( M,N )-Double framed soft ideals of Abel Grassmann’s groupoids

Abstract

The main motivation behind this paper is to study some structural properties of a non-associative structure Abel Grassmann’s groupoid (AG-groupoid) in terms of double-framed soft sets (DFS sets) as it hasn’t attracted much attention compared to associative structures. An AG-groupoid can be referred to as a non-associative semigroup, as the main difference between a semigroup and an AG-groupoid is the switching of an associative law. In this paper, we introduce the concept of (M, N)-double-framed soft ideals (briefly, (M, N)-DFS ideal) of AG-groupoids and investigate some properties of these notions. We have shown that every (M, N)-DFS ideal is (M, N)-DFS AG-groupoid but the converse is not true. This is shown with the help of an example. We also discuss the properties of (M, N)-DFS ideals in regular AG-groupoids. Moreover a decision making algorithm based on DFS-sets is given.

Keywords

DFS set regular AG-groupoid choice values DFS-weighted set

1. Introduction

In many real world problems, there arises uncertainty in the data. For example, the set of regions with cold weather. Two regions with temperature -2 degree Celsius and -10 degree Celsius have different degree of belongness to this set. To deal with such uncertainty, in 1965, Zadeh [34] introduced the concept of a fuzzy set. This concept helped to deal with vagueness and the representation of imperfect knowledge which had troubled the minds of many philosophers, logicians and mathematicians for a long time and became a crucial issue for computer scientists, particularly in the area of artificial intelligence. Since then there have been many extensions of the fuzzy set theory. A few extensions of fuzzy sets are intuitionistic fuzzy sets, rough sets, theory vague sets, bipolar fuzzy sets, picture fuzzy sets. But there are inherent difficulties associated with each of these theories pointed out by Molodtsov [25]. All these techniques lack in the parameterization tool and hence could not be successfully useful in tackling problems especially in areas like economics, environmental sciences and social sciences. Russian researcher Molodtsov conceptualized a framework known as soft set [25] in 1999. Since then, theory of soft sets has undergone rapid growth and has been applied to complex topics like optimization, decision making, data analysis, forecasting etc. (see [5 , 49]. Soft set theory has also been applied to different algebraic structures. We refer the reader to the papers [1–4 , 33].

Commutative law is given by abc = cba in ternary operations. By putting brackets on the left two elements of both hand sides of this equation, i.e. (ab) c = (cb) a, in 1972, Kazim and Naseeruddin [11] introduced a new algebraic structure which they named a left almost semigroup (briefly as LA-semigroup). Some other names have also been used in literature for left almost semigroups. Cho et al. [6] studied this structure under the name of right modular groupoid. Holgate [7] studied it as left invertive groupoid. Similarly, Stevanovic and Protic [28] 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 [27]. For more study of AG-groupoids, the reader is suggested to read [15–17 , 31].

Recently, Jun et al. [8] introduced double-framed soft sets (breifly DFS-sets) and developed ideal theory of BCK/BCI-algebra in terms of DFS-sets. Khan et al. [14] applied the idea of DFS-sets to AG-groupoids.

The present paper is an extension of our previous work [14]. Motivated by the idea of Zhan et al. [20, 35], we introduce (M, N)-DFS ideals of AG-groupoids. The organization of the paper is as follows:

In Section 2 basic concepts of AG-groupoids are given. In Section 3, the concept of soft set and DFS-set and its operations are reviewed. In Section 4, we define (M, N)-DFS AG-groupoids, (M, N)-DFS ideals and discuss some properties of these ideals. In Section 5, we study these notions in regular AG-groupoids. In the last section, we give a decision making algorithm based on DFS-sets and apply it to a real world problem.

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

It is basically a non-associative algebraic structure in 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 contain left identity. If there exists left identity in an AG-groupoid then it is unique [26].

Every AG-groupoid S with left identity satisfies the paramedial law, 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 have 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.

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 ∈ 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. [2] 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. [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. [2] 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. [2] 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 $(f_{A} \tilde{\circ} f_{B}) (x) : = {\begin{matrix} [c] c ⋃_{x = yz} {f_{A} (y) \cap f_{B} (z)} \\ if \exists y, z \in S such that x = yz \\ \emptyset otherwise . \end{matrix}$

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. [8] A double-framed soft pair 〈 (α_A, β_A); A〉 is called a double-framed soft set of A over U (briefly, DFS-set), where α_A and β_A are mappings from A to P (U).

