Double-framed soft generalized bi-ideals of intra-regular AG-groupoids

Abstract

In this paper, we introduce double-framed soft bi-ideals (briefly DFS bi-ideals) and double-framed soft generalized bi-ideals (briefly DFS generalized bi-ideals) in AG-groupoids and some properties of them are investigated. Several characterizations of intra-regular AG-groupoids in terms of DFS left (resp. right) ideals, DFS bi-ideals, DFS generalized bi-ideals and semiprime class of them are provided.

Keywords

DFS-set DFS AG-groupoid DFS generalized bi-ideal DFS bi-ideal DFS int-uni product DFS including set

1 Introduction

The traditional mathematical framework couldn’t allow for the uncertainties that appear in economics, engineering, environmental, medical and social sciences to be resolved within its compass. In 1999, Molodtsov [28], introduced the concept of soft sets as a new mathematical tool for dealing with uncertainties that is free from the difficulties affecting existing methods. Soft set theory has rich potential for applications in several directions, few of which had been demonstrated by Molodtsov in his pioneer work. The 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 in the fields as varied as mentioned above. For example, soft set theory is applied in the field of optimization by Kovkov in [23], decision making problems have been studied in [26, 38]. Soft set theory has been applied to different algebraic structures. We refer the reader to the papers [1–3 , 40]. Some applications in decision making methods can be found in [24 , 43].

The idea of generalization of a commutative semigroup, (known as left almost semigroup) was introduced by Kazim and Naseeruddin in 1972 (see [16]). Several others authors that have contributed on left almost semigroups, can be found in references. Cho et al. [4] studied this structure under the name of right modular groupoid. Holgate [9] studied it as left invertive groupoid. Similarly, Stevanovic and Protic [37] called this structure an Abel-Grassmann groupoid (or simply AG-groupoid), which is the primary name under which this structure is known nowadays. There are many important applications of AG-groupoids in the theory of flocks [31]. For more study of AG-groupoids, the reader is invited to read [6 , 41].

Recently, Jun et al. extended the notions of intersectional and union soft sets into double-framed soft sets and defined double-framed soft subalgebra of a BCK/BCI-algebra and studied the related properties in [11]. 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. [17] applied the idea of double framed soft sets to left almost semigroups.

2 Preliminaries

In this section, we give the basic notions from the theory of AG-groupoids which will be used throughout the paper.

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 [16], that is,

(ab) (cd) = (ac) (bd) for all a, b, c, d ∈ S.

It is a useful non-associative 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 [29].

Every AG-groupoid S with left identity satisfies the paramedial law [29], 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 non-empty subset A of an AG-groupoid S is called sub AG-groupoid of S if A² ⊆ A. A non-empty 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. It is important to note that Sa², a²S and (Sa²) S are ideals of S.

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

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

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

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

A subset A of an AG-groupoid S is called semiprime if for all a ∈ S, whenever a² ∈ A, then a ∈ A.

An AG-groupoid S is called intra-regular if for all a ∈ S, there exist x, y ∈ S such that a = (xa²) y.

Example 2.1. ([6]). Let us consider the set S = {1, 2, 3, 4, 5, 6} with the following multiplication table.

\begin{matrix} . & 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 1 & 2 & 1 & 1 & 1 & 1 \\ 3 & 1 & 1 & 5 & 6 & 3 & 4 \\ 4 & 1 & 1 & 4 & 5 & 6 & 3 \\ 5 & 1 & 1 & 3 & 4 & 5 & 6 \\ 6 & 1 & 1 & 6 & 3 & 4 & 5 \end{matrix}

It is not difficult to verify that (S, ·) is an AG-groupoid and it is intra-regular, since 1 = (1 · 1²) ·1, 2 = (2 · 2²) ·2, 3 = (3 · 3²) ·5, 4 = (6 · 4²) ·3, 5 = (5 · 5²) ·5, 6 = (4 · 6²) ·3.

Lemma 2.2. (cf.[30]). If S is an AG-groupoid with left identity e, then S = S² and S = eS = Se.

Proof. If S is an AG-groupoid with left identity e, then any x ∈ S can be written as x = ex ∈ SS and so S ⊆ SS. Hence S = SS. Now obviously eS = S and using left invertive law, we have Se = (SS) e = (eS) S = SS = S. Hence eS = Se = S. □

Lemma 2.3. (cf. [5]) If S is an AG-groupoid with left identity e, then:

Sa² = a²S.

(Sa) ² = Sa².

(Sa) ² = (Sa²) S.

Proof. (1) Using Eq. 2 and Eq. 1, Sa² = (SS) (aa) = (aa) (SS) = a²S.

(2) Using medial law, (Sa) (Sa) = (SS) (aa) = Sa².

(3) Using Eq. 2, left invertive law, paramedial law and medial law and Lemma 2.2,

(Sa) ² = (Sa) (Sa) = (SS) (aa) = (aa) (SS) = S ((aa) S) = (SS) ((aa) S) = (SS) ((Sa) a) = (SS) (a²S) = (Sa²) (SS) = (Sa²) S. From (1) and (2), we have Sa² = a²S = (Sa²) S. □

Lemma 2.4. (cf. [30]) If S is an AG-groupoid with left identity e, then (xy) ² = x²y² = y²x² for all x, y ∈ S.

Lemma 2.5. If S is an intra-regular AG-groupoid with left identity e, then every a ∈ S can be expressed as

a = (a²t) a², for some t ∈ S.

a = a (ua²), for some u ∈ S.

a = a ((a²q) a²), for some q ∈ S.

a = (ta) a, for some t ∈ S.

a = ((a (xa)) p) a, for some p ∈ S.

Proof. (1). Let a ∈ S. Since S is intra-regular, then there exist x, y ∈ S such that a = (xa²) y. Using Eq. 2, left invertive law, paramedial law and medial law, we can further write, a = (xa²) y = (x (aa)) y = (a (xa)) y = (y (x ((xa²) y))) a = (x (y ((xa²) y))) a = (x ((xa²) y²)) a = ((xa²) (xy²)) a = (x² (a²y²)) a = (a² (x² y²)) a = (a (x²y²)) a² = (((xa²) y) (x²y²)) a² = ((y²y) (x² (xa²))) a² = ((y²x²) (y (xa²))) a² = ((y²x²) ((y₁y₂) (xa²))) a² = ((y²x²) ((a²y₂) (xy₁))) a² = ((y²x²) ((a²x) (y₂y₁))) a² = ((y²x²) (((y₂y₁) x) (aa))) a² = ((y²x²) (a² (x (y₂y₁)))) a² = (a² ((y²x²) (x (y₂y₁))) a² = (a²t) a², where t = ((y²x²) (x (y₂y₁)). Hence a = (a²t) a² for some t ∈ S which is the required result.

(2) Let a ∈ S. Since S is intra-regular, then there exist x, y ∈ S such that a = (xa²) y. Using Eq. 2, left invertive law, paramedial law, Eq. 1 medial law, we can further write

$\begin{matrix} a & = & (x (aa)) y = (a (xa)) y = (y (xa)) a \\ = & (y (xa)) (({xa}^{2}) y) = (y ({xa}^{2})) ((xa) y) \\ = & ((y_{1} y_{2}) ({xa}^{2})) ((xa) (y_{1} y_{2})) \\ = & ((a^{2} x) (y_{2} y_{1})) ((y_{2} y_{1}) (ax)) \\ = & ((a^{2} x) (y_{2} y_{1})) (a ((y_{2} y_{1}) x)) \\ = & a (((a^{2} x) (y_{2} y_{1})) ((y_{2} y_{1}) x)) \\ = & a ((a^{2} (x (y_{2} y_{1})) ((y_{2} y_{1}) x)) \\ = & a ((x (y_{2} y_{1})) ((x (y_{2} y_{1})) a^{2})) = a ({ua}^{2}) \end{matrix}$ where u = (x (y₂y₁)) ((x (y₂y₁)). Hence a = a (ua²) for some u ∈ S.

(3) Let a ∈ S. By using (2), there exists u ∈ S such that a = a (ua²). Now using Eq. 2, left invertive law, paramedial law, Eq. 1, medial law and Lemma 2.4, we have ua² = u ((xa²) y) ² = u ((xa²) ²y²) = (xa²) ² (uy²) = (x² (a²) ²) (uy²) = ((a²) ²x²) (uy²) = ((uy²) x²) (a²) ² = ((uy²) x²) (a²a²) = (a²a²) (x² (uy²)) = ((x² (uy²)) a²) a² = (a² ((uy²) x²)) a² = (a²q) a², where q = (uy²) x². Hence a = a ((a²q) a²), for some q ∈ S. (4) Let a ∈ S. Then there exist x, y ∈ S such that a = (xa²) y. Using Eq. 2, left invertive law, paramedial law, Eq. 1 medial law, we can further write a = (x (aa)) y = (a (xa)) y = (y (xa)) a = ((ey) (xa)) a = ((ax) (ye)) a = (((ye) x) a) a = (ta) a where t = (ye) x. Hence a = (ta) a for some t ∈ S.

