bi-ideal) of S over U if α ((xa) y) ⊇ α (x) ∩ α (y) and β ((xa) y) ⊆ β (x) ∪ β (y) for all a, x, y ∈ A. (v) A DFS-set 〈 (α, β) ; A〉 is called a double-framed soft bi-ideal (briefly, DFS bi-ideal) of S over U if it satisfies:

(a) α (xy) ⊇ α (x) ∩ α (y) and β (xy) ⊆ β (x) ∪ β (y) for all x, y ∈ A.

(b) α ((xa) y) ⊇ α (x) ∩ α (y) and β ((xa) y) ⊆ β (x) ∪ β (y) for all a, x, y ∈ A.

(vi) 〈 (α, β) ; A〉 is called a double-framed soft interior ideal (briefly, DFS interior ideal) of S over U if α ((xa) y) ⊇ α (a) and β ((xa) y) ⊆ β (a) for all a, x, y ∈ A.

Proposition 3.12. (see [21]) A DFS set 〈 (α, β) ; S〉 of an AG-groupoid S over U is DFS left (resp. right) ideal of S if and only if X_S◇ 〈 (α, β) ; S 〉 ⊑ 〈 (α, β) ; S 〉 (resp. 〈 (α, β) ; S〉 ◇ X_S ⊑ 〈 (α, β) ; S 〉).

Proposition 3.13. (see [21]) A DFS set 〈 (α, β) ; S〉 of an AG-groupoid S over U is DFS bi-ideal if and only if 〈 (α, β) ; S〉 ◇ 〈 (α, β) ; S 〉 ⊑ 〈 (α, β) ; S 〉 and (〈 (α, β) ; S 〉 ◇ X_S) ◇ 〈 (α, β) ; S 〉 ⊑ 〈 (α, β) ; S 〉.

Proposition 3.14. (see [24]) A DFS set 〈 (α, β) ; S〉 of an AG-groupoid S over U is DFS interior ideal if and only if (X_S ◇ 〈 (α, β) ; S 〉) ◇ X_S ⊑ 〈 (α, β) ; S 〉 .

Proposition 3.15. (see [24]) If an AG-groupoid S contains left identity, then X_S ◇ X_S = X_S .

Proposition 3.16. (see [21]) If S is an AG-groupoid and A is a non-empty subset of S, then

(i) A is a left (resp. right) ideal of S if and only if the DFS-set $X_{A} = 〈 (χ_{A}, χ_{A}^{c}); A 〉$ is a DFS left (resp. right) ideal of S over U.

(ii) A is an interior ideal of S if and only if the DFS-set $X_{A} = 〈 (χ_{A}, χ_{A}^{c}); A 〉$ is a DFS interior ideal of S over U.

(iii) A is a bi-ideal of S if and only if the DFS-set $X_{A} = 〈 (χ_{A}, χ_{A}^{c}); A 〉$ is a DFS bi-ideal ideal of S over U.

(iv) A is semiprime subset of S if and only if the DFS-set $X_{A} = 〈 (χ_{A}, χ_{A}^{c}); A 〉$ is a DFS semiprime subset of S over U.

(v) A is interior ideal of S if and only if the DFS-set $X_{A} = 〈 (χ_{A}, χ_{A}^{c}); A 〉$ is DFS interior ideal of S over U .

4 Double-framed soft quasi-ideals

In this section, we define double-framed soft quasi-ideals (briefly, DFS quasi-ideals) of AG-groupoids and discuss its properties in AG-groupoids and left regular AG-groupoids.

Definition 4.1. A DFS-set 〈 (α, β) ; A〉 of A over U is called DFS quasi-ideal if $〈 (α, β); A 〉 ◇ X_{S} ⊓ X_{S} ◇ 〈 (α, β); A 〉 ⊑ 〈 (α, β); A 〉$ (3) That is, $(α \tilde{\circ} χ_{S}) \tilde{\cap} (χ_{S} \tilde{\circ} α) \tilde{\subseteq} α$ and $(β \tilde{\circ} χ_{S}^{c}) \tilde{\cup} (χ_{S}^{c} \tilde{\circ} β) \tilde{\supseteq} β .$

Example 4.2. Consider the set S = {e, a, b, c} with the following multiplication table:

·	e	a	b	c
e	e	a	b	c
a	c	b	b	b
b	b	b	b	b
c	a	b	b	b

Define DFS-set 〈 (α, β) ; S〉 of S over a universe U in which α : S → P (U) and β : S → P (U) are given as follows

$α (x) = {\begin{matrix} η_{1} if x = e \\ η_{2} if x = a \\ η_{3} if x = b \\ η_{4} if x = c \end{matrix}$ and $β (x) = {\begin{matrix} ρ_{1} if x = e \\ ρ_{2} if x = a \\ ρ_{3} if x = b \\ ρ_{4} if x = c \end{matrix},$

where η₁, η₂, η₃, η₄ are subsets of U such that η₁ ⊆ η₂ ⊆ η₄ ⊆ η₃ and ρ₁, ρ₂, ρ₃, ρ₄ are subsets of U such that ρ₁ ⊇ ρ₂ ⊇ ρ₄ ⊇ ρ₃ .

Then calculations show that 〈 (α, β) ; S〉 is DFS quasi-ideal of S over U .

Theorem 4.3. Let 〈 (α, β) ; S〉 and 〈 (f, g) ; S〉 be DFS right ideals and DFS left ideal of an AG-groupoid S respectively. Then 〈 (α, β) ; S〉 ⊓ 〈 (f, g) ; S 〉 is a DFS quasi-ideal of S.

Proof. Since $((α \tilde{\cap} f) \tilde{\circ} χ_{S}) \tilde{\cap} (χ_{S} \tilde{\circ} (α \tilde{\cap} f)) \tilde{\subseteq} (α \tilde{\circ} χ_{S}) \tilde{\cap} (χ_{S} \tilde{\circ} α) \tilde{\subseteq} α$ and $((β \tilde{\cup} g) \tilde{\circ} χ_{S}^{c}) \tilde{\cup} (χ_{S}^{c} \tilde{\circ} (β \tilde{\cup} g)) \tilde{\supseteq} (β \tilde{\circ} χ_{S}^{c}) \tilde{\cup} (χ_{S}^{c} \tilde{\circ} α) \tilde{\supseteq} β$ hence 〈 (α, β) ; S〉 ⊓ 〈 (f, g) ; S 〉 is DFS quasi-ideal of S.□

Theorem 4.4. Every DFS quasi-ideal 〈 (α, β) ; S〉 of an AG-groupoid S is a DFS AG-groupoid of S.

Proof. Let 〈 (α, β) ; S〉 be a DFS quasi-ideal of AG-groupoid S. Now 〈 (α, β) ; S 〉 ◇ 〈 (α, β) ; S 〉 ⊑ 〈 (α, β) ; S 〉 ◇ X_S and 〈 (α, β) ; S〉 ◇ 〈 (α, β) ; S 〉 ⊑ X_S ◇ 〈 (α, β) ; S 〉 which means that $α \circ α \subseteq α \circ χ_{S} and α \circ α \subseteq χ_{S} \circ α$ and $β \circ β \supseteq β \circ χ_{S}^{c} and β \circ β \supseteq χ_{S}^{c} \circ β .$ Thus $α \tilde{\circ} α \tilde{\subseteq} α \tilde{\circ} χ_{S} \tilde{\cap} χ_{S} \tilde{\circ} α \tilde{\subseteq} α$ and $β \tilde{\circ} β \tilde{\supseteq} β \tilde{\circ} χ_{S}^{c} \tilde{\cup} χ_{S}^{c} \tilde{\circ} β \tilde{\supseteq} β$ and hence 〈 (α, β) ; S〉 ◇ 〈 (α, β) ; S 〉 ⊑ 〈 (α, β) ; S 〉. Thus 〈 (α, β) ; S〉 is DFS AG-groupoid.□

Theorem 4.5. In an AG-groupoid S, every idempotent DFS quasi-ideal is DFS bi-ideal of S .

Proof. Let 〈 (α, β) ; S〉 be any idempotent DFS quasi-ideal of AG-groupoid S. This means that $〈 (α, β); S 〉 ◇ 〈 (α, β); S 〉 \tilde{=} 〈 (α, β); S 〉$ . That is, $α \tilde{\circ} α = α$ and $β \tilde{\circ} β = β .$ Since 〈 (α, β) ; S〉 is DFS quasi-ideal, then by Theorem 4.4, 〈 (α, β) ; S〉 is DFS AG-groupoid. Now using left invertive law and medial law, we have $(α \tilde{\circ} χ_{S}) \tilde{\circ} α \tilde{\subseteq} (χ_{S} \tilde{\circ} χ_{S}) \tilde{\circ} α \tilde{\subseteq} χ_{S} \tilde{\circ} α$ and $(β \tilde{\circ} χ_{S}^{c}) \tilde{\circ} β \tilde{\supseteq} (χ_{S}^{c} \tilde{\circ} χ_{S}^{c}) \tilde{\circ} β \tilde{\supseteq} χ_{S}^{c} \tilde{\circ} β .$ Now $(α \tilde{\circ} χ_{S}) \tilde{\circ} α = (α \tilde{\circ} χ_{S}) \tilde{\circ} (α \tilde{\circ} α) = (α \tilde{\circ} α) \tilde{\circ} (χ_{S} \tilde{\circ} α) \tilde{\subseteq} α \tilde{\circ} (χ_{S} \tilde{\circ} χ_{S}) \tilde{\subseteq} α \tilde{\circ} χ_{S}$ and $(β \tilde{\circ} χ_{S}^{c}) \tilde{\circ} β = (β \tilde{\circ} χ_{S}^{c}) \tilde{\circ} (β \tilde{\circ} β) = (β \tilde{\circ} β) \tilde{\circ} (χ_{S}^{c} \tilde{\circ} β) \tilde{\supseteq} β \tilde{\circ} (χ_{S}^{c} \tilde{\circ} χ_{S}^{c}) \tilde{\supseteq} β \tilde{\circ} χ_{S} .$