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

We define the order relation ⊑_[M,N] on DFS (U) as follows:

For any 〈 (f_S, g_S); S〉 and 〈 (p_S, q_S); S〉, ∅ ⊆ M ⊂ N ⊆ U, we define;〈 (f_S, g_S); S〉 ⊑ _[M,N] 〈 (p_S, q_S); S 〉 ⇔ (f_S (x) ∩ N) ∪ M ⊆ (p_S (x) ∩ N) ∪ M and (g_S (x) ∪ M) ∩ N ⊇ (q_S (x) ∪ M) ∩ N for all x ∈ S.

If in case 〈 (f_S, g_S); S〉 ⊑ _[M,N] 〈 (p_S, q_S); S 〉 and 〈 (p_S, q_S); B〉 ⊑ _[M,N] 〈 (f_S, g_S); S 〉 then we say 〈 (f_S, g_S); A〉 = _[M,N] 〈 (p_S, q_S); B 〉.

Let 〈 (α_S, β_S); S〉 and 〈 (f_S, g_S); S〉 be two double-framed soft sets of S over U. Then the int-uni soft product [12] is denoted by 〈 (α_S, β_S); S〉 ◊ 〈 (f_S, g_S); S 〉 and is defined as the double framed soft set $〈 (α_{S} \tilde{\circ} f_{S}, β_{S} \tilde{\circ} g_{S}); S 〉$ over U, in which $α_{S} \tilde{\circ} f_{S},$ and $β_{S} \tilde{\circ} g_{S}$ are soft mappings from S to P (U), given as follows: $α s \tilde{o} f_{s} : s \to P (U),$ $x \to {\begin{matrix} \underset{x = y z}{\cup} {α s (y) \cap f_{s} (z)} \\ if \exists y, z \in s s u c h t h a t x = y z,^{,} \\ \emptyset otherwise, \end{matrix}$ $β s \tilde{o} f_{s} : s \to P (U),$ $x \to {\begin{matrix} {β s (y) \cup g_{s} (z)} \\ if \exists y, z \in s such t h a t x = y z, . \\ U otherwise . \end{matrix}$ One can easily see that the operation “◊” is well-defined.

For a DFS-set 〈 (α_S, β_S); S〉 of S and two subsets γ and δ of U, the γ-inclusive set and the δ-exclusive set of 〈 (α_S, β_S); S〉, denoted by i_S (α_S; γ) and e_S (β_S; δ), respectively, are defined as: $i_{S} (α_{S}; γ) = {x \in S | α_{S} (x) \supseteq γ}$ and e_S (β_S; δ) ={ x ∈ S|β_S (x) ⊆ δ } respectively.

The double framed soft including set [8] of the DFS-set 〈 (α_S, β_S); S〉 is defined as:

DF_S (α_S, β_S) _(γ,δ) : ={ x ∈ S|α_S (x) ⊇ γ, β_S (x) ⊆ δ }.

It is clear that DF_S (α_S, β_S) _(γ,δ) : = i_S (α_S; γ) ∩ e_S (β_S; δ).

For any two DFS sets 〈 (α_S, β_S); S〉 and 〈 (f_S, g_S); S〉 of S over U, the DFS intersection [8] of 〈 (α_S, β_S); S〉 and 〈 (f_S, g_S); S 〉, is defined to be the DFS set $〈 (α_{S} \tilde{\cap} f_{S}, β_{S} \tilde{\cup} g_{S}); S 〉$ where $α_{S} \tilde{\cap} f_{S},$ and $β_{S} \tilde{\cup} g_{S}$ are mappings given by $α_{S} \tilde{\cap} f_{S} : S \to P (U), x \to α_{S} (x) \cap f_{S} (x)$ and $β_{S} \tilde{\cup} g_{S} : S \to P (U), x \to β_{S} (x) \cup g_{S} (x)$ for all x ∈ S. It is denoted by $〈 (α_{S}, β_{S}); S 〉 ⊓ 〈 (f_{S}, g_{S}); S 〉 = 〈 (α_{S} \tilde{\cap} f_{S}, β_{S} \tilde{\cup} g_{S}); S 〉$ . For any two DFS sets 〈 (α_S, β_S); S〉 and 〈 (f_S, g_S); S〉 of S over U, the DFS union of 〈 (α_S, β_S); S〉 and 〈 (f_S, g_S); S 〉, is defined the be the DFS set $〈 (α_{S} \tilde{\cup} f_{S}, β_{S} \tilde{\cap} g_{S}); S 〉$ where $α_{S} \tilde{\cup} f_{S},$ and $β_{S} \tilde{\cap} g_{S}$ are mappings given by $α_{S} \tilde{\cup} f_{S} : S \to P (U), x \to α_{S} (x) \cup f_{S} (x)$ and $β_{S} \tilde{\cap} g_{S} : S \to P (U), x \to β_{S} (x) \cap g_{S} (x)$ for all x ∈ S. It is denoted by $〈 (α_{S}, β_{S}); S 〉 ⊔ 〈 (f_{S}, g_{S}); S 〉 = 〈 (α_{S} \tilde{\cup} f_{S}, β_{S} \tilde{\cap} g_{S}); S 〉$ . We have the following lemmas.