(5) Let a ∈ S. Then by using (iv), there exists t ∈ S such that a = (ta) a. Thus using Eq. 2, left invertive law, paramedial law, Eq. 1 and medial law, we can further write a = (ta) a = (t ((xa²) y)) a = ((xa²) (ty)) a = ((x (aa)) (ty)) a = ((a (xa)) (ty)) a = ((a (xa)) p) a where p = ty. Hence a = ((a (xa)) p) a for some p ∈ 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 set, E a set of parameters, P (U) the power set of U and A ⊆ E. Then a 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 ∈ E, f_A (x) ∈ 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. [39] 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. [11] A double-framed soft pair 〈 (α, β); A〉 is called a double-framed soft set of A 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 ∈ A|α (x) ⊇ γ }

and e_A (β; δ) : ={ x ∈ A|β (x) ⊆ δ }

respectively. The set DF_A (α, β) _(γ,δ) : ={ x ∈ A|α (x) ⊇ γ, β (x) ⊆ δ }

is called a double-framed soft including set [11] 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 over U. Then the int-uni soft product [11] 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: $α \tilde{\circ} f : S ⟶ P (U)$ $x ⟼ {\begin{matrix} ⋃_{x = yz} {α (y) \cap f (z)} if \exists y, z \in S such that x = yz, \\ \emptyset otherwise, \end{matrix}$ $β \tilde{\circ} g : S ⟶ P (U)$ $x ⟼ {\begin{matrix} ⋂_{x = yz} {β (y) \cup g (z)} if \exists y, z \in S such that x = yz, \\ U otherwise . \end{matrix}$

It can be easily seen that the operation "◊" is well-defined.

Let 〈 (α, β) A〉 and 〈 (f, g) B〉 be two double-framed soft sets of A and B respectively over a common universe U. Then 〈 (α, β) A〉 is called a double-framed soft subset [11] (briefly, DFS subset) of 〈 (f, g) B〉, denoted by 〈 (α, β) A〉 ⊑ 〈 (f, g) B 〉, if:

(i) A ⊆ B,

(ii) (∀ e ∈ A)

$(\begin{matrix} α and f are identicalapproximations . i . e . α (e) \subseteq f (e) \\ β^{c} and g^{c} are identicalapproximations . i . e . β (e) \supseteq g (e) \end{matrix}) .$

Two DFS sets 〈 (α, β) A〉 and 〈 (f, g) B〉 of S over U are double-framed soft equalif 〈 (α, β) A〉 ⊑ 〈 (f, g) B 〉 and 〈 (f, g) B 〉 ⊑ 〈 (α, β) A 〉.

For any two DFS sets 〈 (α, β) A〉 and 〈 (f, g) A〉 of A over U, the DFS int-uni set [11] 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 $α \tilde{\cap} f : A \to P (U), x \to α (x) \cap f (x)$ , $β \tilde{\cup} g : A \to P (U), x \to β (x) \cup g (x)$ .

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 [17] 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 state the following lemmas without proof.

Lemma 3.6. (cf. [17]) If S is an AG-groupoid, then the set (DFS (U), ◊) is an AG-groupoid.

Lemma 3.7. (cf. [17]) If S is an AG-groupoid, then the medial law holds in DFS (U). That is, for all 〈 (α, β) S 〉, 〈 (f, g) S 〉, 〈 (h, k) S〉 and 〈 (p, q) S 〉 ∈ DFS (U), $(α \tilde{\circ} f) \tilde{\circ} (h \tilde{\circ} p) = (α \tilde{\circ} h) \tilde{\circ} (f \tilde{\circ} p)$ 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. [17]) 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. Let A and B be two non-empty subsets of an AG-groupoid S. Then the following properties hold:

A ⊆ B if and only if X_A⊑ X_B.

X_A ⊓ X_B = X_A∩B.

X_A ◊ X_B = X_AB.

Proof. Straightforward. □ The following definitions are available in [17].

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

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

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

(3) 〈 (α, β) A〉 is called a double-framed soft semiprime if α (a) ⊇ α (a²) and β (a) ⊆ β (a²).

Proposition 3.10. (cf. [17]) A DFS set 〈 (α, β) S〉 of an AG-groupoid S over U is DFS left (resp. right) ideal of S if X_S◊ 〈 (α, β) S 〉 ⊑ 〈 (α, β) S 〉 (resp. 〈 (α, β) S〉 ◊ X_S ⊑ 〈 (α, β) S 〉).

Proposition 3.11. (cf. [17]) If S is an AG-groupoid, then X_S ◊ X_S ⊑ X_S. If a left identity adjoined to S, then X_S ◊ X_S = X_S.

Proposition 3.12. (cf. [17]) If S is an AG-groupoid with left identity e, then every DFS right ideal of S over U is DFS ideal of S.

Proposition 3.13. (cf. [19]) For any DFS set 〈 (α, β) S〉 of an AG-groupoid S over U, the following statements are equivalent:

〈 (α, β) S〉 is a DFS AG-groupoid.

The non-empty γ-inclusive set and δ-exclusive set of 〈 (α, β) S〉 are sub AG-groupoids of S for any subsets γ and δ of U.

Theorem 3.14. If S is an intra-regular AG-groupoid with left identity e, then for any DFS set 〈 (α, β) S〉 of S over U, 〈 (α, β) S 〉 ⊑ X_S ◊ 〈 (α, β) S 〉.

Proof. Suppose S is intra-regular AG-groupoid with left identity e and 〈 (α, β) S〉 be a DFS set of S over U. By Lemma 2.5(4) for any a ∈ S, there exists t ∈ S such that a = (ta) a or a = ua for some u ∈ S.

Now $(χ_{S} \tilde{\circ} α) (a) = ⋃_{a = mn} {χ_{S} (m) \cap α (n)} \supseteq χ_{S} (u)$ ∩α (a) = α (a), so $α \tilde{\subseteq} χ_{S} \tilde{\circ} α$ , and $(χ_{S}^{c} \tilde{\circ} β) (a) = ⋂_{a = mn} {χ_{S}^{c} (m) \cup β (n)} \subseteq χ_{S}^{c} (u) \cup β (a) = β (a)$ , so $β \tilde{\supseteq} χ_{S}^{c} \tilde{\circ} β .$ Hence 〈 (α, β) S〉 ⊑ X_S ◊ 〈 (α, β) S 〉. □

Theorem 3.15. If S is an intra-regular AG-groupoid with left identity e, then for any DFS set 〈 (f, g) S〉 of S over U, 〈 (f, g) S 〉 ⊑ 〈 (f, g) S 〉 ◊ X_S.

Proof. Suppose S is intra-regular AG-groupoid with left identity e and 〈 (f, g) S〉 be a DFS set of S over U. By Lemma 2.5(2) for any a ∈ S, there exists u ∈ S such that a = a (ua²) or a = at where t = ua² ∈ S. Now $(f \tilde{\circ} χ_{S}) (a) = ⋃_{a = mn} {f (m) \cap χ_{S} (n)} \supseteq f (a) \cap$ χ_S (t) = f (a), so $f \tilde{\subseteq} f \tilde{\circ} χ_{S}$ , and $(g \tilde{\circ} χ_{S}^{c}) (a) = ⋂_{a = mn} {g (m) \cup χ_{S}^{c} (n)} \subseteq g (a) \cup χ_{S}^{c} (t) = g (a)$ , so $g \tilde{\supseteq}$ $g \tilde{\circ} χ_{S}^{c} .$ Hence 〈 (f, g) S 〉 ⊑ 〈 (f, g) S 〉 ◊ X_S. □ As a consequence of Proposition 3.10, Theorem 3.14 and Theorem 3.15, we have the following result.

Theorem 3.16. If S is an intra-regular AG-groupoid with left identity e, then for any DFS left (resp. right ideal) 〈 (α, β) S〉 of S over U, we have $X_{S} ◊ 〈 (α, β); S 〉 \tilde{=} 〈 (α, β); A 〉$ (resp. $〈 (α, β); S 〉 ◊ X_{S} \tilde{=} 〈 (α, β); S 〉$ ).

4 Double-framed soft generalized bi-ideals

In this section, we introduce the notions of DFS bi-ideals and DFS generalized bi-ideals. Some properties of them are provided.