Hence $(α \tilde{\circ} χ_{S}) \tilde{\circ} α \tilde{\subseteq} χ_{S} \tilde{\circ} α \tilde{\cap} α \tilde{\circ} χ_{S} \tilde{\subseteq} α$ and $(β \tilde{\circ} χ_{S}^{c}) \tilde{\circ} β \tilde{\supseteq} χ_{S}^{c} \tilde{\circ} β \tilde{\cup} β \tilde{\circ} χ_{S} \tilde{\supseteq} β$ . Thus (〈 (α, β) ; S 〉 ◇ X_S) ◇ 〈 (α, β) ; S 〉 ⊑ 〈 (α, β) ; S 〉 and so 〈 (α, β) ; S〉 is DFS bi-ideal of S.□

Theorem 4.6. Every DFS left (resp. right) ideal of an AG-groupoid is DFS quasi-ideal.

Proof. Straightforward.□ The converse of this theorem is not true in general.

Example 4.7. Consider the set S = {a, b, c, d, e} with the following multiplication table

·	a	b	c	d	f
a	d	a	a	b	d
b	a	a	a	a	a
c	a	a	a	a	a
d	a	a	a	b	a
f	d	a	a	a	d

Define DFS-set 〈 (α, β) ; S〉 of S over a universe U in which α : S → P (U) and β : S → P (U) are given as follows

$α (x) = {\begin{matrix} η if x = a \\ η_{2} if x = b \\ η_{3} if x = c \\ η_{4} if x = d \\ η if x = f \end{matrix}$ and $β (x) = {\begin{matrix} ρ if x = a \\ ρ_{2} if x = b \\ ρ_{3} if x = c \\ ρ_{4} if x = d \\ ρ if x = f \end{matrix},$

where η, η₂, η₃, η₄ are subsets of U such that η₄ ⊆ η₃ ⊆ η and η₃ ⊆ η and ρ₁, ρ₂, ρ₃, ρ₄ are subsets of U such that ρ₄ ⊇ ρ₂ ⊇ ρ and ρ ⊇ ρ₃ . Then 〈 (α, β) ; S〉 is DFS quasi-ideal of S over U .

Now α (b) = α (ad) nsupseteqα (a) and/or β (b) = β (ad) ⊈ β (a) which shows that 〈 (α, β) ; S〉 is not DFS right ideal of S. Also α (d) = α (aa) ⊈ α (a) and/or β (d) = β (aa) nsupseteqβ (a) which shows that 〈 (α, β) ; S〉 is not DFS left ideal of S .

Theorem 4.8. Intersection of two DFS quasi-ideals of an AG-groupoid is a DFS quasi-ideal.

Proof. It follows from Lemma 3.10 and Definition 4.1.□

Theorem 4.9. For a DFS-set 〈 (α, β) ; S〉 of an AG-groupoid S, the following statements are equivalent:

(i) 〈 (α, β) ; S〉 is a DFS quasi-ideal of S .

(ii) The non-empty γ-inclusive set and δ-exclusive set of 〈 (α, β) ; S〉 are quasi-ideals of S for any subsets γ and δ of U.

Proof. (i)⇒(ii) Suppose that 〈 (α, β) ; S〉 is DFS quasi-ideal of S. Let γ and δ be subsets of U such that i_S (α ; γ) ¬ = ∅ ¬ = e_S (β ; δ). Let a ∈ i_S (α ; γ) S ∩ Si_S (α ; γ). Then a ∈ i_S (α ; γ) S and a ∈ Si_S (α ; γ). Thus there exist p, q ∈ i_S (α ; γ) and x, y ∈ S such that a = px ∈ i_S (α ; γ) S and a = yq ∈ Si_S (α ; γ) .

Now

$\begin{matrix} (α \tilde{\circ} χ_{S}) (a) & = & ⋃_{a = mn} {α (m) \cap χ_{S} (n)} \supseteq α (p) \cap χ_{S} (x) \\ = & α (p) \supseteq γ \end{matrix}$ (4) and

$\begin{matrix} (χ_{S} \tilde{\circ} α) (a) & = & ⋃_{a = mn} {χ_{S} (m) \cap α (n)} \supseteq χ_{S} (y) \cap α (q) \\ = & α (q) \supseteq γ \end{matrix}$ (5)

Since 〈 (α, β) ; S〉 is DFS quasi-ideal of S, then using its definition along with (4) and (5), we get $α (a) \supseteq ((α \tilde{\circ} χ_{S}) \circ (χ_{S} \tilde{\circ} α)) (a) \supseteq γ .$ Thus a ∈ i_S (α ; γ) which shows that i_S (α ; γ) is quasi-ideal of S . Now let b ∈ e_S (β ; δ) S ∩ Se_S (β ; δ), so there exist k, l ∈ e_S (β ; δ) and c, d ∈ S such that b = kc ∈ e_S (β ; δ) S and b = dl ∈ Se_S (β ; δ) . Now

$\begin{matrix} (β \tilde{\circ} χ_{S}^{c}) (b) & = & ⋂_{b = mn} {β (m) \cup χ_{S}^{c} (n)} \\ \subseteq & β (k) \cup χ_{S}^{c} (c) = β (k) \subseteq δ \end{matrix}$ (6) and

$\begin{matrix} (χ_{S}^{c} \tilde{\circ} β) (b) & = & ⋂_{b = mn} {χ_{S}^{c} (m) \cup β (n)} \\ \subseteq & χ_{S}^{c} (d) \cup β (l) = β (q) \subseteq δ \end{matrix}$ (7)

Since 〈 (α, β) ; S〉 is DFS quasi-ideal of S, then using its definition along with (6) and (7), we get $β (b) \subseteq ((β \tilde{\circ} χ_{S}^{c}) \tilde{\cup} (χ_{S}^{c} \tilde{\circ} β)) (b) \subseteq δ .$ Thus b ∈ e_S (β ; δ) which shows that e_S (β ; δ) is quasi-ideal of S .

(ii)⇒(i) Suppose the non-empty γ-inclusive set and δ-exclusive set of 〈 (α, β) ; S〉 are quasi-ideals of S for any subsets γ and δ of U. Let x ∈ S . If x is not product of elements of S, then the result is proved. Suppose that there exist a, b ∈ S such that x = ab and let us consider a subset of U as γ = α (a) ∩ α (b), δ = β (a) ∪ β (b) . Since γ = α (a) ∩ α (b), then a, b ∈ i_S (α ; γ) . Thus x = ab∈ Si_S (α ; γ) ∩ i_S (α ; γ) S ⊆ i_S (α ; γ), since i_S (α ; γ) is quasi-ideal of S. Hence α (a) ∩ α (b) = γ ⊆ α (x). Also, as δ = β (a) ∪ β (b), then a, b ∈ e_S (β ; δ) and thus x = ab ∈ Se_S (β ; δ) ∩ e_S (β ; δ) S ⊆ e_S (β ; δ), since e_S (β ; δ) is quasi-ideal of S. Hence β (a) ∪ β (b) = δ ⊇ β (x) .

Now $\begin{matrix} ((α \tilde{\circ} χ_{S}) \tilde{\cap} (χ_{S} \tilde{\circ} α)) (x) = (α \tilde{\circ} χ_{S}) (x) \cap (χ_{S} \tilde{\circ} α) (x) \\ = {⋃_{x = pq} {α (p) \cap χ_{S} (q)}} \cap {⋃_{x = pq} {χ_{S} (p) \cap α (q)}} \\ = {⋃_{x = pq} {α (p)}} \cap {⋃_{x = pq} {α (q)}} \\ = ⋃_{x = pq} {α (p) \cap α (q)} \\ \subseteq & α (x) \end{matrix}$ and $\begin{matrix} ((β \tilde{\circ} χ_{S}^{c}) \tilde{\cup} (χ_{S}^{c} \tilde{\circ} β)) (x) = (β \tilde{\circ} χ_{S}^{c}) (x) \cup (χ_{S}^{c} \tilde{\circ} β) (x) \\ = {⋂_{x = pq} {β (p) \cup χ_{S}^{c} (q)}} \cup {⋂_{x = pq} {χ_{S}^{c} (p) \cup β (q)}} \\ = {⋂_{x = pq} {β (p)}} \cup {⋂_{x = pq} {β (q)}} \\ = ⋂_{x = pq} {β (p) \cup β (q)} \\ \supseteq & β (x) \end{matrix}$ Hence 〈 (α, β) ; A〉 ◇ X_S ⊓ X_S ◇ 〈 (α, β) ; A 〉 ⊑ 〈 (α, β) ; A 〉 which implies 〈 (α, β ; S)〉 is DFS quasi-ideal of S.□

Theorem 4.10. Let A be a non-empty subset of an AG-groupoid S. Then A is an quasi-ideal of S if and only if the DFS-set $X_{A} = 〈 (χ_{A}, χ_{A}^{c}); A 〉$ is a DFS quasi-ideal of S over U.