Lemma 3.6. The set (DFS (U), ◊) forms an AG-groupoid.

Proof. Obviously, the operation ”◊” is well-defined.Let 〈 (f_S, g_S); S 〉, 〈 (h_S, k_S); S 〉, and 〈 (p_S, q_S); S 〉 ∈ DFS (U) and take an element x of S. If there does not exist any elements in S such that x is their product then clearly, $((((f_{S} \tilde{\circ} h_{S}) \tilde{\circ} p_{S}) (x)) \cap N) \cup M = ((((p_{S} \tilde{\circ} h_{S}) \tilde{\circ} f_{S}) (x)) \cap N) \cup M$ and $((((g_{S} \tilde{\circ} k_{S}) \tilde{\circ} q_{S}) (x)) \cup M) \cap N = ((((q_{S} \tilde{\circ} k_{S}) \tilde{\circ} g_{S}) (x))$ ∪ M ) ∩ N. Let x can be expressed as the product of two elements y, z of S. Then, we have $\begin{matrix} (((f_{S} \tilde{\circ} h_{S}) \tilde{\circ} p_{S}) (x) \cap N) \cup M \\ = ((⋃_{x = st} {(f_{S} \tilde{\circ} h_{S}) (s) \cap p_{S} (t)}) \cap N) \cup M \\ = ((⋃_{x = st} {⋃_{s = mn} {f_{S} (m) \cap h_{S} (n)} \cap p_{S} (t)}) \cap N) \cup M \\ = ((⋃_{x = (mn) t} {f_{S} (m) \cap h_{S} (n) \cap p_{S} (t)}) \cap N) \cup M \\ = ((⋃_{x = (tn) m} {p_{S} (t) \cap h_{S} (n) \cap f_{S} (m)}) \cap N) \cup M \\ \subseteq ((⋃_{x = wm} (⋃_{w = tn} p_{S} (t) \cap h_{S} (n)} \cap f_{S} (m))) \cap N) \cup M \\ = ((⋃_{x = wm} {(p_{S} \tilde{\circ} h_{S}) (w) \cap f_{S} (m)}) \cap N) \cup M \\ = ((((p_{S} \tilde{\circ} h_{S}) \tilde{\circ} f_{S}) (x)) \cap N) \cup M \end{matrix}$ Similarly we can prove that $((((p_{S} \tilde{\circ} h_{S}) \tilde{\circ} f_{S}) (x))$ $\cap N) \cup M \subseteq (((f_{S} \tilde{\circ} h_{S}) \tilde{\circ} p_{S}) (x) \cap N) \cup M .$ Now $\begin{matrix} (((g_{S} \tilde{\circ} k_{S}) \tilde{\circ} q_{S}) (x) \cup M) \cap N \\ = ((⋂_{x = st} {(g_{S} \tilde{\circ} k_{S}) (s) \cup q_{S} (t)}) \cup M) \cap N \\ = ((⋂_{x = st} {⋂_{s = mn} {g_{S} (m) \cup k_{S} (n)} \cup q_{S} (t)}) \cup M) \cap N \\ = ((⋂_{x = (mn) t} {g_{S} (m) \cup k_{S} (n) \cup q_{S} (t)}) \cup M) \cap N \\ = ((⋂_{x = (tn) m} {q_{S} (t) \cup k_{S} (n) \cup g_{S} (m)}) \cup M) \cap N \\ \supseteq ((⋂_{x = wm} {⋂_{w = tm} {q_{S} (t) \cup k_{S} (n)} \cup g_{S} (m)}) \cup M) \cap N \\ = ((⋂_{x = wm} {(q_{S} \tilde{\circ} k_{S}) (w) \cup g_{S} (m)}) \cup M) \cap N \\ = (((q_{S} \tilde{\circ} k_{S}) \tilde{\circ} g_{S}) (x) \cup M) \cap N \end{matrix}$ Similarly we can prove that $((((q_{S} \tilde{\circ} k_{S}) \tilde{\circ} g_{S}) (x))$ $\cup M) \cap N \supseteq ((((g_{S} \tilde{\circ} k_{S}) \tilde{\circ} q_{S}) (x)) \cup M) \cap N$ . Hence (DFS (U), ◊) is an AG-groupoid. □