Definition 4.1. A double-framed soft set 〈 (α, β) A〉 of A over U is called a double-framed soft generalized bi-ideal (briefly, DFS generalized bi-ideal) of A over U if it satisfies:

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

Definition 4.2. A double-framed soft set 〈 (α, β) A〉 of A over U is called a double-framed soft bi-ideal (briefly, DFS bi-ideal) of A over U if it satisfies:

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

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

Example 4.3. Let us consider S = {a, b, c} with the following multiplication table:

\begin{matrix} . & a & b & c \\ a & b & c & b \\ b & b & b & b \\ c & b & b & b \end{matrix}

Then (S, ·) is an AG-groupoid. Clearly S is non-commutative and non-associative since ab ≠ ba and (ab) b ≠ a (bb). Consider a DFS set 〈 (α, β) S〉 of S over $U = ℤ$ defined by $α (a) = 8 ℤ$ , $α (b) = 2 ℤ$ , $α (c) = 4 ℤ$ and $β (a) = 4 ℤ$ , $β (b) = 16 ℤ$ , $β (c) = 8 ℤ$ .

By routine checking, 〈 (α, β) S〉 is DFS bi-ideal of S over U.

It is important to note that every DFS bi-ideal is a DFS generalized bi-ideal, but the converse is not true in general.

Let us consider a DFS set 〈 (f, g) S〉 of S over $U = ℤ$ . defined by $f (a) = 8 ℤ$ , $f (b) = 4 ℤ$ , $f (c) = 16 ℤ$ and $g (a) = 16 ℤ$ , $g (b) = 32 ℤ$ , $g (c) = 8 ℤ$ .

Clearly 〈 (f, g) S〉 is DFS generalized bi-ideal but it is not DFS bi-ideal because f (c) = f (ab) nsupseteqf (a) ∩ f (b) and/or g (c) = g (ab) ⊈ g (a) ∪ g (b).

It is interesting to note that if an AG-groupoid is intra-regular and contains left identity, then every DFS generalized bi-ideal becomes DFS bi-ideal. We will prove this in Theorem 4.13.

Theorem 4.4. If 〈 (α, β) S〉 is a DFS-set, then the following statements are equivalent:

〈 (α, β) S〉 is a DFS generalized bi-ideal.

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

Proof. Suppose 〈 (α, β) S〉 is a DFS generalized bi-ideal over U. Let γ and δ be subsets of U such that i_S (α γ) ¬ = ∅ ¬ = e_S (β δ). Let x, y, z ∈ S and x, z ∈ i_S (α γ). Then α (x) ⊇ γ and α (z) ⊇ γ. Since 〈 (α, β) S〉 is DFS generalized bi-ideal of S, then α ((xy) z) ⊇ α (x) ∩ α (z) ⊇ γ ∩ γ = γ. Thus (xy) z ∈ i_A (α γ) and so i_A (α γ) is generalized bi-ideal of S. Now suppose v, u, w ∈ S and v, w ∈ e_S (β δ) then β (v) ⊆ δ and β (w) ⊆ δ. Since 〈 (α, β) S〉 is DFS generalized bi-ideal of S, then β ((vu) w) ⊆ β (v) ∪ β (w) ⊆ δ ∪ δ = δ. Thus (vu) w ∈ e_S (β δ) and so e_S (β δ) is generalized bi-ideal of S.

Conversely, suppose the non-empty γ-inclusive set and δ-exclusive set of 〈 (α, β) S〉 are generalized bi-ideal of S for any subsets γ and δ of U. Let x, y, z ∈ S such that α (x) = γ₁, α (z) = γ₂, β (x) = δ₁, β (z) = δ₂. Let us take γ = γ₁ ∩ γ₂ and δ = δ₁ ∪ δ₂. Now α (x) = γ₁ ⊇ γ₁ ∩ γ₂ = γ and so x ∈ i_S (α γ). Similarly z ∈ i_S (α γ). By hypothesis, i_S (α γ) is generalized bi-ideal of S, hence (xy) z ∈ i_S (α γ) and so α ((xy) z) ⊇ γ = γ₁ ∩ γ₂ = α (x) ∩ α (z). Also as β (x) = δ₁ ⊆ δ₁ ∪ δ₂ = δ then x ∈ e_S (β δ). Similarly z ∈ e_S (β δ). By hypothesis, e_S (β δ) is a generalized bi-ideal of S, hence (xy) z ∈ e_S (β δ) and so β ((xy) z) ⊆ δ = δ₁ ∪ δ₂ = β (x) ∪ β (z). Therefore 〈 (α, β S)〉 is a DFS generalized bi-ideal of S. □

Example 4.5. Let S = {a, b, c, d} with the following multiplication table:

\begin{matrix} . & a & b & c & d \\ a & c & c & c & d \\ b & d & d & c & c \\ c & d & d & d & d \\ d & d & d & d & d \end{matrix}

(S, ·) is an AG-groupoid. Generalized bi-ideals of S are {d}, {a, d}, {b, d}, {c, d}, {a, c, d}, {b, c, d} and {a, b, d}.

Define a DFS 〈 (f, g) S〉 of S over U = {1, 2, 3} as follows:

f (a) = {1}, f (b) = {1, 2}, f (c) = {2, 3}, f (d) = {1, 2, 3}

g (a) = {1, 2, 3}, g (b) = {1, 3}, g (c) = {1, 2}, g (d) = {1}.

Then

$i_{S} (f; γ) = {\begin{matrix} {a, b, d} if γ = {1}, \\ {b, c, d} if γ = {2}, \\ {c, d} if γ = {3}, \\ {b, d} if γ = {1, 2}, \\ {d} if γ = {1, 3}, \\ {c, d} if γ = {2, 3}, \\ {d} if γ = {1, 2, 3} \end{matrix}$ and $e_{S} (g; δ) = {\begin{matrix} {d} if δ = {1}, \\ {} if δ = {2}, \\ {} if δ = {3}, \\ {c, d} if δ = {1, 2}, \\ {b, d} if δ = {1, 3}, \\ {} if δ = {2, 3}, \\ {a, b, c, d} if δ = {1, 2, 3} \end{matrix}$

We note that the non-empty γ-inclusive set and δ-exclusive set of 〈 (f, g) S〉 are generalized bi-ideals of S for any subsets γ and δ of U. Hence 〈 (f, g) S〉 is a DFS generalized bi-ideal of S over U.

Theorem 4.6. If 〈 (α, β) S〉 is a DFS-set, then the following statements are equivalent:

〈 (α, β) S〉 is a DFS bi-ideal of S.

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

Proof. It follows from Theorem 4.4 and Proposition 3.13. □

Theorem 4.7. (cf. [17]) Let 〈 (f, g) S〉 be a DFS-set of S over U. Then 〈 (f, g) S〉 is a DFS bi-ideal of S over U if and only if 〈 (f, g) S〉 ◊ 〈 (f, g) S 〉 ⊑ 〈 (f, g) S 〉

and (〈 (f, g) S 〉 ◊ X_S) ◊ 〈 (f, g) S 〉 ⊑ 〈 (f, g) S 〉.

Theorem 4.8. Every DFS right ideal of an AG-groupoid S is DFS bi-ideal.

Proof. Assume that S is an AG-groupoid and 〈 (f, g) S〉 is a DFS right ideal of S. Now $(f \tilde{\circ} χ_{S}) \tilde{\circ} f \tilde{\subseteq} f \tilde{\circ} f \tilde{\subseteq} f \tilde{\circ} χ_{S} \tilde{\subseteq} f$ and $f \tilde{\circ} f \tilde{\subseteq} f \tilde{\circ} χ_{S} \tilde{\subseteq} f$ . Also $(g \tilde{\circ} χ_{S}) \tilde{\circ} g \tilde{\supseteq} g \tilde{\circ} g \tilde{\supseteq} g \tilde{\circ} χ_{S} \tilde{\supseteq} g$ and $g \tilde{\circ} g \tilde{\supseteq} g \tilde{\circ} χ_{S} \tilde{\supseteq} g .$ Thus 〈 (f, g) S〉 ◊ 〈 (f, g) S 〉 ⊑ 〈 (f, g) S 〉and (〈 (f, g) S 〉 ◊ X_S) ◊ 〈 (f, g) S 〉 ⊑ 〈 (f, g) S 〉 and so by Theorem 4.7, 〈 (f, g) S〉 is DFS bi-ideal. □

Theorem 4.9. Every DFS ideal of an AG-groupoid is DFS bi-ideal.

Proof. Straightforward. □

Theorem 4.10. Let A be a non-empty subset of an AG-groupoid S. Then A is a bi-ideal of S if and only if the DFS-set X_A is a DFS bi-ideal of S over U.