Proof. Assume that A is quasi-ideal of S. We have $X_{A} = 〈 (χ_{A}, χ_{A}^{c}); A 〉$ is DFS characteristic function of A. Let x ∈ S.

If x ∉ A, then x ∉ SA ∩ AS. It means that x ∉ SA or x ∉ AS or x ∉ SA as well as AS. In this case, χ_A (x) =∅ and $χ_{A}^{c} (x) = U .$ Now if x ∉ AS, then $χ_{AS} (x) = \emptyset = (χ_{A} \tilde{\circ} χ_{S}) (x)$ and $χ_{AS}^{c} (x) = U = (χ_{A}^{c} \tilde{\circ} χ_{S}^{c}) (x) .$ Thus $(χ_{A} \tilde{\circ} χ_{S}) (x) \cap (χ_{S} \tilde{\circ} χ_{A}) (x) = \emptyset = χ_{A} (x)$ and $(χ_{A}^{c} \tilde{\circ} χ_{S}^{c}) (x) \cup (χ_{S}^{c} \tilde{\circ} χ_{A}^{c}) (x) = U = χ_{A}^{c} (x) .$

Similarly, if x ∉ SA or x ∉ AS as well as SA, then we have $(χ_{A} \tilde{\circ} χ_{S}) (x) \cap (χ_{S} \tilde{\circ} χ_{A}) (x) = \emptyset = χ_{A} (x)$ and $(χ_{A}^{c} \tilde{\circ} χ_{S}^{c}) (x) \cup (χ_{S}^{c} \tilde{\circ} χ_{A}^{c}) (x) = U = χ_{A}^{c} (x) .$

If x ∈ A, then x ∈ AS ∩ SA or x ∉ AS ∩ SA .

Case (i) If x ∈ SA ∩ AS, then x ∈ SA as well as x ∈ AS. Thus χ_A (x) = U, $χ_{SA} (x) = U = (χ_{S} \tilde{\circ} χ_{A}) (x), χ_{AS} (x) = U = (χ_{A} \tilde{\circ} χ_{S}) (x)$ . Hence $(χ_{A} \tilde{\circ} χ_{S}) (x) \cap (χ_{S} \tilde{\circ} χ_{A}) (x) = U = χ_{A} (x)$ and $χ_{A}^{c} (x) = \emptyset$ , $χ_{SA}^{c} (x) = \emptyset = (χ_{S} \tilde{\circ} χ_{A}) (x)$ , $χ_{AS}^{c} (x) = \emptyset = (χ_{A}^{c} \tilde{\circ} χ_{S}^{c}) (x) .$ Therefore $(χ_{A}^{c} \tilde{\circ} χ_{S}^{c}) (x) \cup (χ_{S}^{c} \tilde{\circ} χ_{A}^{c}) (x) = \emptyset = χ_{A}^{c} (x) .$

Case (ii) If x ∉ SA ∩ AS, then x ∉ SA or x ∉ AS.

If x ∉ SA, then $χ_{SA} (x) = \emptyset = (χ_{S} \tilde{\circ} χ_{A}) (x)$ and so $(χ_{A} \tilde{\circ} χ_{S}) (x) \cap (χ_{S} \tilde{\circ} χ_{A}) (x) = \emptyset \subseteq χ_{A} (x) .$ Similarly, when x ∉ AS then $(χ_{A} \tilde{\circ} χ_{S}) (x) \cap (χ_{S} \tilde{\circ} χ_{A}) (x) = \emptyset \subseteq χ_{A} (x) .$ Similarly, the other case also yields the same result. Hence $X_{A} = 〈 (χ_{A}, χ_{A}^{c}); A 〉$ is a DFS quasi-ideal of S over U.

Conversely, assume that $X_{A} = 〈 (χ_{A}, χ_{A}^{c}); A 〉$ is a DFS quasi-ideal of S over U. Let x ∈ AS ∩ SA. Then $U = χ_{AS \cap SA} (x) = χ_{SA} (x) \cap χ_{AS} (x) = (χ_{S} \tilde{\circ} χ_{A}) (x) \cap (χ_{A} \tilde{\circ} χ_{S}) (x) \subseteq χ_{A} (x) .$ Hence χ_A (x) = U and so x ∈ A . Also $\emptyset = χ_{AS \cap SA}^{c} (x) = {(χ_{AS} (x) \cap χ_{SA} (x))}^{c} = χ_{AS}^{c} (x) \cup χ_{SA}^{c} (x) = (χ_{A}^{c} \tilde{\circ} χ_{S}^{c}) (x) \cup (χ_{S}^{c} \tilde{\circ} χ_{A}^{c}) (x) \supseteq χ_{A}^{c} (x)$ , hence $χ_{A}^{c} (x) = \emptyset$ and thus x ∈ A. Therefore A is quasi-ideal of S.□

5 Left regular AG-groupoids

An element a of an AG-groupoid S is left regular if there exists x ∈ S such that a = xa² . An AG-groupoid S is called left regular if every element of S is left regular.

Example 5.1. Let us consider the set S = {0, 1, 2, 3, 4} with the following multiplication table

·	1	2	3	4
0	0	0	0	0
1	1	1	1	1
2	1	3	4	2
3	1	2	3	4
4	1	4	2	3

Obviously S is an AG-groupoid and it is non-associative as well as non-commutative.

Note that 0 = 1 ·0², 1 = 2 ·1², 2 = 4 ·2², 3 = 3 ·3², 4 = 2 ·4², which shows that S is left regular.

Lemma 5.2. In a left regular AG-groupoid S with left identity, for every DFS-set 〈 (α, β) ; S〉 of S over U, the followings hold:

(i) 〈 (α, β) ; S 〉 ⊑ 〈 (α, β) ; S 〉 ◇ X_S .

(ii) 〈 (α, β) ; S〉 ⊑ X_S ◇ 〈 (α, β) ; S 〉.

Proof. (i) Let S be an AG-groupoid with left identity say e and 〈 (α, β) ; S〉 be a DFS-set of S over U and a ∈ S . Since S is left regular, then there exists x ∈ S such that a = xa² = x (aa) = a (xa) .

We have $(α \tilde{\circ} χ_{S}) (a) = ⋃_{a = xy} {α (x) \cap χ_{S} (y)} \supseteq α (a) \cap χ_{S} (xa) = α (a)$ ,

and $(β \tilde{\circ} χ_{S}^{c}) (a) = ⋂_{a = xy} {β (x) \cup χ_{S}^{c} (y)} \subseteq β (a) \cup χ_{S}^{c} (xa) = β (a)$ .

Thus $α \tilde{\subseteq} α \tilde{\circ} χ_{S}$ and $β \tilde{\supseteq} β \tilde{\circ} χ_{S}^{c}$ . Hence 〈 (α, β) ; S 〉 ⊑ 〈 (α, β) ; S 〉 ◇ X_S .

(ii) Let S be an AG-groupoid with left identity say e and 〈 (α, β) ; S〉 be a DFS-set of S over U and a ∈ S . Since S is left regular, then there exists x ∈ S such that a = xa² = x (aa) = (ex) (aa) = (ea) (xa) = a (xa) = (xa²) (xa) = (ax) (a²x) = (ax) ((aa) x)) = (ax) ((xa) a) = (xa) ((ax) a) = (a (ax)) (ax) = ((ax) (ax)) a = ((ax) ²) a = ta where t = (ax) ².

We have $(χ_{S} \tilde{\circ} α) (a) = ⋃_{a = xy} {χ_{S} (y) \cap α (x)} \supseteq χ_{S} (t) \cap α (a) = α (a)$ ,

and $(χ_{S}^{c} \tilde{\circ} β) (a) = ⋂_{a = xy} {χ_{S} (y) \cup β (x)} \subseteq χ_{S} (t) \cup β (a) = β (a) .$

Thus $α \tilde{\subseteq} χ_{S} \tilde{\circ} α$ and $β \tilde{\supseteq} χ_{S}^{c} \tilde{\circ} β$ . Hence 〈 (α, β) ; S 〉 ⊑ X_S ◇ 〈 (α, β) ; S 〉 .□

Corollary 5.3. In a left regular AG-groupoid S with left identity, for every DFS-set 〈 (α, β) ; S〉 of S over U, we have 〈 (α, β) ; S〉 ⊑ 〈 (α, β) ; S 〉 ◇ X_S ⊓ X_S ◇ 〈 (α, β) ; S 〉.

As a consequence of Lemma 4.3 and Definition 4.1, we have the following theorem.

Theorem 5.4. In a left regular AG-groupoid S with left identity, the following conditions are equivalent:

(i) 〈 (α, β) ; S〉 is DFS quasi-ideal of S over U .

(ii) $(〈 (α, β); S 〉 ◇ X_{S}) ⊓ (X_{S} ◇ 〈 (α, β); S 〉) \tilde{=} 〈 (α, β); S 〉 .$

Proof. (i)⇒(ii) It follows from Definition 4.1 and Corollary 4.3.

(ii)⇒(i) It is obvious.□

Theorem 5.5. In a left regular AG-groupoid S with left identity, the followings are equivalent:

(i) DFS-set 〈 (α, β) ; S〉 of S over U is a DFS right ideal.

(ii) DFS-set 〈 (α, β) ; S〉 of S over U is a DFS left ideal.

(iii) DFS-set 〈 (α, β) ; S〉 of S over U is a DFS bi-ideal.

(iv) DFS-set 〈 (α, β) ; S〉 of S over U is a DFS generalized bi-ideal.