The proof of the following lemmas are based on Lemma 3.6.

Lemma 3.7. If S is an AG-Groupoid then the medial law holds in DFS (U). i.e. for 〈 (α_S, β_S); S 〉, 〈 (f_S, g_S); S 〉, 〈 (h_S, k_S); S〉 and 〈 (p_S, q_S); S 〉 ∈ DFS (U), we have ( 〈 (α_S, β_S); S 〉 ◊ 〈 ( f_S, g_S); S 〉) ◊ (〈 (h_S, k_S); S 〉 ◊ 〈 (p_S, q_S); S 〉 ) = _[M,N] (〈 (α_S, β_S); S 〉 ◊ 〈 (h_S, k_S); S 〉) ◊ (〈 (f_S, g_S); S 〉 ◊ 〈 (p_S, q_S); S 〉).

Lemma 3.8. If S is an AG-groupoid with left identity then the paramedial law holds in DFS (U). That is for all 〈 (α_S, β_S); S 〉, 〈 (f_S, g_S); S 〉, 〈 (h_S, k_S); S〉 and 〈 (p_S, q_S); S 〉 ∈ DFS (U), (〈 (α_S, β_S); S 〉 ◊ 〈 (f_S, g_S); S 〉) ◊ (〈 (h_S, k_S); S 〉 ◊ 〈 (p_S, q_S); S 〉) = _[M,N] (〈 (p_S, q_S); S 〉 ◊ 〈 (f_S, g_S); S 〉) ◊ (〈 (h_S, k_S); S 〉 ◊ 〈 (α_S, β_S); S 〉).

Lemma 3.9. If S is an AG-groupoid with left identity then for all 〈 (f_S, g_S); S 〉, 〈 (h_S, k_S); S〉 and 〈 (p_S, q_S); S 〉 ∈ DFS (U), we have〈 (f_S, g_S); S 〉 ◊ (〈 (h_S, k_S); S 〉 ◊ 〈 (p_S, q_S); S 〉) = _[M,N] 〈 (h_S, k_S); S 〉 ◊ (〈 (f_S, g_S); S 〉 ◊ 〈 (p_S, q_S); S 〉).

Lemma 3.10. If S is an AG-groupoid and 〈 (α_S, β_S); S〉, 〈 (f_S, g_S); S〉,〈 (h_S, k_S); S〉 and 〈 (p_S, q_S); S 〉 ∈ DFS (U), then the following hold. (i) 〈 (α_S, β_S); S 〉 ◊ (〈 (f_S, g_S); S 〉 ⊓ 〈 (h_S, k_S); S 〉) = _[M,N] 〈 (α_S, β_S); S 〉 ◊ 〈 (f_S, g_S); S 〉 ⊓ 〈 (α_S, β_S); S 〉 ◊ 〈 (h_S, k_S); S 〉). (ii) 〈 (α_S, β_S); S 〉 ◊ (〈 (f_S, g_S); S 〉 ⊔ 〈 (h_S, k_S); S 〉) = _[M,N] 〈 (α_S, β_S); S 〉 ◊ 〈 (f_S, g_S); S 〉 ⊔ 〈 (α_S, β_S); S 〉 ◊ 〈 (h_S, k_S); S 〉). (iii) If 〈 (α_S, β_S); S〉 ⊑ _[M,N] 〈 (f_S, g_S); S 〉 then 〈 (α_S, β_S); S 〉 ◊ 〈 (h_S, k_S); S 〉 ⊑ _[M,N] 〈 (f_S, g_S); S 〉 ◊ 〈 (h_S, k_S); S 〉. (iv) If 〈 (α_S, β_S); S〉 ⊑ _[M,N] 〈 (f_S, g_S); S 〉 and 〈 (h_S, k_S); S〉 ⊑ _[M,N] 〈 (p_S, q_S); S 〉 then 〈 (α_S, β_S); S 〉 ◊ 〈 (h_S, k_S); S 〉 ⊑ _[M,N] 〈 (f_S, g_S); S 〉 ◊ 〈 (p_S, q_S); S 〉.