Proof. Assume that A is bi-ideal of S. Let x, y, z ∈ S. If x, z ∈ A, then (xy) z ∈ A, since A is bi-ideal of S. Therefore χ_A ((xy) z) = U = χ_A (x) ∩ χ_A (z). Now if one of x or z does not belong to A, for definiteness let x ∉ A, then (xy) z may or may not belong to A. If (xy) z ∈ A, then χ_A ((xy) z) = U ⊇ ∅ = χ_A (x) ∩ χ_A (z). If (xy) z ∉ A, then χ_A ((xy) z) = ∅ = χ_A (x) ∩ χ_A (z). Hence in any case χ_A ((xy) z) ⊇ χ_A (x) ∩ χ_A (z) for any x, y, z ∈ S. Now take a, b ∈ S. If a, b ∈ A, then ab ∈ A and so χ_A (ab) = U = χ_A (a) ∩ χ_A (b). Now if one of a or b does not belong to A, for definiteness let a ∉ A, then ab may or may not belong to A. If ab ∈ A, then χ_A (ab) = U ⊇ ∅ = χ_A (a) ∩ χ_A (b). If ab ∉ A, then χ_A (ab) = ∅ = ∅ ∩ U = χ_A (x) ∩ χ_A (y). Hence χ_A (ab) ⊇ χ_A (a) ∩ χ_A (b).

Now again let x, y, z ∈ S. If x, z ∈ A, then (xy) z ∈ A, since A is bi-ideal of S. Therefore $χ_{A}^{c} ((xy) z) = \emptyset$ $= χ_{A}^{c} (x) \cup χ_{A}^{c} (z) .$ Now if one of x or z does not belong to A, for definiteness let x ∉ A, then (xy) z may or may not belong to A. If (xy) z ∈ A, then $χ_{A}^{c} ((xy) z) = \emptyset \subseteq χ_{A}^{c} (x) \cup χ_{A}^{c} (z)$ . If (xy) z ∉ A, then $χ_{A}^{c} ((xy) z) = U = χ_{A}^{c} (x) \cup χ_{A}^{c} (z)$ . Hence $χ_{A}^{c} ((xy) z) \subseteq χ_{A}^{c} (x) \cup χ_{A}^{c} (z)$ for any x, y, z ∈ S. Now take a, b ∈ S. If a, b ∈ A, then ab ∈ A and so $χ_{A}^{c} (ab) = \emptyset = χ_{A}^{c} (a) \cup χ_{A}^{c} (b)$ . Now if one of a or b does not belong to A, for definiteness let a ∉ A, then ab may or may not belong to A. If ab ∈ A, then $χ_{A}^{c} (ab) = \emptyset \subseteq χ_{A}^{c} (a) \cup χ_{A}^{c} (b)$ . If ab ∉ A, then $χ_{A}^{c} (ab) = U = χ_{A}^{c} (a) \cup χ_{A}^{c} (b) .$ Hence $χ_{A}^{c} (ab) \subseteq χ_{A}^{c} (a) \cup χ_{A}^{c} (b)$ . Hence X_A is DFS bi-ideal of S over U.

Conversely, suppose that X_A is DFS bi-ideal of S over U. Let x, y, z ∈ A such that x, z ∈ A. Since X_A is DFS bi-ideal, then χ_A ((xy) z) ⊇ χ_A (x) ∩ χ_A (z) = U. Thus χ_A ((xy) z) = U and so (xy) z ∈ A. Now let x, y ∈ A then χ_A (xy) ⊇ χ_A (x) ∩ χ_A (y) = U. Hence χ_A (xy) = U and so xy ∈ A. Similarly we have the same result if we choose $χ_{A}^{c}$ part. Hence A is bi-ideal of S. □

Corollary 4.11. Let A be a non-empty subset of an AG-groupoid S. Then A is a generalized bi-ideal of S if and only if the DFS-set X_A is a DFS generalized bi-ideal of S over U.

Theorem 4.12. (cf. [19]) Let A be a non-empty subset of an AG-groupoid S. Then the following statements hold:

A is a left (resp. right) of S if and only if the DFS-set X_A is a DFS left (resp. right ideal) of S over U.

A is a semiprime if and only if the DFS-set X_A is a DFS semiprime.

Theorem 4.13. If S is an intra-regular AG-groupoid with left identity e, then the following statements are equivalent:

〈 (f, g) S〉 is DFS bi-ideal of S over U.

〈 (f, g) S〉 is DFS generalized bi-ideal of S over U.

Proof. (1)⇒(2) is obvious.

(2)⇒(1). Let 〈 (f, g) S〉 is DFS generalized bi-ideal of S over U. Let p, q ∈ S. Since S is intra-regular, then for p ∈ S, there exist u, v in S such that p = (up²) v. By using medial law, paramedial law, Eq. 1 and Eq. 2, we have f (pq) = f ((up²) v) q) = f (((up²) (ev)) q) = f (((ve) (p²u)) q) = f ((p² ((ve) u)) q) = f (((pp) ((ve) u)) q) = f (((u (ve)) (pp)) q) = f ((p ((u (ve)) p)) q) ⊇ f (p) ∩ f (q). Now g (pq) = g ((up²) v) q) = g (((up²) (ev)) q) = g (((ve) (p²u)) q) = g ((p² ((ve) u)) q) = g (((pp) ((ve) u)) q) = g (((u (ve)) (pp)) q) = g ((p ((u (ve)) p)) q) ⊆ g (p) ∪ g (q). Since p, q are arbitrary elements of S, then we conclude that 〈 (f, g) S〉 is DFS bi-ideal. □

Theorem 4.14. If S is an intra-regular AG-groupoid with left identity e, then the following statements are equivalent:

〈 (f, g) S〉 is DFS left (resp. right) ideal of S over U.

〈 (f, g) S〉 is DFS bi-ideal of S over U.

Proof. (1)⇒(2). Let 〈 (f, g) S〉 is DFS left ideal of S over U. Let a, b, c ∈ S then using left invertive law, we have f ((ab) c) = f ((cb) a) ⊇ f (a) ⊇ f (a) ∩ f (c), and g ((ab) c) = g ((cb) a) ⊆ g (a) ⊆ g (a) ∪ g (c). Hence 〈 (f, g) S〉 is DFS generalized bi-ideal. By the previous theorem, 〈 (f, g) S〉 is DFS bi-ideal of S over U.

(2)⇒(1). Suppose that 〈 (f, g) S〉 is DFS bi-ideal of S over U and let a, b ∈ S. Since S is intra-regular, then there exist x, y and u, v in S such that a = (xa²) y and b = (ub²) v. Therefore by using left invertive law, medial law, paramedial law, Eq. 1 and Eq. 2, we have f (ab) = f (a ((ub²) v)) = f ((ub²) (av)) = f ((va) (b²u)) = f (b² ((va) u)) = f ((bb) ((va) u)) = f ((((va) u) b) b) = f ((((va) u) ((ub²) v)) b) = f (((ub²) (((va) u) v)) b) = f (((v ((va) u)) (bu²)) b) = f ((b² ((v ((va) u)) u)) b) = f (((bb) ((v ((va) u)) u)) b) = f (((u (v ((va) u))) (bb)) b) = f ((b ((u (v ((va) u))) b)) b) ⊇f (b) ∩ f (b) = f (b) Now g (ab) = g (a ((ub²) v)) = g ((ub²) (av)) = g ((va) (b²u)) = g (b² ((va) u)) = g ((bb) ((va) u)) = g ((((va) u) b) b) = g ((((va) u) ((ub²) v)) b) = g (((ub²) (((va) u) v)) b) = g (((v ((va) u)) (bu²)) b) = g ((b² ((v ((va) u)) u)) b) = g (((bb) ((v ((va) u)) u)) b) = g (((u (v ((va) u))) (b b)) b) = g ((b ((u (v ((va) u))) b)) b) ⊆ g (b) ∪ g (b) = g (b). Hence f (ab) ⊇ f (b) and g (ab) ⊆ g (b). Since a, b are arbitrary elements of S, then 〈 (f, g) S〉 is DFS left ideal of S over U.□

Corollary 4.15. If S is an intra-regular AG-groupoid with left identity e, then the following statements are equivalent:

〈 (f, g) S〉 is DFS ideal of S over U.

〈 (f, g) S〉 is DFS bi-ideal of S over U.

Theorem 4.16. If S is an AG-groupoid with left identity e, then the following statements are equivalent:

S is intra-regular.

Every DFS bi-ideal is idempotent.