(v) DFS-set 〈 (α, β) ; S〉 of S over U is a DFS interior ideal.

(vi) DFS-set 〈 (α, β) ; S〉 of S over U is a DFS quasi-ideal.

(vii) $〈 (α, β); S 〉 ◇ X_{S} \tilde{=} 〈 (α, β); S 〉$ and $X_{S} ◇ 〈 (α, β); S 〉 \tilde{=} 〈 (α, β); S 〉$ .

Proof. (i)⇒(vii) Let S be a left regular AG-groupoid with left identity say e and 〈 (α, β) ; S〉 be a DFS right ideal of S over U . Then by Proposition 3.11, 〈 (α, β) ; S〉 ◇ X_S ⊑ 〈 (α, β) ; S 〉 and by Lemma 5.2 (i), 〈 (α, β) ; S 〉 ⊑ 〈 (α, β) ; S 〉 ◇ X_S .. Thus $〈 (α, β); S 〉 ◇ X_{S} \tilde{=} 〈 (α, β); S 〉$ . Also by Lemma 5.2 (ii), 〈 (α, β) ; S 〉 ⊑ X_S ◇ 〈 (α, β) ; S 〉 .

Now $χ_{S} \tilde{\circ} α = (χ_{S} \tilde{\circ} χ_{S}) \tilde{\circ} α = (α \tilde{\circ} χ_{S}) \tilde{\circ} χ_{S} \tilde{\subseteq} α \tilde{\circ} χ_{S} \tilde{\subseteq} α$ and $χ_{S}^{c} \tilde{\circ} β = (χ_{S}^{c} \tilde{\circ} χ_{S}^{c}) \tilde{\circ} β = (β \tilde{\circ} χ_{S}^{c}) \tilde{\circ} χ_{S}^{c} \tilde{\supseteq} β \tilde{\circ} χ_{S}^{c} \tilde{\supseteq} β$ . Thus X_S◇ 〈 (α, β) ; S 〉 ⊑ 〈 (α, β) ; S 〉 and hence $X_{S} ◇ 〈 (α, β); S 〉 \tilde{=} 〈 (α, β); S 〉 .$

(vii)⇒(vi) It is obvious.

(vi)⇒(v) Let S be a left regular AG-groupoid with left identity say e and 〈 (α, β) ; S〉 be a DFS quasi-ideal of S. Let a, x, y ∈ S. Since S is left regular, then for a ∈ S, there exists t ∈ S such that a = ta². Now (xa) y = (xa) (ey) = (ye) (ax) = a ((ye) x) .

Also (xa) y = (x (ta²)) y = ((ex) (ta²)) y = ((a²t) (xe)) y = (x ((a²t) e)) y = (x ((et) a²)) y = (x (ta²)) y = (x (t (aa))) y = (x (a (ta))) y = (a (x (ta))) y = (y (x (ta))) a.

Now $(α \tilde{\circ} χ_{S}) ((xa) y) = ⋃_{(xa) y = mn} {α (m) \cap χ_{S} (n)} \supseteq α (a) \cap χ_{S} ((ye) x) = α (a)$

and $(χ_{S} \tilde{\circ} α) ((xa) y) = ⋃_{(xa) y = mn} {χ_{S} (m) \cap α (n)} \supseteq χ_{S} (y (x (ta))) \cap α (a) = α (a)$ .

Since $α ((xa) y) \supseteq (α \tilde{\circ} χ_{S}) ((xa) y) \cap (χ_{S} \tilde{\circ} α) ((xa) y)$ and $(α \tilde{\circ} χ_{S}) ((xa) y) \cap (χ_{S} \tilde{\circ} α) ((xa) y) \supseteq α (a)$ , then α ((xa) y) ⊇ α (a) .

Also $(β \tilde{\circ} χ_{S}^{c}) ((xa) y) = ⋂_{(xa) y = mn} {β (m) \cup χ_{S}^{c} (n)} \subseteq β (a) \cup χ_{S}^{c} ((ye) x) = β (a)$ and $(χ_{S}^{c} \tilde{\circ} β) ((xa) y) = ⋂_{(xa) y = mn} {χ_{S}^{c} (m) \cup β (n)} \subseteq χ_{S}^{c} (y (x (ta))) \cup β (a) = β (a)$ . Since $β ((xa) y) \subseteq (β \tilde{\circ} χ_{S}^{c}) ((xa) y) \cup (χ_{S}^{c} \tilde{\circ} β) ((xa) y)$ and $(β \tilde{\circ} χ_{S}^{c}) ((xa) y) \cup (χ_{S}^{c} \tilde{\circ} β) ((xa) y) \subseteq β (a)$ , then β ((xa) y) ⊆ β (a) .

Hence 〈 (α, β) ; S〉 is DFS interior ideal of S.

(v)⇒(iv) Let S be a left regular AG-groupoid with left identity say e and 〈 (α, β) ; S〉 be a DFS interior ideal of S. Let a, x, y ∈ S. Since S is left regular, then there exist u, v, w ∈ S such that a = ua², x = vx², y = wy².

Now (xa) y = ((vx²) a) y = ((v (xx)) a) y = ((x (vx)) a) y = ((a (vx)) x) y = (tx) y where t = a (vx) and (xa) y = (xa) (wy²) = (xa) (w (yy)) = (xa) (y (wy)) = (xy) (a (wy)) = (xy) p where p = a (wy) .

Thus α ((xa) y) = α ((tx) y) ⊇ α (x) and α ((xa) y) = α ((xy) p) ⊇ α (y), so α ((xa) y) ⊇ α (x) ∩ α (y) .

Also β ((xa) y) = β ((tx) y) ⊆ β (x) and β ((xa) y) = β ((xy) p) ⊆ β (y), so β ((xa) y) ⊆ β (x) ∪ β (y) . Hence 〈 (α, β) ; S〉 is DFS generalized bi-ideal.

(iv)⇒(iii) Let S be a left regular AG-groupoid with left identity say e and 〈 (α, β) ; S〉 be a DFS generalized bi-ideal of S. Let a, b ∈ S. Since S is left regular, then there exist u, m ∈ S such that a = ua² and b = mb² .

Now ab = (ua²) b = ((eu) (aa)) b = ((ea) (ua)) b = (b (ua)) a = (b (ua)) (ua²) = (b (ua)) (u (aa)) = (bu) ((ua) (aa)) = ((aa) u) ((ua) b) = ((ua) a) k where k = (ua) b, and ab = a (mb²) = a (m (bb)) = a (b (mb)) = b (a (mb)) = (mb²) (a (mb)) = (m (bb)) (m (ab)) = (b (mb)) (m (ab)) = ((ab) m) ((mb) b)) = (mb) (((ab) m) b) = (mb) p where p = (ab) m) b .

Thus α (ab) = α (((ua) a) k) ⊇ α (a) and α (ab) = α ((mb) p) ⊇ α (b) and so α (ab) ⊇ α (a) ∩ α (b) . Similarly, β (ab) ⊆ β (a) ∪ β (b) . Hence 〈 (α, β) ; S〉 is DFS bi-ideal.

(iii)⇒(ii) Let S be a left regular AG-groupoid with left identity say e and 〈 (α, β) ; S〉 be a DFS bi-ideal of S and a, b ∈ S. Since S is left regular, then there exists y ∈ S such that b = yb² .

Now ab = a (yb²) = a ((ey) (bb)) = a ((bb) (ye)) = a ((bb) t) = (bb) (at) = ((at) b) b = ((at) (yb²)) b = ((at) ((ey) (bb))) b = ((at) ((bb) (ye))) b = ((at) ((bb) t)) b = ((at) ((tb) b)) b = ((a (tb)) (tb)) b = ((bt) ((tb) a)) b = ((tb) ((bt) a)) b = ((a (bt)) (bt)) b = (b ((a (bt)) t)) b = (bp) b where p = (a (bt)) t) .

Thus α (ab) = α ((bp) b) ⊇ α (b) ∩ α (b) = α (b) and β (ab) = β ((bp) b) ⊆ β (b) ∪ β (b) = β (b) . Hence 〈 (α, β) ; S〉 is DFS left ideal of S.

(ii)⇒(i) Let S be a left regular AG-groupoid with left identity say e and 〈 (α, β) ; S〉 be a DFS left ideal of S and a, b ∈ S. Since S is left regular, then there exists z ∈ S such that a = za² .

Now ab = (xa²) b = (x (aa)) b = (a (xa)) b = (b (xa)) a .

Thus α (ab) = α ((b (xa)) a) ⊇ α (a) and β (ab) = β ((b (xa)) a) ⊆ β (a). Hence 〈 (α, β) ; S〉 is DFS right ideal of S.□

Lemma 5.6. (cf. [8]) If S is an AG-groupoid with left identity, then Sa, Sa², Sa ∩ aS are quasi-ideals of S for all a ∈ S .

Theorem 5.7. If S is an AG-groupoid with left identity e, then the following statements are equivalent:

(i) S is left regular.

(ii) Every DFS quasi-ideal is idempotent.

Proof. (i)⇒(ii) Assume that S is left regular. Let 〈 (f, g) ; S〉 be a DFS quasi-ideal of S. By Theorem 5.5, 〈 (f, g) ; S〉 is a DFS ideal of S .