Definition 3.11. [12] Let S be an AG-groupoid and 〈 (f_S, g_S); S〉 be a DFS-set of S over U. Then 〈 (f_S, g_S); S〉 is called double-framed soft AG-groupoid (briefly, DFS AG-groupoid) of S over U if f_S (xy) ⊇ f_S (x) ∩ f_S (y) and g_S (xy) ⊆ g_S (x) ∪ g_S (y) for all x, y ∈ S.

4. (M, N)-double framed soft ideals

In this section, we give concepts of (M, N)-DFS AG-groupoid, (M, N)-DFS ideal of an AG-groupoid and discuss their properties. From now on ∅⊆ M ⊂ N ⊆ U.

Definition 4.1. Let S be an AG-groupoid and 〈 (f_A, g_A); A〉 be a DFS-set of S over U. Then 〈 (f_A, g_A); A〉 is called (M, N)-double-framed soft AG-groupoid (briefly, (M, N)-DFS AG-groupoid) of S over U if f_A (xy) ∪ M ⊇ f_A (x) ∩ f_A (y) ∩ N and f_A (xy) ∪ M ⊇ f_A (x) ∩ f_A (y) ∩ N and g_A (xy) ∩ N ⊆ g_A (x) ∪ g_A (y) ∪ M for all x, y ∈ A.

Example 4.2. Consider an AG-groupoid S = {a, b, c, d} with the following multiplication table: $\begin{array}{l} . & a & b & c & d \\ a & c & c & c & d \\ b & d & d & c & c \\ c & d & d & d & d \\ d & d & d & d & d \end{array}$ Let $U = ℤ$ , $M = 32 ℤ$ , $N = 16 ℤ$ and a double-framed soft 〈 (f_S, g_S); S〉 of S over $U = ℤ$ defined by $f_{S} (a) = 16 ℤ,$ $f_{S} (b) = 4 ℤ,$ $f_{S} (c) = 2 ℤ,$ $f_{S} (d) = 8 ℤ$ and $g_{S} (a) = 16 ℤ,$ $g_{S} (b) = 64 ℤ,$ $g_{S} (c) = 128 ℤ,$ $g_{S} (d) = 32 ℤ .$

By routine checking it is easy to verify that 〈 (f_S, g_S); S〉 is an (M, N)-DFS AG-groupoid of S over U.

Remark 4.3. If 〈 (f_S, g_S); S〉 is a DFS AG-groupoid of S over U, then 〈 (f_S, g_S); S〉 is an (∅, U)-DFS AG-groupoid of S over U. Therefore,every DFS AG-groupoid of S is an (M, N)-DFS AG-groupoid, but the converse is not true.

Example 4.4. Reconsider Example 4.2, 〈 (f_S, g_S); S〉 is (M, N)-DFS AG-groupoid which is not DFS AG-groupoid because f_S (d) nsupseteqf_S (b) ∩ f_S (b) and g_S (d) ⊈ g_S (c) ∪ g_S (d).

For any DFS-set 〈 (f_S, g_S); S〉 of S, let $〈 (f_{S}^{*}, g_{S}^{*}); S 〉$ be a DFS-set of S defined by $\begin{matrix} f_{S}^{*} & : S \to P (U), x \to {\begin{matrix} [c] c f_{S} (x) if x \in i_{E} (f_{S}; γ) \\ η otherwise \end{matrix} \\ g_{S}^{*} & : S \to P (U), x \to {\begin{matrix} [c] c g_{S} (x) if x \in e_{E} (g_{S}; δ) \\ ρ otherwise \end{matrix} \end{matrix}$

where γ, δ, η and ρ are subsets of U with η ⊊ f_S (x) and ρsupsetneqg_S (x).