Proof. (1)⇒(2). Let S be intra-regular AG-groupoid with left identity e then S = S². That is, for all x ∈ S, there exist u, v ∈ S such that x = uv. Assume that 〈 (f, g) S〉 is DFS bi-ideal of S, so by Theorem 4.7 $f \tilde{\circ} f \tilde{\subseteq} f$ and $g \tilde{\circ} g \tilde{\supseteq} g$ . Using left invertive law, medial law, paramedial law, Eq. 1 and Eq. 2, for any a ∈ S, there exist x, y ∈ S such that a = (xa²) ya = ((uv) a²) y = ((uv) (aa)) y = ((aa) (vu)) y = (y (vu)) (aa) = (ya) ((vu) a) = (a (vu)) (ay) = ((ay) (vu)) a. We have $(f \tilde{\circ} f) (a) = ⋃_{a = pq} {f (p) \cap f (q)} \supseteq f ((ay) (vu)) \cap f$ (a) ⊇f (ay) ∩ f (a) (Corollary 4.15) ⊇f (a) ∩ f (a) (Corollary 4.15) = f (a) and so $f \tilde{\circ} f \tilde{\supseteq} f .$ Thus $f \tilde{\circ} f \tilde{=} f$ . Now $(g \tilde{\circ} g) (a) = ⋂_{a = st} {g (s) \cup g (t)} \subseteq g ((ay) (vu)) \cup g (a)$ ⊆g (ay) ∪ g (a) (Corollary 4.15) ⊆g (a) ∪ g (a) (Corollary 4.15) = g (a) and so $g \tilde{\circ} g \tilde{\subseteq} \dot{g} .$ Thus $g \tilde{\circ} g \tilde{=} \dot{g}$ . Hence $〈 (f, g); S 〉 ◊ 〈 (f, g); S 〉 \tilde{=} 〈 (f, g); S 〉 .$ This proves that the DFS bi-ideal 〈 (f, g) S〉 is idempotent.

(2)⇒(1). Assume that every DFS bi-ideal is idempotent. Let a ∈ S. Since Sa is bi-ideal of S so, by Theorem 4.10, X_Sa is DFS bi-ideal of S. By hypothesis; X_Sa = X_Sa ◊ X_Sa = X_(Sa)(Sa) = X_{Sa
²}. Now a ∈ Sa, so χ_Sa (a) = U = χ_{Sa
²} (a). Thus a ∈ Sa² = (Sa²) S (Lemma 2.3). If we take $χ_{Sa}^{c}$ part, we get a ∈ (Sa²) S. Since a is an arbitrary element of S, hence S is intra-regular. □

Theorem 4.17. If S is an AG-groupoid with left identity e, then the following statements are equivalent:

S is intra-regular.

For every DFS bi-ideal 〈 (f, g) S〉 of S, f (a) = f (a²) ⊇ f (e) and g (a) = g (a²) ⊆ g (e) for all a ∈ S.

Proof. (1)⇒(2). Suppose S is intra-regular and 〈 (f, g) S〉 is DFS bi-ideal of S. Let a ∈ S. Then by Lemma 2.5 (i), there exists t such that a = (a²t) a². Now f (a) = f ((a²t) a²) ⊇ f (a²) ∩ f (a²) = f (a²), so f (a) ⊇ f (a²) and g (a) = g ((a²t) a²) ⊆ g (a²) ∪ g (a²) = g (a²), so g (a) ⊆ g (a²). Since 〈 (f, g) S〉 is DFS bi-ideal of S, by Corollary 4.15, 〈 (f, g) S〉 is DFS ideal of S, thus f (a²) = f (aa) ⊇ f (a) and g (a²) = g (aa) ⊆ g (a). Hence f (a) = f (a²) and g (a) = g (a²) for all a ∈ S. Now f (a²) = f (aa) = f ((ee) (aa)) = f ((aa) (ee)) = f ((e (aa)) e) ⊇ f (e) ∩ f (e) = f (e) and so f (a²)⊇ f (e). Also g (a²) = g (aa) = g ((ee) (aa)) = g ((aa) (ee)) = g ((e (aa)) e) ⊆ g (e) ∪ g (e) = g (e) and thus g (a²) ⊆ g (e).

(2)⇒(1). Assume that for every DFS bi-ideal 〈 (f, g) S〉 of S, f (a) = f (a²) and g (a) = g (a²) for all a ∈ S. Let a ∈ S. Then Sa² is bi-ideal of S and a² ∈ Sa². Since Sa² is bi-ideal of S, then by Theorem 4.10, X_{Sa
²} is DFS bi-ideal of S. Since a² ∈ Sa², then U = χ_{Sa
²} (a²) = χ_{Sa
²} (a). Thus a ∈ Sa² = (Sa²) S.(Lemma 2.3). Since a is arbitrary element of S, then S is intra-regular. □

Theorem 4.18. Let 〈 (f, g) S〉 be a DFS bi-ideal of an intra-regular AG groupoid S with left identity e. Then f (ab) = f (ba) and g (ab) = g (ba) for all a, b ∈ S.

Proof. Suppose S is intra-regular AG-groupoid with left identity e, and 〈 (f, g) S〉 a DFS bi-ideal of S over U then, $\begin{matrix} f (ab) & = & f ((ab)^{2}) (By Theorem 4.17) \\ = & f ((ab) (ab)) \\ = & f ((ba) (ba)) (using Eq . 1) \\ \supseteq & f (ba) (By Corollary 4.15) \end{matrix}$ and $\begin{matrix} f (ba) & = & f ((ba)^{2}) (By Theorem 4.17) \\ = & f ((ba) (ba)) \\ = & f ((ab) (ab)) (using Eq . 1) \\ \supseteq & f (ab) (By Corollary 4.15) \end{matrix}$ Thus f (ab) = f (ba). Now $\begin{matrix} g (ab) & = & g ((ab)^{2}) (By Theorem 4.17) \\ = & g ((ab) (ab)) \\ = & g ((ba) (ba)) (using Eq . 1) \\ \supseteq & g (ba) (By Corollary 4.15) \end{matrix}$ and $\begin{matrix} g (ba) & = & g ((ba)^{2}) (By Theorem 4.17) \\ = & g ((ba) (ba)) \\ = & g ((ab) (ab)) (using Eq . 1) \\ \supseteq & g (ab) (By Corollary 4.15) \end{matrix}$ Thus g (ab) = g (ba) and hence the proof is completed. □

5 Characterizations of intra-regular AG-groupoids in terms of DFS ideals

In this section, we give some characterizations of intra-regular AG-groupoids using DFS ideals and DFS bi-ideals.

Theorem 5.1. If S is an AG-groupoid with left identity e, then the following statements are equivalent:

S is intra-regular.

A ∩ B ⊆ AB for every subset A and bi-ideal B of S.

〈 (f, g) S〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉 for every DFS subset 〈 (f, g) S〉 and DFS bi-ideal 〈 (α, β) S〉 of S over U.

Proof. (1)⇒(3). Assume S is intra-regular. Let 〈 (f, g) S〉 be a DFS set of S and 〈 (α, β) S〉 a DFS bi-ideal of S over U. Now let a ∈ S. Then by Lemma 2.5(3), there exists q ∈ S, such that a = a ((a²q) a²). Now $(f \tilde{\circ} α) (a) = ⋃_{a = st} {f (s) \cap α (t)} \supseteq$ f (a) ∩ α ((a²q) a²) ⊇ f (a) ∩ α (a²) ⊇ f (a) ∩ α (a) [Corollary 4.15]. Hence $f \cap α \tilde{\subseteq} f \tilde{\circ} α$ and $(g \tilde{\circ}$ β) (a) = ⋂ _a=mn { g (m) ∪ β (n) } ⊆ g (a) ∪ β ((a²q) a²) ⊆ g (a) ∪ β (a²) ⊆ g (a) ∪ β (a) [Corollary 4.15]. Hence $g \cup β \tilde{\supseteq} g \tilde{\circ} β$ . Thus 〈 (f, g) S〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉.

(3)⇒(2). Assume for every DFS subset 〈 (f, g) S〉 and DFS bi-ideal 〈 (α, β) S〉 of S over U, we have 〈 (f, g) S 〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉. Let A be a subset of S and B a bi-ideal. Then by Theorem 4.10, X_B is DFS bi-ideal of S over U. By hypothesis X_A ⊓ X_B ⊑ X_AB. Let a ∈ A ∩ B, so a ∈ L as well as a ∈ B and so χ_A∩B (a) = U = χ_A (a) = χ_B (a). By hypothesis, $U = χ_{A \cap B} (a) = (χ_{A} \cap χ_{B}) (a) \subseteq (χ_{A} \tilde{\circ} χ_{B}) (a) = χ_{AB} (a)$ . Hence a ∈ AB. Similarly result is obtained if we choose uni-part of characteristic function. Thus A ∩ B ⊆ AB.

(2)⇒(1). Let a ∈ S. Since Sa is a bi-ideal of S and a ∈ Sa, then by hypothesis a ∈ Sa ∩ Sa ⊆ (Sa) (Sa) = (Sa²) S (by Lemma 2.3). Hence a ∈ (Sa²) S, which shows that S is intra-regular. □

Corollary 5.2. If S is an AG-groupoid with left identity e, then the following statements are equivalent:

S is intra-regular.