Now $f \tilde{\circ} f \tilde{\subseteq} f \tilde{\circ} χ_{S}$ and $f \tilde{\circ} f \tilde{\subseteq} χ_{S} \tilde{\circ} f$ , so $f \tilde{\circ} f \tilde{\subseteq} f \tilde{\circ} χ_{S} \tilde{\cap} χ_{S} \tilde{\circ} f \tilde{\subseteq} f .$ Also $g \tilde{\circ} g \tilde{\supseteq} g \tilde{\circ} χ_{S}^{c}$ and $g \tilde{\circ} g \tilde{\supseteq} χ_{S}^{c} \tilde{\circ} g$ , so $g \tilde{\circ} g \tilde{\supseteq} g \tilde{\circ} χ_{S}^{c} \tilde{\cup} χ_{S}^{c} \tilde{\circ} g \tilde{\supseteq} g .$ Thus 〈 (f, g) ; S 〉 ◇ 〈 (f, g) ; S 〉 ⊑ 〈 (f, g) ; S 〉 .

Now let a ∈ S. Since S is left regular, then there exists x ∈ S such that a = xa² .

We have a = x ((xa²) ²) = x ((xa²) (xa²)) = (xa²) (x (xa²)) = (x (aa)) (x (xa²)) = (a (xa)) (x (xa²)) = (ax) ((xa) (xa²)) = (xa) ((ax) (xa²)).

Now $(f \tilde{\circ} f) (a) = ⋃_{a = mn} {f (m) \cap f (n)} \supseteq f (xa) \cap f ((ax) ({xa}^{2})) \supseteq f (a) \cap f (ax) \supseteq f (a) \cap f (a) = f (a) .$ Thus $f \tilde{\subseteq} f \tilde{\circ} f .$

Also, $(g \tilde{\circ} g) (a) = ⋂_{a = mn} {g (m) \cup g (n)} \subseteq g (xa) \cup g ((ax) ({xa}^{2})) \subseteq g (a) \cup g (ax) \subseteq g (a) \cup g (a) = g (a) .$ Thus $g \tilde{\supseteq} g \tilde{\circ} g$ and so 〈 (f, g) ; S 〉 ⊑ 〈 (f, g) ; S 〉 ◇ 〈 (f, g) ; S 〉 .

Hence $〈 (f, g); S 〉 ◇ 〈 (f, g); S 〉 \tilde{=} 〈 (f, g); S 〉$ which shows that 〈 (f, g) ; S〉 is idempotent.

(ii)⇒(i) Suppose every DFS quasi-ideal is idempotent. Let a ∈ S. Since Sa is a quasi-ideal, then X_Sa is DFS quasi-ideal and by hypothesis, X_(Sa)(Sa) = X_{Sa
²} = X_Sa ◇ X_Sa = X_Sa . Hence X_Sa = X_{Sa
²} . Since a ∈ Sa = Sa², then S is left regular.□

Theorem 5.8. In an AG-groupoid S with left identity, the following statements are equivalent:

(i) S is left regular.

(ii) Every DFS quasi-ideal of semiprime.

Proof. (i)⇒(ii) Assume that S is left regular. Let 〈 (f, g) ; S〉 be a DFS quasi-ideal of S. By Theorem 5.5, 〈 (f, g) ; S〉 is a DFS ideal of S . Now let a ∈ S. Since S is left regular, then there exists x ∈ S such that a = xa² . We have a = x ((xa²) ²) = x ((xa²) (xa²)) = (xa²) (x (xa²)) .

Now f (a) = f ((xa²) (x (xa²))) ⊇ f (xa²) ⊇ f (a²) and g (a) = g ((xa²) (x (xa²))) ⊆ g (xa²) ⊆ g (a²). Hence 〈 (f, g) ; S〉 is DFS semiprime.

(ii)⇒(i) Assume every DFS quasi-ideal is DFS semiprime. Let a ∈ S . Since Sa² is quasi-ideal, then X_{Sa
²} is DFS quasi-ideal and by hypothesis, it is DFS semiprime.

Now a² ∈ Sa², so U = χ_{Sa
²} (a²) ⊆ χ_{Sa
²} (a) . Hence a ∈ Sa² showing that S is left regular. Similarly the χ_{Sa
²} part of X_{Sa
²} gives S is left regular.□

Theorem 5.9. In an AG-groupoid with left identity, the following statements are equivalent:

(i) S is left regular.

(ii) L ∩ Q ⊆ LQ for any left ideal L and any quasi-ideal Q of S .

(iii) 〈 (f, g) ; S〉 ⊓ 〈 (α, β) ; S 〉 ⊑ 〈 (f, g) ; S 〉 ◇ 〈 (α, β) ; S 〉 for any DFS left ideal 〈 (f, g) ; S〉 and DFS quasi-ideal 〈 (α, β) ; S〉 of S.

Proof. (i)⇒(iii) Assume S is left regular. Let 〈 (f, g) ; S〉 be a DFS left ideal and 〈 (α, β) ; S〉 be a DFS quasi-ideal of S. By Theorem 5.5, 〈 (α, β) ; S〉 becomes DFS ideal. Let a ∈ S. Since S is left regular, then there exists x ∈ S such that a = xa² .

Then a = x ((xa²) ²) = x ((xa²) (xa²)) = (xa²) (x (xa²)) = (x (aa)) (x (xa²)) = (a (xa)) (x (xa²)) = (ax) ((xa) (xa²)) = (xa) ((ax) (xa²)) .

Now $(f \tilde{\circ} α) (a) = ⋃_{a = mn} {f (m) \cap α (n)} \supseteq f (xa) \cap α ((ax) ({xa}^{2})) \supseteq f (a) \cap α (ax) \supseteq f (a) \cap α (a) = (f \cap α) (a) .$

Thus $f \tilde{\cap} g \tilde{\subseteq} f \tilde{\circ} α .$

Also, $(g \tilde{\circ} β) (a) = ⋂_{a = mn} {g (m) \cup β (n)} \subseteq g (xa) \cup β ((ax) ({xa}^{2})) \subseteq g (a) \cup β (ax) \subseteq g (a) \cup β (a) = (g \cup β) (a) .$

Thus $g \tilde{\cup} β \tilde{\supseteq} g \tilde{\circ} β .$

Hence 〈 (f, g) ; S〉 ⊓ 〈 (α, β) ; S 〉 ⊑ 〈 (f, g) ; S 〉 ◇ 〈 (α, β) ; S 〉 for any DFS left ideal 〈 (f, g) ; S〉 and DFS quasi-ideal 〈 (α, β) ; S〉 of S.

(iii)⇒(ii) Suppose 〈 (f, g) ; S〉 ⊓ 〈 (α, β) ; S 〉 ⊑ 〈 (f, g) ; S 〉 ◇ 〈 (α, β) ; S 〉 for any DFS left ideal 〈 (f, g) ; S〉 and DFS quasi-ideal 〈 (α, β) ; S〉 of S. Let L is a left ideal of S and Q be a quasi-ideal of S. Then X_L is DFS left ideal and X_Q is DFS quasi-ideal of S. By hypothesis, X_L ⊓ X_Q = X_L∩Q ⊑ X_L ◇ X_Q = X_LQ . Hence L ∩ Q ⊆ LQ .

(ii)⇒(i) Assume L ∩ Q ⊆ LQ, for any left ideal L and quasi-ideal Q . Let a ∈ S . Since S has left identity, then Sa is left ideal as well as quasi-ideal. Hence a ∈ Sa ∩ Sa ⊆ (Sa) (Sa) = (SS) (aa) = Sa² . Thus S is left regular.□ The following theorems can be proved easily and hence their proof is omitted.

Theorem 5.10. In a left regular AG-groupoid, the following statements are equivalent:

(i) S is left regular.

(ii) Q ∩ L ⊆ QL for any quasi-ideal Q and any left ideal L of S .

(iii) 〈 (f, g) ; S〉 ⊓ 〈 (α, β) ; S 〉 ⊑ 〈 (f, g) ; S 〉 ◇ 〈 (α, β) ; S 〉 for any DFS quasi-ideal 〈 (f, g) ; S〉 and DFS left ideal 〈 (α, β) ; S〉 of S.

Theorem 5.11. In a left regular AG-groupoid, the following statements are equivalent:

(i) S is left regular.

(ii) Q ∩ B = QB for any quasi-ideal Q and bi-ideal B of S .

(iii) $〈 (f, g); S 〉 ⊓ 〈 (α, β); S 〉 \tilde{=} 〈 (f, g); S 〉 ◇ 〈 (α, β); S 〉$ for any DFS quasi-ideals 〈 (f, g) ; S〉 and DFS bi-ideal 〈 (α, β) ; S〉 of S.

Proof. (i)⇒(iii) Assume S is left regular with left identity e. Let 〈 (f, g) ; S〉 is a DFS quasi-ideal and 〈 (α, β) ; S〉 is DFS bi-ideal of S. By Theorem 5.5, 〈 (f, g) ; S〉 and 〈 (α, β) ; S〉 becomes DFS ideals of S. Let a ∈ S. Since S is left regular, then there exists x ∈ S such that a = xa² .

We have a = x ((xa²) ²) = x ((xa²) (xa²)) = (xa²) (x (xa²)) = (x (aa)) (x (xa²)) = (a (xa)) (x (xa²)) = (ax) ((xa) (xa²)) = (xa) ((ax) (xa²)) .