Theorem 4.5. If 〈 (f_A, g_A); A〉 is an (M, N)-DFS AG-groupoid of A over U then so is $〈 (f_{A}^{*}, g_{A}^{*}); A 〉$ .

Proof. Suppose that 〈 (f_A, g_A); A〉 is an (M, N)-DFS AG-groupoid of A over U then non-empty γ-inclusive set and δ-exclusive set of 〈 (f_A, g_S); A〉 are sub AG-groupoids of S for any subsets γ and δ of U with M ⊆ γ ⊆ N, M ⊆ δ ⊆ N. Let x, y ∈ A. Case (i). If x, y ∈ i_A (f_A; γ) then xy ∈ i_A (f_A; γ) and hence $f_{A}^{*} (xy) \cup M = f_{A} (xy) \cup M \supseteq f_{A} (xy) \supseteq f_{A} (x) \cap f_{A} (y) \cap N = f_{A}^{*} (x) \cap f_{A}^{*} (y) \cap N .$ Case (ii). If x ∉ i_A (f_A; γ) and y ∈ i_A (f_A; γ) then $f_{A}^{*} (x) = η$ and $f_{A}^{*} (y) = f_{A} (y)$ . Here two subcases arise.

Case (ii-a) If xy ∈ i_A (f_A; γ) then $f_{A}^{*} (xy) \cup M = f_{A} (xy) \cup M \supseteq f_{A} (xy) \supseteq f_{A} (x) \cap f_{A} (y) \cap N \supseteq η \cap f_{A} (y) \cap N = f_{A}^{*} (x) \cap f_{A}^{*} (y) \cap N .$

Case (ii-b) If xy ∉ i_A (f_A; γ) then $f_{A}^{*} (xy) \cup M = η \cup M \supseteq η \supseteq η \cap N = f_{A}^{*} (x) \cap f_{A}^{*} (y) \cap N$ .

Case (iii) If x ∉ i_A (f_A; γ) and y ∉ i_A (f_A; γ) then two subcases arise.

Case (iii-a) If xy ∈ i_A (f_A; γ) then $f_{A}^{*} (xy) \cup M = f_{A} (xy) \cup M \supseteq f_{A} (xy) \supseteq f_{A} (x) \cap f_{A} (y) \cap N \supset η \cap η \cap N = f_{A}^{*} (x) \cap f_{A}^{*} (y) \cap N .$

Case (iii-b) If xy ∉ i_A (f_A; γ) then $f_{A}^{*} (xy) \cup M = η \cup M \supseteq η \supseteq η \cap N = f_{A}^{*} (x) \cap f_{A}^{*} (y) \cap N .$

Now let x, y ∈ A. We discuss similar cases for the e_A (g_A; δ).

Case (i) If x, y ∈ e_A (g_A; δ) then xy ∈ e_A (g_A; δ) and hence $g_{A}^{*} (xy) \cap N \subseteq g_{A}^{*} (xy) \subseteq g_{A} (xy) \subseteq g_{A} (x) \cup g_{A} (y) \cup M = g_{A}^{*} (x) \cup g_{A}^{*} (y) \cup M .$ Case (ii) If x ∉ e_A (g_A; δ) and y ∈ e_A (g_A; δ) then two subcases arise.

Case (ii-a) If xy ∈ e_A (g_A; δ) then $g_{A}^{*} (xy) \cap N = g_{A} (xy) \cap N \subseteq g_{A} (xy) \subseteq g_{A} (x) \cup g_{A} (y) \cup M \subset ρ \cup g_{A}^{*} (y) \cup M = g_{A}^{*} (x) \cup g_{A}^{*} (y) \cup M$ .

Case (ii-b) If xy ∉ e_A (g_A; δ) then $g_{A}^{*} (xy) \cap N = ρ \cap N \subseteq ρ \subseteq ρ \cup M \subseteq g_{A}^{*} (x) \cup g_{A}^{*} (y) \cup M .$

Case (iii) If x ∉ e_A (g_A; δ) and y ∉ e_A (g_A; δ) then two subcases arise.