B₁ ∩ B₂ ⊆ B₁B₂ for all bi-ideals B₁ and B₂ of S.

〈 (f, g) S〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉 for all DFS bi-ideals 〈 (f, g) S〉 and 〈 (α, β) S〉 of S over U.

〈 (f, g) S〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉 for all DFS generalized bi-ideals 〈 (f, g) S〉 and 〈 (α, β) S〉 of S over U.

Theorem 5.3. If S is an AG-groupoid with left identity e, then the following statements are equivalent:

S is intra-regular.

B ∩ L ⊆ BL for every left ideal L and every bi-ideal B of S.

〈 (f, g) S〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉 for all DFS bi-ideal 〈 (f, g) S〉 and DFS left ideal 〈 (α, β) S〉 of S over U.

〈 (f, g) S〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉 for all DFS generalized bi-ideal 〈 (f, g) S〉 and DFS left ideal 〈 (α, β) S〉 of S over U.

Proof. (1)⇒(4). Let S is intra-regular AG-groupoid with left identity e. Let 〈 (f, g) S〉 be a DFS generalized bi-ideal and 〈 (α, β) S〉 be DFS left ideal of S over U. For any a ∈ S. By Lemma 2.5(5), there exists p such that a = ((a (xa)) p) a. Since DFS generalized bi-ideal is DFS ideal in intra-regular AG-groupoid with left identity, then we have $(f \tilde{\circ} α) (a) = ⋃_{a = uv} {f (u) \cap α (v)} \supseteq f ((a (xa)) p) \cap α (a)$ ⊇f (a (xa)) ∩ α (a) ⊇ f (a) ∩ α (a) = (f ∩ α) (a), so $f \cap α \tilde{\subseteq} f \tilde{\circ} α .$ Similarly, $(g \tilde{\circ} β) (a) = ⋂_{a = uv} {g (u) \cup β$ (v)} ⊆ g (a (xa)) ∪ β (a) ⊆ g (a) ∪ β (a) = (g ∪ β) (a), so $g \cup β \tilde{\supseteq} g \tilde{\circ} β .$ Hence 〈 (f, g) S〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉 for all DFS generalized bi-ideal 〈 (f, g) S〉 and DFS left ideal 〈 (α, β) S〉 of S over U.

(4)⇒(3). Straightforward.

(3)⇒(2). Assume that in AG-groupoid S with left identity, 〈 (f, g) S〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉 for all DFS bi-ideal 〈 (f, g) S〉 and DFS left ideal 〈 (α, β) S〉 of S over U. Let B is a bi-ideal and L left ideal of S. Then by Theorem 4.10 and Theorem 4.12, X_B is DFS bi-ideal and X_L is DFS left ideal of S over U, then X_B ⊓ X_L = X_B∩L ⊑ X_B ◊ X_L = X_BL. Hence by Lemma 3.9(1), B∩ L ⊆ BL.

(2)⇒(1). Let a ∈ S. Since Sa is bi-ideal as well as left ideal of AG-groupoid S with left identity, then a ∈ Sa ∩ Sa = (Sa) (Sa) = (Sa²) S (Lemma 2.3). Thus S is intra-regular. □

Theorem 5.4. If S is an AG-groupoid with left identity e, then the following statements are equivalent:

S is intra-regular.

Every ideal is semiprime.

Every bi-ideal is semiprime.

Every DFS bi-ideal is semiprime.

Every DFS generalized bi-ideal is semiprime.

Proof. (1)⇒(2). Assume S is intra-regular and 〈 (f, g) S〉 be DFS generalized bi-ideal of S over U. Let a ∈ S, since S is intra-regular so by Lemma 2.5(1), there exists t ∈ S such that a = (a²t) a². Now f (a) = f ((a²t) a²) ⊇ f (a²) ∩ f (a²) = f (a²) and g (a) = g ((a²t) a²) ⊆ g (a²) ∪ g (a²) = g (a²). Hence DFS generalized bi-ideal 〈 (f, g) S〉 is semiprime.

(5)⇒(4) is obvious.

(4)⇒(3). Suppose every DFS bi-ideal of S is semiprime. Let B is bi-ideal of S. By Theorem 4.10, X_B is DFS bi-ideal of S. Let a² ∈ B. Then χ_B (a²) = U and $χ_{B}^{c} (a^{2}) = \emptyset$ , since X_B is DFS bi-ideal of S. So by hypothesis X_B is DFS semiprime and thus U = χ_B (a²) ⊆ χ_B (a) or χ_B (a) ⊇ U. Hence χ_B (a) = U and so a ∈ B. Now $χ_{B}^{c} (a^{2}) = \emptyset$ , then $\emptyset = χ_{B}^{c} (a^{2}) \supseteq χ_{B}^{c} (a)$ and so $χ_{B}^{c} (a) = \emptyset .$ Hence a ∈ B. In any case, a² ∈ B implies a ∈ B. Hence B is semiprime.

(3)⇒(2) is obvious.

(2)⇒(1) Assume every ideal of S is semiprime. Let a ∈ S. Then a²S is ideal of S. By Lemma 2.3, Sa² = a²S, hence Sa² is ideal of S. Thus, by hypothesis, Sa² is semiprime. For every a ∈ S, a² ∈ Sa² implies a ∈ Sa² = (Sa²) S (Lemma 2.3). Hence S is intra-regular. □

Theorem 5.5. If S is an AG-groupoid with left identity e, then the following statements are equivalent:

S is intra-regular.

Every ideal is semiprime.

Every DFS ideal is semiprime.

Every DFS right ideal is semiprime.

Every DFS bi-ideal is semiprime.

Proof. (1)⇒(5). Assume that S is intra-regular and 〈 (f, g) S〉 is a DFS bi-ideal of S. Let a ∈ S. By Lemma 2.5(1), there exists u ∈ S such that a = (a²q) a². Now f (a) = f ((a²q) a²) ⊇ f (a²) ∩ f (a²) = f (a²) and g (a) = g ((a²q) a²) ⊆ g (a²) ∪ g (a²) = g (a²). Hence f (a) ⊇ f (a²) and g (a) ⊆ g (a²) and thus the DFS bi-ideal 〈 (f, g) S〉 of is DFS semiprime.

(5)⇒(4). It follows from Theorem 4.8.

(4)⇒(3). Obvious.

(3)⇒(2). Assume every DFS bi-ideal is semiprime. Let I be an ideal of S, so by Theorem 4.12, X_I is DFS ideal of S. By Theorem 4.9, X_I is DFS bi-ideal and by hypothesis, X_I is DFS semiprime. Let a² ∈ I. Then χ_I (a²) = U and $χ_{I}^{c} (a^{2}) = \emptyset$ . Since X_I is DFS semiprime, then U = χ_I (a²) ⊆ χ_I (a) and $\emptyset = χ_{I}^{c} (a^{2}) \supseteq χ_{I}^{c} (a)$ . Thus χ_I (a) = U and $χ_{I}^{c} (a) = \emptyset$ . Hence a ∈ I and so I is semiprime.

(2)⇒(1). Assume every ideal of S is semiprime. Let a ∈ S then a²S is ideal of S. By Lemma 2.3, Sa² = a²S, hence Sa² is ideal of S. Thus, by hypothesis, Sa² is semiprime. For every a ∈ S, a² ∈ Sa² implies a ∈ Sa² = (Sa²) S (Lemma 2). Hence S is intra-regular. □

Theorem 5.6. If S is an AG-groupoid with left identity e, then the following statements are equivalent:

S is intra-regular.

〈 (f, g) S〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (f, g) S 〉 for every DFS bi-ideal 〈 (f, g) S〉 of S over U.

〈 (f, g) S〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉 for all DFS bi-ideals 〈 (f, g) S〉 and 〈 (α, β) S〉 of S over U.

Proof. (1)⇒(3). Assume that S is intra-regular AG-groupoid with left identity e. Let 〈 (f, g) S〉 and 〈 (α, β) S〉 are DFS bi-ideals of S over U. For any a ∈ S, there exist x, y ∈ S such that a = (xa²) y = (x (aa)) y = (a (xa)) y = (y (xa)) a. Now y (xa) = y (x ((xa²) y)) = y ((xa²) (xy)) = (xa²) (xy²) = (a (xa)) (xy²) = ((xy²) (xa)) a = ((xy²) (x ((xa²) y)) a = ((xy²) ((xa²) (xy))) a = ((xa²) ((xy²)) (a²x)) a = (((xy) (xy²)) (a²x)) a = (a² (((xy) (xy²)) x)) a = ((x ((xy) (xy²))) (aa)) a = (a (x ((xy) (xy²))) a) a. Thus a = (y (xa)) a = (a (x ((xy) (xy²))) a) a. We have $(f \tilde{\circ} α) (a) = ⋃_{a = pq} f (p) \cap α (q) \supseteq f (a (x ((xy) ({xy}^{2})))$ a) ∩ α (a) ⊇ f (a) ∩ f (a) ∩ α (a) = f (a) ∩ α (a) = (f ∩α) (a). Thus $f \cap α \tilde{\subseteq} f \tilde{\circ} α$ and $(g \tilde{\circ} β) (a) = ⋂_{a = uv} {g (u) \cup β (v)} \subseteq g (a (x ((xy) ({xy}^{2}))) a) \cup β (a) \subseteq g (a) \cup g (a) \cup β (a) = g (a) \cup β (a) = (g \cup β) (a)$ , so $g \cup β \tilde{\supseteq} g \tilde{\circ} β .$ Thus 〈 (f, g) S〉 ⊓ 〈 (α, β) S 〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (α, β) S 〉 for all DFS bi-ideals 〈 (f, g) S〉 and 〈 (α, β) S〉 of S over U.