Now $(f \tilde{\circ} α) (a) = ⋃_{a = mn} {f (m) \cap α (n)} \supseteq f (xa) \cap α ((ax) ({xa}^{2})) \supseteq f (a) \cap α (ax) \supseteq f (a) \cap α (a) = (f \cap α) (a),$ and $(g \tilde{\circ} β) (a) = ⋂_{a = mn} {g (m) \cup β (n)} \subseteq g (xa) \cup β ((ax) ({xa}^{2})) \subseteq g (a) \cup β (ax) \subseteq g (a) \cup β (a) = (g \cup β) (a) .$

Hence 〈 (f, g) ; S 〉 ⊓ 〈 (α, β) ; S 〉 ⊑ 〈 (f, g) ; S 〉 ◇ 〈 (α, β) ; S 〉 .

Now $(f \tilde{\circ} α) (a) = ⋃_{a = mn} {f (m) \cap α (n)} \subseteq ⋃_{a = mn} {f (mn) \cap α (mn)} = (f \cap α) (a),$ and $(g \tilde{\circ} β) (a) = ⋂_{a = mn} {g (m) \cup β (n)} \supseteq ⋂_{a = mn} {g (mn) \cup β (mn)} = (g \cap β) (a) .$

Hence 〈 (f, g) ; S〉 ◇ 〈 (α, β) ; S 〉 ⊑ 〈 (f, g) ; S 〉 ⊓ 〈 (α, β) ; S 〉 and so $〈 (f, g); S 〉 ⊓ 〈 (α, β); S 〉 \tilde{=} 〈 (f, g); S 〉 ◇ 〈 (α, β); S 〉$ for any DFS quasi-ideals 〈 (f, g) ; S〉 and DFS bi-ideal 〈 (α, β) ; S〉 of S.

(iii)⇒(ii) Assume $〈 (f, g); S 〉 ⊓ 〈 (α, β); S 〉 \tilde{=} 〈 (f, g); S 〉 ◇ 〈 (α, β); S 〉$ for any DFS quasi-ideals 〈 (f, g) ; S〉 and DFS bi-ideal 〈 (α, β) ; S〉 of S. Let Q be a quasi-ideal and B a bi-ideal. Then X_Q is DFS quasi-ideal and X_B is a DFS bi-ideal of S and by hypothesis, $X_{Q} ⊓ X_{B} \tilde{=} X_{Q \cap B} \tilde{=} X_{Q} ◇ X_{B} \tilde{=} X_{QB} .$ Hence Q ∩ B = QB .

(ii)⇒(i) Assume Q ∩ B = QB for any quasi-ideal Q and bi-ideal B of S . Let a ∈ S. Then we have a ∈ Q [a] ∩ B [a] = Q [a] B [a] = (a ∪ (Sa ∩ aS)) (a ∪ a² ∪ (aS) a)

⊆Sa (a ∪ a² ∪ (aS) a) = (Sa) a ∪ (Sa) a² ∪ (Sa) ((aS) a) ⊆ (Sa) (ea) ∪ Sa² ∪ (aS) ((Sa) a) = (aa) (eS) ∪ Sa² ∪ S ((Sa) (ea))

= (aa) (SS) ∪ Sa² ∪ S ((aa) (eS)) = (SS) (aa) ∪ Sa² ∪ S ((aa) S) = Sa² ∪ Sa² ∪ a²S = Sa² ∪ Sa² = Sa². Hence S is left regular.□

6 Application of double-framed fuzzy soft expert sets in decision making

As stated earlier in the definition of soft set, parameter set is actually the set of properties of the elements in the universal set. There are many real life situations, where the parameters are multi-valued. For example, if the universal set contains cars and a parameter is say colour. Colour is multi-valued parameter because colour may mean that car contains blue, red, green, black, white colours. Similarly if a universal set contains dresses then a parameter may be size. Size means small, medium, large, extra large. There are many instances like these where the parameters are not single-valued.

Moreover, it may be noted that the parameters in certain cases are such that decision makers are interested in the degree of satisfaction of a property p and degree of satisfaction of the property notp or degree of satisfaction of some counter property of p. In these cases intuitionistic fuzzy soft sets and bipolar fuzzy soft sets are useful. In our research, parameters are two-valued and the two values of the parameters may not be opposite of each other. For example, consider a set of newspapers and a parameter is quality of the news. Quality is a multi-valued variable which mean true news, unbiased news, updated news etc. The three values of the parameters are not counter properties of each other. In such cases multi-framed soft set (extension of DFS-set) may play a vital role in decision making. Our current paper is dealing with cases where parameters are two valued. For example, in this case a parameter may be coverage, which means international coverage and national coverage.

In this paper, we develop a decision making scheme where the parameters are two-valued. A concrete example, where two-valued parameters are taken, is explained here. If the universal set is containing books on “Calculus & Analytic Geometry” then the parameters may be calculus contents and geometry contents. Here the parameter, calculus contents, is intended to be two-valued. i.e. Calculus contents means single variable calculus and several-variable calculus. Other parameter, geometry contents of the book mean 2-dimensional geometry and 3-dimensional geometry. In situations, as exemplified, the concept of DFS-sets may play a vital role in decision making. Recently, in [18, 20], we have presented some applications of DFS-sets and DFS expert sets in decision making. In continuation to our previous work, we present another decision making scheme based on double-framed fuzzy soft expert set (briefly, DFFSE-set) in the following. The reader is suggested to read the papers [22 , 53–57] on different recently kinds of soft and fuzzy soft applied on decision making.

From now on, since the parameters are intended to be two-valued, we shall consider write a parameters e_i as a pair $(e_{i}^{(1)}, e_{i}^{(2)})$ .

Definition 6.1. Let U = {u₁, u₂, . . . , u_n} be a universe, E = {e₁, e₂, . . . , e_m} a set of parameters and X = {x₁, x₂, . . . , x_t} a set of experts (agents). Let O = {0 =disagree, 1 =agree} be a set of opinions, Z = E × X × O and A ⊆ Z . An object of the form $〈 (\tilde{α}, \tilde{β}); A 〉$ is called a DFFSE-set over U, where $\tilde{α}$ and $\tilde{β}$ are mappings from A to FP (U) (set of all fuzzy subsets of U).

The tabular representation of a DFFSE-sets has the form

${\begin{matrix} [c] c ((e_{i}, x_{j}, 1), {\frac{u_{k}}{α (e_{i}^{(1)}) (u_{k}), β (e_{i}^{(2)}) (u_{k})}}), ((e_{i}, x_{j}, 0), {\frac{u_{k}}{α (e_{i}^{(1)}) (u_{k}), β (e_{i}^{(2)}) (u_{k})}}), \\ i \in {1, 2, . . ., m}, j \in {1, 2, . . ., t}, k \in {1, 2, . . ., n} \end{matrix}}$

Example 6.2. Assume that a company wants to subscribe a newspaper from a set U = {n₁ . n₂, n₃, n₄, n₅} of available newspapers. The company considers the set of parameters, A = {e₁, e₂, e₃} = {coverage, print, quality} . The parameters involved here are two-valued. e₁ stands for coverage which includes national news and international news, e₂ stands for print which includes coloured print and online print, e₃ stands for quality which includes unbaised news and true news. A set of experts X = {p, q} have been appointed to give there expert opinion. The following DFFSE-set is established.

$〈 (\tilde{α}, \tilde{β}); A 〉 = {\begin{matrix} [c] l ((e_{1}, p, 1), {\frac{n_{1}}{0.55, 0.40}, \frac{n_{2}}{0.60, 0.70}, \frac{n_{3}}{0.35, 0.55}, \frac{n_{4}}{0.45, 0.45}, \frac{n_{5}}{0.70, 0.80}}), \\ ((e_{2}, p, 1), {\frac{n_{1}}{0.40, 0.55}, \frac{n_{2}}{0.50, 0.50}, \frac{n_{3}}{0.60, 0.10}, \frac{n_{4}}{0.30, 0.25}, \frac{n_{5}}{0.60, 0.70}}), \\ ((e_{3}, p, 1), {\frac{n_{1}}{0.30, 0.35}, \frac{n_{2}}{0.60, 0.50}, \frac{n_{3}}{0.10, 0.10}, \frac{n_{4}}{0.40, 0.45}, \frac{n_{5}}{0.70, 0.60}}), \\ ((e_{1}, q, 1), {\frac{n_{1}}{0.60, 0.50}, \frac{n_{2}}{0.65, 0.60}, \frac{n_{3}}{0.30, 0.60}, \frac{n_{4}}{0.50, 0.55}, \frac{n_{5}}{0.65, 0.75}}), \\ ((e_{2}, q, 1), {\frac{n_{1}}{0.50, 0.45}, \frac{n_{2}}{0.40, 0.50}, \frac{n_{3}}{0.55, 0.30}, \frac{n_{4}}{0.40, 0.35}, \frac{n_{5}}{0.70, 0.60}}), \\ ((e_{3}, q, 1), {\frac{n_{1}}{0.40, 0.40}, \frac{n_{2}}{0.65, 0.45}, \frac{n_{3}}{0.20, 0.20}, \frac{n_{4}}{0.45, 0.50}, \frac{n_{5}}{0.60, 0.70}}), \\ ((e_{1}, p, 0), {\frac{n_{1}}{0.15, 0.20}, \frac{n_{2}}{0.20, 0.15}, \frac{n_{3}}{0.65, 0.75}, \frac{n_{4}}{0.15, 0.20}, \frac{n_{5}}{0.20, 0.25}}), \\ ((e_{2}, p, 0), {\frac{n_{1}}{0.10, 0.50}, \frac{n_{2}}{0.15, 0.15}, \frac{n_{3}}{0.60, 0.75}, \frac{n_{4}}{0.60, 0.55}, \frac{n_{5}}{0.25, 0.25}}), \\ ((e_{3}, p, 0), {\frac{n_{1}}{0.70, 0.90}, \frac{n_{2}}{0.10, 0.10}, \frac{n_{3}}{0.90, 0.80}, \frac{n_{4}}{0.15, 0.10}, \frac{n_{5}}{0.10, 0.10}}), \\ ((e_{1}, q, 0), {\frac{n_{1}}{0.20, 0.25}, \frac{n_{2}}{0.25, 0.25}, \frac{n_{3}}{0.60, 0.70}, \frac{n_{4}}{0.25, 0.30}, \frac{n_{5}}{0.30, 0.35}}), \\ ((e_{2}, q, 0), {\frac{n_{1}}{0.20, 0.55}, \frac{n_{2}}{0.25, 0.25}, \frac{n_{3}}{0.65, 0.70}, \frac{n_{4}}{0.55, 0.60}, \frac{n_{5}}{0.35, 0.35}}), \\ ((e_{3}, q, 0), {\frac{n_{1}}{0.60, 0.80}, \frac{n_{2}}{0.20, 0.20}, \frac{n_{3}}{0.80, 0.85}, \frac{n_{4}}{0.25, 0.20}, \frac{n_{5}}{0.20, 0.20}}), \end{matrix}}$