Case (iii-a) If xy ∈ e_A (g_A; δ) then $g_{A}^{*} (xy) \cap N = g_{A} (xy) \cap N \subseteq g_{A} (xy) \subseteq g_{A} (x) \cup g_{A} (y) \cup M \subset ρ \cup ρ \cup M = g_{A}^{*} (x) \cup g_{A}^{*} (y) \cup M$ .

Case (iii-b) If xy ∉ e_A (g_A; δ) then $g_{A}^{*} (xy) \cap N = ρ \cap N \subseteq ρ \subseteq ρ \cup M \subseteq g_{A}^{*} (x) \cup g_{A}^{*} (y) \cup M .$ Therefore $〈 (f_{A}^{*}, g_{A}^{*}); A 〉$ is an (M, N)-DFS AG-groupoid of A.

The converse of this theorem is not true in general.

Example 4.6. Suppose the initial universe U given by $U = ℤ_{10}$ , the non-negative integers modulo 10. Let S = {a, b, c, d} be the set with the following multiplication table $\begin{array}{l} . & a & b & c & d \\ a & c & c & c & d \\ b & d & d & c & c \\ c & d & d & d & d \\ d & d & d & d & d \end{array}$ Let M = {1, 2}, N = {1, 2, 3, 5} and $U = ℤ_{10}$ be the non-negative integers modulo 10. Define a DFS set 〈 (f_S, g_S); S〉 by f_S (a) = {1, 3, 5}, f_S (b) = {2, 3, 5}, f_S (c) = {1, 2, 3, 4}, f_S (d) = {1, 2, 3} and g_S (a) = {1, 2}, g_S (b) = {2, 3}, g_S (c) = {2}, g_S (d) = {1, 3} then i_S (f_S; γ) = {c, d} for γ = {1, 2} and e_S (g_S; δ) = {c} for δ = {2}.

According to the definition, we have $〈 (f_{S}^{*}, g_{S}^{*}); S 〉$ is defined as; $f_{S}^{*} (a) = {3},$ $f_{S}^{*} (b) = {3},$ $f_{S}^{*} (c) = {1, 2, 3, 4},$ $f_{S}^{*} (d) = {1, 2, 3}$ and $g_{S}^{*} (a) = {1, 2, 3},$ $g_{S}^{*} (b) = {1, 2, 3},$ $g_{S}^{*} (c) = {2},$ $g_{S}^{*} (d) = {1, 2, 3}$ . By routine checking, we have $〈 (f_{S}^{*}, g_{S}^{*}); S 〉$ is (M, N)-DFS AG-groupoid. But 〈 (f_S, g_S); S〉 is not (M, N)-DFS AG-groupoid because f_S (c) ∪ M = {1, 2, 3, 4} nsupseteq {3, 5} = f_S (a) ∩ f_S (b) ∩ N or g_S (d) ∩ N = {1, 2, 3} ⊈ {1, 2} = g_S (a) ∪ g_S (a) ∪ M.

Theorem 4.7. If 〈 (f_S, g_S); S〉 and 〈 (h_S, k_S); S〉 are (M, N)-DFS AG groupoids then their intersection 〈 (f_S, g_S); S〉 ⊓ 〈 (h_S, k_S); S 〉 is (M, N)-DFS AG groupoid.

Proof. Let x, y ∈ S. Since 〈 (f_S, g_S); S〉 and 〈 (h_S, k_S); S〉 are (M, N)-DFS AG groupoids then $\begin{matrix} (f_{S} \tilde{\cap} h_{S}) (xy) \cup M \\ = (f_{S} (xy) \cap h_{S} (xy)) \cup M \\ = (f_{S} (xy) \cup M) \cap (h_{S} (xy) \cup M) \\ \supseteq (f_{S} (x) \cap f_{S} (y) \cap N) \cap (h_{S} (x) \cap h_{S} (y) \cap N) \\ = (f_{S} (x) \cap h_{S} (x) \cap f_{S} (y) \cap h_{S} (y)) \cap N \\ = (f_{S} \tilde{\cap} h_{S}) (x) \cap (f_{S} \tilde{\cap} h_{S}) (y) \cap N \end{matrix}$

and $\begin{matrix} (g_{S} \tilde{\cup} k_{S}) (xy) \cap N \\ = (g_{S} (xy) \cup k_{S} (xy)) \cap N \\ = (g_{S} (xy) \cap N) \cup (k_{S} (xy) \cap N) \\ \subseteq (g_{S} (x) \cup g_{S} (y) \cup M) \cup (k_{S} (x) \cup k_{S} (y) \cup M) \\ = (g_{S} (x) \cup k_{S} (x) \cup g_{S} (y) \cup k_{S} (y)) \cup M \\ = (g_{S} \tilde{\cup} k_{S}) (x) \cup (g_{S} \tilde{\cup} k_{S}) (y) \cup M \end{matrix}$