(3)⇒(2). Straightforward.

(2)⇒(1). Assume that 〈 (f, g) S〉 ⊑ 〈 (f, g) S 〉 ◊ 〈 (f, g) S 〉 for every DFS bi-ideal 〈 (f, g) S〉 of S over U. Let B is a bi-ideal of S. Then by Theorem 4.10, X_B is DFS bi-ideal of S and by hypothesis, X_B ⊑ X_B ◊ X_B = X_{B
²}. Thus X_B ⊑ X_{B
²}. Also since B is bi-ideal, then B² ⊆ B and so X_{B
²} ⊑ X_B. Thus X_{B
²} = X_B, which implies that X_B is idempotent. By Theorem 4.16, S is intra-regular. □

6 Some applications

The important of our results and in general, of results on double-framed soft sets can be seen in using them in different application in decision making situations.

Example 6.1. Let we denote by U = {p₁, p₂, p₃} the initial universe, where p_i, i = 1, …, 3 are three patients. Let E = {e₁, e₂, e₃, e₄} be the set of parameters showing status of patients in which

e₁ stands for the parameter "chest pain"

e₂ stands for the parameter "head ache"

e₃ stands for the parameter "tooth ache"

e₄ stands for the parameter "back pain"

with the following multiplication table:

\begin{matrix} . & e_{1} & e_{2} & e_{3} & e_{4} \\ e_{1} & e_{3} & e_{3} & e_{3} & e_{4} \\ e_{2} & e_{4} & e_{4} & e_{3} & e_{3} \\ e_{3} & e_{4} & e_{4} & e_{4} & e_{4} \\ e_{4} & e_{4} & e_{4} & e_{4} & e_{4} \end{matrix}

(E, ·) is an AG-groupoid. Generalized bi-ideals of E are {e₄},{e₁, e₄},{e₂, e₄},{e₃, e₄},{e₁, e₃, e₄}, {e₂, e₃, e₄} and {e₁, e₂, e₄}.

Define a DFS 〈 (f, g) S〉 of E over U = {p₁, p₂, p₃} as follows:

f (e₁) = {p₁}, f (e₂) = {p₁, p₂}, f (e₃) = {p₂, p₃}, f (e₄) = {p₁, p₂, p₃}

g (e₁) = {p₁, p₂, p₃}, g (e₂) = {p₁, p₃}, g (e₃) = {p₁, p₂}, g (e₄) = {p₁}. Then

$i_{S} (f; γ) = {\begin{matrix} {e_{1}, e_{2}, e_{4}} if γ = {p_{1}}, \\ {e_{2}, e_{3}, e_{4}} if γ = {p_{2}}, \\ {e_{3}, e_{4}} if γ = {p_{3}}, \\ {e_{2}, e_{4}} if γ = {p_{1}, p_{2}}, \\ {e_{4}} if γ = {p_{1}, p_{3}}, \\ {e_{3}, e_{4}} if γ = {p_{2}, p_{3}}, \\ {e_{4}} if γ = {p_{1}, p_{2}, p_{3}} \end{matrix}$ and $e_{S} (g; δ) = {\begin{matrix} {e_{4}} if δ = {p_{1}}, \\ {} if δ = {p_{2}}, \\ {} if δ = {p_{3}}, \\ {e_{3}, e_{4}} if δ = {p_{1}, p_{2}}, \\ {e_{2}, e_{4}} if δ = {p_{1}, p_{3}}, \\ {} if δ = {p_{2}, p_{3}}, \\ {e_{1}, e_{2}, e_{3}, e_{4}} if δ = {p_{1}, p_{2}, p_{3}} \end{matrix}$

Also, we can apply these notions and results for studying DFS expert sets and their application in Decision making, as we can see below, and which will be furthermore, one of our aim to the future.

Assume that a team of experts express their views by way of double-framed soft expert set (briefly, DFS expert-set) and take the advantage of DFS expert-sets to model the situation and present a method to select the best optimal choice.

Definition 6.2. Let U be a universe, E a set of parameters, and X a set of experts (agents). Let O = {0 =disagree, 1 =agree} be a set of opinions, Z = E × X × O and A ⊆ Z. A double-framed pair 〈 (α, β) A〉 is called a DFS expert-set over U, where α and β are mappings from A to P (U) (power set of U).

Example 6.3. Let U ={ h₁, h₂, h₃, h₄, h₅ } be a set of houses and E = { e₁, e₂, e₃ } = { in green surroundings, cheap, in good location}. Let X = {p, q, r} be a set of experts. Suppose that $(α, Z) = {\begin{matrix} ((e_{1}, p, 1), {h_{2}, h_{3}, h_{4}}), ((e_{1}, q, 1), {h_{1}, h_{3}, h_{4}}), ((e_{1}, r, 1), {h_{1}, h_{2}, h_{4}}), \\ ((e_{2}, p, 1), {h_{1}, h_{2}, h_{3}}), ((e_{2}, q, 1), {h_{3}, h_{4}}), ((e_{2}, r, 1), {h_{2}, h_{4}}), \\ ((e_{3}, p, 1), {h_{2}, h_{3}}), ((e_{3}, q, 1), {h_{1}, h_{4}}), ((e_{3}, r, 1), {h_{1}, h_{3}}), \\ ((e_{1}, p, 0), {h_{4}, h_{5}}), ((e_{1}, q, 0), {h_{3}, h_{5}}), ((e_{1}, r, 0), {h_{3}, h_{4}}), \\ ((e_{2}, p, 0), {h_{2}, h_{5}}), ((e_{2}, q, 0), {h_{2}, h_{4}}), ((e_{2}, r, 0), {h_{3}, h_{4}}), \\ ((e_{3}, p, 0), {h_{2}, h_{3}}), ((e_{3}, q, 0), {h_{1}, h_{5}}), ((e_{3}, r, 0), {h_{1}, h_{4}}), \end{matrix}}$ $(β, Z) = {\begin{matrix} ((e_{1}, p, 1), {h_{1}}), ((e_{1}, q, 1), {h_{2}}), ((e_{1}, r, 1), {h_{3}}), \\ ((e_{2}, p, 1), {h_{4}}), ((e_{2}, q, 1), {h_{5}}), ((e_{2}, r, 1), {h_{1}, h_{2}, h_{5}}), \\ ((e_{3}, p, 1), {h_{2}, h_{4}, h_{5}}), ((e_{3}, q, 1), {h_{1}, h_{3}, h_{5}}), ((e_{3}, r, 1), {h_{1}, h_{4}, h_{5}}), \\ ((e_{1}, p, 0), {h_{2}, h_{3}, h_{4}}), ((e_{1}, q, 0), {h_{2}, h_{4}, h_{5}}), ((e_{1}, r, 0), {h_{3}, h_{4}, h_{5}}), \\ ((e_{2}, p, 0), {h_{1}, h_{2}, h_{3}, h_{4}}), ((e_{2}, q, 0), {h_{1}, h_{2}, h_{3}, h_{5}}), ((e_{2}, r, 0), {h_{1}, h_{2}, h_{4}, h_{5}}), \\ ((e_{3}, p, 0), {h_{1}, h_{3}, h_{4}, h_{5}}), ((e_{3}, q, 0), {h_{2}, h_{3}, h_{4}, h_{5}}), ((e_{3}, r, 0), U), \end{matrix}}$

In Tables 1 & 2, we present the agree-soft expert set (α₁, Z) and disagree-soft expert set (α₀, Z), respectively. Similarly, in Tables 3 & 4 we represent the agree-soft expert set (β₁, Z) and disagree-soft expert set (β₀, Z) such that $h_{ij} = {\begin{matrix} 1, if h_{i} \in α_{1} (e), \\ 0, otherwise \end{matrix}$ and

$\begin{array}{l} h_{i j} = {{1, if h}_{i} \in α_{0} (e), \\ 0, otherwise \end{array}$

$\begin{array}{l} h_{i j}^{'} = {\begin{matrix} 1, if h_{i}^{'} \in β_{1} (e), \\ 0, otherwise \end{matrix} \end{array}$ and

$\begin{array}{l} h_{i j}^{'} = {\begin{matrix} 1, if h_{i}^{'} \in β_{0} (e), \\ 0, otherwise \end{matrix} \end{array}$

where h_ij are the entries of tables 1 & 2 and $h_{i j}^{'}$ are the entries of tables 3 & 4, respectively.