Definition 6.3. Let $ℑ = 〈 (\tilde{α}, \tilde{β}); A 〉$ be a DFFSE-set over U where A ⊆ Z . The agree-DFFSE-set of $〈 (\tilde{α}, \tilde{β}); A 〉$ is denoted by $\overset{(1)}{ℑ} = {〈 (\tilde{α}, \tilde{β}); A 〉}_{1}$ and is defined to be a set of the form

$\begin{matrix} {〈 (\tilde{α}, \tilde{β}); A 〉}_{1} & = & {(\underset{(1)}{\tilde{α}} (a), \underset{(1)}{\tilde{β}} (a)) : a \in E \times X \times {1}} \\ = & {((e_{i}, x_{j}, 1), \frac{u_{k}}{α (e_{i}^{(1)}) (u_{k}), β (e_{i}^{(2)}) (u_{k})})} \end{matrix}$

Similarly the disagree DFFSE-set of $〈 (\tilde{α}, \tilde{β}); A 〉$ is denoted by $\overset{(0)}{ℑ} = {〈 (\tilde{α}, \tilde{β}); A 〉}_{0}$ and is defined as

$\begin{matrix} {〈 (\tilde{α}, \tilde{β}); A 〉}_{0} & = & {(\underset{(0)}{\tilde{α}} (a), \underset{(0)}{\tilde{β}} (a)) : a \in E \times X \times {0}} \\ = & {((e_{i}, x_{j}, 0), \frac{u_{k}}{α (e_{i}^{(1)}) (u_{k}), β (e_{i}^{(2)}) (u_{k})})} . \end{matrix}$

Example 6.4. Consider Example 6.

The agree DFFSE-set is given below.

${〈 (\tilde{α}, \tilde{β}); A 〉}_{1} = {\begin{matrix} [c] l ((e_{1}, p, 1), {\frac{n_{1}}{0.55, 0.40}, \frac{n_{2}}{0.60, 0.70}, \frac{n_{3}}{0.35, 0.55}, \frac{n_{4}}{0.45, 0.45}, \frac{n_{5}}{0.70, 0.80}}), \\ ((e_{2}, p, 1), {\frac{n_{1}}{0.40, 0.55}, \frac{n_{2}}{0.50, 0.50}, \frac{n_{3}}{0.60, 0.10}, \frac{n_{4}}{0.30, 0.25}, \frac{n_{5}}{0.60, 0.70}}), \\ ((e_{3}, p, 1), {\frac{n_{1}}{0.30, 0.35}, \frac{n_{2}}{0.60, 0.50}, \frac{n_{3}}{0.10, 0.10}, \frac{n_{4}}{0.40, 0.45}, \frac{n_{5}}{0.70, 0.60}}), \\ ((e_{1}, q, 1), {\frac{n_{1}}{0.60, 0.50}, \frac{n_{2}}{0.65, 0.60}, \frac{n_{3}}{0.30, 0.60}, \frac{n_{4}}{0.50, 0.55}, \frac{n_{5}}{0.65, 0.75}}), \\ ((e_{2}, q, 1), {\frac{n_{1}}{0.50, 0.45}, \frac{n_{2}}{0.40, 0.50}, \frac{n_{3}}{0.55, 0.30}, \frac{n_{4}}{0.40, 0.35}, \frac{n_{5}}{0.70, 0.60}}), \\ ((e_{3}, q, 1), {\frac{n_{1}}{0.40, 0.40}, \frac{n_{2}}{0.65, 0.45}, \frac{n_{3}}{0.20, 0.20}, \frac{n_{4}}{0.45, 0.50}, \frac{n_{5}}{0.60, 0.70}}), \end{matrix}}$

The disagree DFFSE-set is given as,

${〈 (\tilde{α}, \tilde{β}); A 〉}_{0} = {\begin{matrix} [c] l ((e_{1}, p, 0), {\frac{n_{1}}{0.15, 0.20}, \frac{n_{2}}{0.20, 0.15}, \frac{n_{3}}{0.65, 0.75}, \frac{n_{4}}{0.15, 0.20}, \frac{n_{5}}{0.20, 0.25}}), \\ ((e_{2}, p, 0), {\frac{n_{1}}{0.10, 0.50}, \frac{n_{2}}{0.15, 0.15}, \frac{n_{3}}{0.60, 0.75}, \frac{n_{4}}{0.60, 0.55}, \frac{n_{5}}{0.25, 0.25}}), \\ ((e_{3}, p, 0), {\frac{n_{1}}{0.70, 0.90}, \frac{n_{2}}{0.10, 0.10}, \frac{n_{3}}{0.90, 0.80}, \frac{n_{4}}{0.15, 0.10}, \frac{n_{5}}{0.10, 0.10}}), \\ ((e_{1}, q, 0), {\frac{n_{1}}{0.20, 0.25}, \frac{n_{2}}{0.25, 0.25}, \frac{n_{3}}{0.60, 0.70}, \frac{n_{4}}{0.25, 0.30}, \frac{n_{5}}{0.30, 0.35}}), \\ ((e_{2}, q, 0), {\frac{n_{1}}{0.20, 0.55}, \frac{n_{2}}{0.25, 0.25}, \frac{n_{3}}{0.65, 0.70}, \frac{n_{4}}{0.55, 0.60}, \frac{n_{5}}{0.35, 0.35}}), \\ ((e_{3}, q, 0), {\frac{n_{1}}{0.60, 0.80}, \frac{n_{2}}{0.20, 0.20}, \frac{n_{3}}{0.80, 0.85}, \frac{n_{4}}{0.25, 0.20}, \frac{n_{5}}{0.20, 0.20}}), \end{matrix}}$

Definition 6.5. Let $ℑ = 〈 (\tilde{α}, \tilde{β}); A 〉$ be a DFFSE-set over U where A ⊆ Z . For (t₁, t₂) ∈ [0, 1] × [0, 1], the (t₁, t₂)-level cut of the agree-DFFSE-set $\overset{(1)}{ℑ}$ is denoted by $L ((\overset{(1)}{ℑ}, A), (t_{1}, t_{2}))$ and is defined as $\begin{matrix} L ((\overset{(1)}{ℑ, A}), (t_{1}, t_{2})) \\ = ((\underset{(1)}{α_{t_{1}}}, A) = ⋃_{x \in X} {u \in U : \underset{(1)}{\tilde{α}} (a) (u) \geq t_{1}}, \\ (\underset{(1)}{β_{t_{2}}}, A) = ⋃_{x \in X} {u \in U : \underset{(1)}{\tilde{β}} (a) (u) \geq t_{2}}), \end{matrix}$ and the (t₁, t₂)-level cut of the disagree-DFFSE-set $\overset{(0)}{ℑ}$ is denoted by $L ((\overset{(0)}{ℑ}, A), (t_{1}, t_{2}))$ and is defined as

$\begin{matrix} L ((\overset{(0)}{ℑ, A}), (t_{1}, t_{2})) \\ = ((\underset{(0)}{α_{t_{1}}}, A) = ⋃_{x \in X} {u \in U : \underset{(0)}{\tilde{α}} (a) (u) \geq t_{1}}, \\ (\underset{(0)}{β_{t_{2}}}, A) = ⋃_{x \in X} {u \in U : \underset{(0)}{\tilde{β}} (a) (u) \geq t_{2}}) . \end{matrix}$

Clearly, $L ((\overset{(1)}{ℑ, A}), (t_{1}, t_{2}))$ and $L ((\overset{(0)}{ℑ, A}), (t_{1}, t_{2}))$ are elements of U × U .

In this definition, t₁, t₂ ∈ [0, 1] are thresholds on the membership values. In real-life decision making problems based on DFFSE-sets, usually these thresholds are chosen by decision makers in advance.