The following algorithm may be applied for the best optimal choice of a house.

Algorithm

Step 1. Input the DFS expert-set 〈 (α, β) Z 〉.

Step 2. find agree and disagree expert sets for (α, Z) and (β, Z).

Step 3. find c_j = ∑_ih_ij for agree soft expert sets and k_j = ∑_ih_ij for disagree soft expert sets of (α, Z).

Step 4. find $c_{j}^{'} = {\sum_{i} h_{i j}}^{'}$ for agree soft expert sets and $k_{j}^{'} = {\sum_{i} h_{i j}}^{'}$ for disagree soft expert sets of (β, Z).

Step 5. find o_j = |c_j - k_j| and $o_{j}^{'} = | c_{j}^{'} - k_{j}^{'} |$

Step 6. calculate $s_{j} = \frac{o_{j} + o_{j}^{'}}{2}$

Step 7. find m, for wihch s_m =maxs_j.

Thus, s_m is the best optimal choice for a house. If m has more than one values, the any one may be chosen by agents.

Table 1
Agree-soft expert set of (α, Z)

U h ₁ h ₂ h ₃ h ₄ h ₅

(e₁, p) 0 1 1 1 0

(e₂, p) 1 1 1 0 0

(e₃, p) 0 1 1 0 0

(e₁, q) 1 0 1 1 0

(e₂, q) 0 0 1 1 0

(e₃, q) 1 0 0 1 0

(e₁, r) 1 1 0 1 0

(e₂, r) 0 1 0 1 0

(e₃, r) 1 0 1 0 0

c_j = ∑_ih_ij c₁ = 5 c₂ = 5 c₃ = 6 c₄ = 6 c₅ = 0

U	h ₁	h ₂	h ₃	h ₄	h ₅
(e₁, p)	0	1	1	1	0
(e₂, p)	1	1	1	0	0
(e₃, p)	0	1	1	0	0
(e₁, q)	1	0	1	1	0
(e₂, q)	0	0	1	1	0
(e₃, q)	1	0	0	1	0
(e₁, r)	1	1	0	1	0
(e₂, r)	0	1	0	1	0
(e₃, r)	1	0	1	0	0
c_j = ∑_ih_ij	c₁ = 5	c₂ = 5	c₃ = 6	c₄ = 6	c₅ = 0

Table 2

Disagree-soft expert set of (α, Z)

U	h ₁	h ₂	h ₃	h ₄	h ₅
(e₁, p)	0	0	0	1	1
(e₂, p)	0	1	0	0	1
(e₃, p)	0	1	1	0	0
(e₁, q)	0	0	1	0	1
(e₂, q)	0	1	0	1	0
(e₃, q)	1	0	0	0	1
(e₁, r)	0	0	1	1	0
(e₂, r)	0	0	1	1	0
(e₃, r)	1	0	0	1	0
k_j = ∑_ih_ij	k₁ = 2	k₂ = 2	k₃ = 4	k₄ = 5	k₅ = 4

Table 3

Agree-soft expert set of (β, Z)

U	h ₁	h ₂	h ₃	h ₄	h ₅
(e₁, p)	1	0	0	0	0
(e₂, p)	0	0	0	1	0
(e₃, p)	0	1	0	1	1
(e₁, q)	0	1	0	0	0
(e₂, q)	0	0	0	0	1
(e₃, q)	1	0	1	0	1
(e₁, r)	0	0	1	0	0
(e₂, r)	1	1	0	0	1
(e₃, r)	1	0	0	1	1
$c_{j}^{'} = \sum_{i} h_{i j}$	$c_{1}^{'} = 4$	$c_{2}^{'} = 3$	$c_{3}^{'} = 2$	$c_{4}^{'} = 3$	$c_{5}^{'} = 5$

Table 4

Disagree-soft expert set of (β, Z)

U	h ₁	h ₂	h ₃	h ₄	h ₅
(e₁, p)	0	1	1	1	0
(e₂, p)	1	1	1	1	0
(e₃, p)	1	0	1	1	1
(e₁, q)	0	1	0	1	1
(e₂, q)	1	1	1	0	1
(e₃, q)	0	1	1	1	1
(e₁, r)	0	0	1	1	1
(e₂, r)	1	1	0	1	1
(e₃, r)	1	1	1	1	1
$k_{j}^{'} = \sum_{i} h_{i j}$	$k_{1}^{'} = 5$	$k_{2}^{'} = 7$	$k_{3}^{'} = 7$	$k_{4}^{'} = 8$	$k_{5}^{'} = 7$

From Tables 1 & 2, we have Table 5 and from Tables 3 & 4, we have Table 6.

Table 5

Table5

c_j = ∑_ih_ij	k_j = ∑_ih_ij	o_j = \|c_j - k_j\|
c₁ = 5	k₁ = 2	3
c₂ = 5	k₂ = 2	3
c₃ = 6	k₃ = 4	2
c₄ = 6	k₄ = 5	1
c₅ = 0	k₅ = 4	4

Table 6

Table6

$c_{j}^{'} = \sum_{i} h_{i j}$	$k_{j}^{'} = \sum_{i} h_{i j}$	$o_{j}^{'} = \| c_{j}^{'} - k_{j}^{'} \|$
$c_{1}^{'} = 4$	$k_{1}^{'} = 5$	1
$c_{2}^{'} = 3$	$k_{2}^{'} = 7$	4
$c_{3}^{'} = 2$	$k_{3}^{'} = 7$	5
$c_{4}^{'} = 3$	$k_{4}^{'} = 8$	5
$c_{5}^{'} = 5$	$k_{5}^{'} = 7$	2

From Tables 5 & 6, we get Table 7.

Table 7

Table7

o = \|c_j - k_j\|	$o^{'} = \| c_{j}^{'} - k_{j}^{'} \|$	$s_{j} = \frac{o_{j} + o_{j}^{'}}{2}$
3	1	2
3	4	3.5
2	5	3.5
1	5	3
4	2	3

Thus max s_j = s₂ or s₃ and the optimal choice of house is h₂ or h₃.

Also, we can apply the results:

- to define DFS linguistic sets and their application in multi-attribute decision making.

- to define DFS neutrosophic set and it’s application in financial mathematics

- in other algebraic structures e.g. rings, hemirings, semiring and others and their application in studying the structural properties of these notions.

7 Conclusion

In this paper, we studied DFS generalized bi-ideals and DFS bi-ideals in intra-regular AG-groupoids. Our future work, except those mentioned to last above section, is to study these ideals in left regular AG-groupoids, right regular AG-groupoids, regular AG-groupoids, completely regular AG-groupoids, V-regular AG-groupoids. An important role is played by Lemma 2.3, that is, the relation Sa² = a²S = (Sa²) S for all a in an AG-groupoid S, in obtaining of several characertizations of intra-regular AG-groupoids. We will try to establish relation like this to obtain some new characterizations of left regular, right regular, regular, V-regular, completely regular AG-groupoids. As a future objective, it will be the characterization of intra-regular AG-groupoids into Pythagorean fuzzy set [35, 36].

The results obtained in this paper, will serve also as a base for future work, combining double-framed soft sets, soft sets, rough sets, fuzzy rough sets and their hybrid and applying them in real-life applications inspired by [24 , 42–49].

Footnotes

Acknowledgments

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 Professor Xueling Ma for editing and communicating the paper.

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Double-framed soft generalized bi-ideals of intra-regular AG-groupoids

Abstract

Keywords

1 Introduction

2 Preliminaries

4 Double-framed soft generalized bi-ideals

5 Characterizations of intra-regular AG-groupoids in terms of DFS ideals

6 Some applications

Table 1 Agree-soft expert set of (α, Z) U h 1 h 2 h 3 h 4 h 5 (e1, p) 0 1 1 1 0 (e2, p) 1 1 1 0 0 (e3, p) 0 1 1 0 0 (e1, q) 1 0 1 1 0 (e2, q) 0 0 1 1 0 (e3, q) 1 0 0 1 0 (e1, r) 1 1 0 1 0 (e2, r) 0 1 0 1 0 (e3, r) 1 0 1 0 0 c j = ∑ i h ij c1 = 5 c2 = 5 c3 = 6 c4 = 6 c5 = 0

Footnotes

Acknowledgments

References