Example 6.6. Reconsider Example 6, and setting t₁ = 0.5, t₂ = 0.5we have, $\begin{matrix} L (\overset{(1)}{ℑ} (e_{1})) & = (\underset{(1)}{α_{0.5}} (e_{1}) = {n_{1}, n_{2}, n_{4}, n_{5}}, \underset{(1)}{β_{0.5}} (e_{1}) = {n_{1}, n_{2}, n_{3}, n_{4}, n_{5}}) \\ L (\overset{(1)}{ℑ} (e_{2})) & = (\underset{(1)}{α_{0.5}} (e_{2}) = {n_{1}, n_{2}, n_{3}, n_{5}}, \underset{(1)}{β_{0.5}} (e_{2}) = {n_{1}, n_{2}, n_{5}}) \\ L (\overset{(1)}{ℑ} (e_{3})) & = (\underset{(1)}{α_{0.5}} (e_{3}) = {n_{2}, n_{5}}, \underset{(1)}{β_{0.5}} (e_{3}) = {n_{2}, n_{4}, n_{5}}) \end{matrix}$ and $\begin{matrix} L (\overset{(0)}{ℑ} (e_{1})) & = (\underset{(0)}{α_{0.5}} (e_{1}) = {n_{3}}, \underset{(0)}{β_{0.5}} (e_{1}) = {n_{3}}) \\ L (\overset{(0)}{ℑ} (e_{2})) & = (\underset{(0)}{α_{0.5}} (e_{2}) = {n_{3}, n_{4}}, \underset{(0)}{β_{0.5}} (e_{2}) = {n_{1}, n_{3}, n_{4}}) \\ L (\overset{(0)}{ℑ} (e_{3})) & = (\underset{(0)}{α_{0.5}} (e_{3}) = {n_{1}, n_{3}}, \underset{(0)}{β_{0.5}} (e_{3}) = {n_{1}, n_{3}}) \end{matrix}$

To represent the (t₁, t₂)-level cut of the agree-DFFSE-set $\overset{(1)}{ℑ}$ , we form a table which has entries of the form (a_ij, b_ij) where a_ij, b_ij ∈ {0, 1} and the components are defined as $a_{ij} (u_{i}) = {\begin{matrix} [c] c 1 if u_{i} \in \underset{(1)}{α_{t_{1}}} (e_{j}) \\ 0 if u_{i} \notin \underset{(1)}{α_{t_{1}}} (e_{j}) \end{matrix} and b_{ij} = {\begin{matrix} [c] c 1 if u_{i} \in \underset{(1)}{β_{t_{2}}} (e_{j}) \\ 0 if u_{i} \notin \underset{(1)}{β_{t_{2}}} (e_{j}) \end{matrix}$ Similarly the tabular representation of the (t₁, t₂)-level cut of the disagree-DFFSE-set $\overset{(0)}{ℑ}$ , the entries are of the form (m_ij, n_ij) where m_ij, n_ij ∈ {0, 1} and the components are defined as $m_{ij} (u_{i}) = {\begin{matrix} [c] c 1 if u_{i} \in \underset{(0)}{α_{t_{1}}} (e_{j}) \\ 0 if u_{i} \notin \underset{(0)}{α_{t_{1}}} (e_{j}) \end{matrix} and n_{ij} = {\begin{matrix} [c] c 1 if u_{i} \in \underset{(0)}{β_{t_{2}}} (e_{j}) \\ 0 if u_{i} \notin \underset{(0)}{β_{t_{2}}} (e_{j}) \end{matrix}$

Consider Example 6, the tabular representation of the (0.5, 0.5)-level cut of the agree-DFFSE-set

and the tabular representation of the level (0.5, 0.5)-level cut of the disagree-DFFSE-set

6.1 Score values of an object

The agree scores $A_{i}^{(1)}$ and $A_{i}^{(2)}$ of an object u_i ∈ U are calculated from the tabular representation of the (t₁, t₂)-level cut of the agree-DFFSE-set by the formulas given below, $A_{i}^{(1)} = \sum_{j} a_{ij} and A_{i}^{(1)} = \sum_{j} b_{ij} .$

Similarly the disagree scores $D_{i}^{(1)}$ and $D_{i}^{(2)}$ of an object u_i ∈ U are calculated from the tabular representation of the (t₁, t₂)-level cut of the disagree-DFFSE-set by the formulas given below, $D_{i}^{(1)} = \sum_{j} m_{ij} and D_{i}^{(1)} = \sum_{j} n_{ij} .$

6.2 Decision making algorithm based on DFFSE-sets

Here a decision making algorithm is propsed based on DFFSE-sets The following steps may be followed by the decision makers for the best possible decision.

Algorithm

1. Input the DFFSE-set $〈 (\tilde{α}, \tilde{β}); A 〉 .$

2. Find the agree-DFFSE-set and disagree-DFFSE-set of the given DFFSE-set $〈 (\tilde{α}, \tilde{β}); A 〉$ .

3. Set the pair of thresholds (t₁, t₂) ∈ [0, 1] × [0, 1] and find the (t₁, t₂)-level cut of the agree-DFFSE-set and the (t₁, t₂)-level cut of the disagree-DFFSE-set.

4. Present the (t₁, t₂)-level cut of the agree-DFFSE-set and the (t₁, t₂)-level cut of the disagree-DFFSE-set in tabular form.

5. Compute the agree scores $A_{i} = A_{i}^{(1)} + A_{i}^{(2)}$ and the disagree scores $D_{i} = D_{i}^{(1)} + D_{i}^{(2)}$ of each element u_i ∈ U.

6. Find the values S_i = A_i - D_i for each u_i .

7. If S_k = max S_i then u_k is the best possible decision. If there are two or more values of k then any u_k may be choosen as the best optimal solution.

Let us apply the algorithm to Example 6. From Tables 1 and 2, the agree scores A_i and disagree scores D_i in the following tables.

Table 1
Table1

U e ₁ e ₂ e ₃

n ₁ (1, 1) (1, 1) (0, 0)

n ₂ (1, 1) (1, 1) (1, 1)

n ₃ (0, 1) (1, 0) (0, 0)

n ₄ (1, 1) (0, 0) (0, 1)

n ₅ (1, 1) (1, 1) (1, 1)

U	e ₁	e ₂	e ₃
n ₁	(1, 1)	(1, 1)	(0, 0)
n ₂	(1, 1)	(1, 1)	(1, 1)
n ₃	(0, 1)	(1, 0)	(0, 0)
n ₄	(1, 1)	(0, 0)	(0, 1)
n ₅	(1, 1)	(1, 1)	(1, 1)

Table 2

Table2

U	e ₁	e ₂	e ₃
n ₁	(0, 0)	(0, 1)	(1, 1)
n ₂	(0, 0)	(0, 0)	(0, 0)
n ₃	(1, 1)	(1, 1)	(1, 1)
n ₄	(0, 0)	(1, 1)	(0, 0)
n ₅	(0, 0)	(0, 0)	(0, 0)

Agree scores of each u_i
U	e1	e2	e3	Ai(1)	Ai(2)	Ai
n1	(1,1)	(1,1)	(0,0)	2	2	4
n2	(1,1)	(1,1)	(1,1)	3	3	6
n3	(0,1)	(1,0)	(0,0)	1	1	2
n4	(1,1)	(0,0)	(0,1)	1	2	3
n5	(1,1)	(1,1)	(1,1)	3	3	6
Disagree scores of each u_i
U	e1	e2	e3	Di(1)	Di(2)	Di
n1	(0,0)	(0,1)	(1,1)	1	2	3
n2	(0,0)	(0,0)	(0,0)	0	0	0
n3	(1,1)	(1,1)	(1,1)	3	3	6
n4	(0,0)	(1,1)	(0,0)	1	1	2
n5	(0,0)	(0,0)	(0,0)	0	0	0

The scores for each u_i ∈ U are calculated by using Step 6 of the proposed algoritm, and the scores are S₁ = 4 -3 = 1, S₂ = 6 -0 = 6, S₃ = 2 -6 = -4, S₄ = 3 -2 = 1, S₅ = 6 -0 .

Hence n₂ and n₅ are the best newspapers and the company may subscribe any one of these.

7 Conclusion

In this paper, DFS-quasi ideals have been defined alongwith some examples and some properties of them were discussed in AG-groupoids. Moreover, application of double-framed fuzzy soft expert- sets(briefly, DFFSE-sets) in a decision making problem was presented and a decision making algorithm was proposed based on DFFSE-sets. In our future work, we intend to extend the results of this paper introducing (M, N) DFS-quasi ideals of AG-groupoids. Moreover, the development of multi-framed soft sets and their applications to decision making problems will be our next research project. In [22 , 53–57] multi-attribute decision making schemes are designed. Our proposed multi-framed soft sets will play useful role in multi-attribute decision making problems.

Acknowledgements. The authors are highly grateful to the referees for their valuable comments and suggestions which were helpful in improving this paper, and to the Assoc. Editor of the journal for editing and communicating the paper.

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On algebraic properties of DFS sets and its application in decision making problems

Abstract

Keywords

1 Introduction

2 Preliminaries

4 Double-framed soft quasi-ideals

6 Application of double-framed fuzzy soft expert sets in decision making

6.1 Score values of an object

6.2 Decision making algorithm based on DFFSE-sets

Table 1 Table1 U e 1 e 2 e 3 n 1 (1, 1) (1, 1) (0, 0) n 2 (1, 1) (1, 1) (1, 1) n 3 (0, 1) (1, 0) (0, 0) n 4 (1, 1) (0, 0) (0, 1) n 5 (1, 1) (1, 1) (1, 1)

References

Table 1
Table1

U e ₁ e ₂ e ₃

n ₁ (1, 1) (1, 1) (0, 0)

n ₂ (1, 1) (1, 1) (1, 1)

n ₃ (0, 1) (1, 0) (0, 0)

n ₄ (1, 1) (0, 0) (0, 1)

n ₅ (1, 1) (1, 1) (1, 1)