Hence 〈 (f_S, g_S); S〉 ⊓ 〈 (h_S, k_S); S 〉 is (M, N)-DFS AG groupoid.

It is important to note that the union of two (M, N)-DFS AG groupoids may or may not be an (M, N)-DFS AG groupoid.

Example 4.8. Suppose the initial universe U given by $U = ℤ_{10}$ , the non-negative integers modulo 10. Let S = {a, b, c, d} be the set with the following multiplication table $\begin{array}{l} . & a & b & c & d \\ a & c & c & c & d \\ b & d & d & c & c \\ c & d & d & d & d \\ d & d & d & d & d \end{array}$ Let M = {1, 4}, N = {1, 2, 3, 4, 5} and $U = ℤ_{10}$ be the non-negative integers modulo 10. Define a DFS set 〈 (f_S, g_S); S〉 by f_S (a) = {1, 4}, f_S (b) = {4, 5}, f_S (c) = {1, 4, 7}, f_S (d) = {1, 3, 5} and g_S (a) = {1, 2, 3, 5, 7}, g_S (b) = {1, 3, 4, 5}, g_S (c) = {1, 2, 3, 5}, g_S (d) = {1, 3, 5, 7}.

Define another DFS set 〈 (h_S, k_S); S 〉 by h_S (a) = {1, 4}, h_S (b) = {4, 7}, h_S (c) = {1, 7, 9}, h_S (4) = {3, 5, 7} and k_S (a) = {2, 3, 4, 5}, k_S (b) = {1, 3, 4, 5}, k_S (c) = {1, 2, 3, 4}, k_S (d) = {3, 5, 7}

One can easily check that 〈 (f_S, g_S); S〉 and 〈 (h_S, k_S); S〉 are (M, N)-DFS AG groupoids of S over U. But their union 〈 (p_S, q_S); S〉 given by p_S (a) = {1, 4}, p_S (b) = {4, 5, 7}, p_S (c) = {1, 4, 7, 9}, p_S (d) ={ 1, 3, 5, 7 } and q_S (a) = {2, 3, 5}, q_S (b) = {1, 3, 4, 5}, q_S (c) = {1, 2, 3}, q_S (d) = {3, 5, 7} is not (M, N)-DFS AG groupoid since p_S (c) ∪ M = {1, 4, 7, 9} nsupseteq {5} = p_S (b) ∩ p_S (d) ∩ N and/or q_S (c) ∩ N = {1, 2, 3} ⊈ {1, 3, 4, 5, 7} = q_S (b) ∪ q_S (d) ∪ M.

Theorem 4.9. A DFS-set 〈 (f_S, g_S); S〉 of an AG groupoid S over U is an (M, N)-DFS AG groupoid if and only if

$〈 (f_{S}, g_{S}); S 〉 ◊ 〈 (f_{S}, g_{S}); S 〉 ⊑_{[M, N]} 〈 (f_{S}, g_{S}); S 〉 .$

Proof. Assume that the DFS-set 〈 (f_S, g_S); S〉 of an AG groupoid S over U is (M, N)-DFS AG groupoid and take x ∈ S. If there are no y, z ∈ S such that x = yz then $\begin{matrix} ((f_{S} \tilde{\circ} f_{S}) (x) \cap N) \cup M & = M \subseteq (f_{S} (x) \cap N) \cup M \\ ((g_{S} \tilde{\circ} g_{S}) (x) \cup M) \cap N & = N \supseteq (g_{S} (x) \cup M) \cap N \end{matrix}$

Suppose there exist y, z ∈ S such that x = yz then $\begin{matrix} ((f_{S} \tilde{\circ} f_{S}) (x) \cap N) \cup M \\ = ((⋃_{x = yz} {f_{S} (y) \cap f_{S} (z)}) \cap N) \cup M \\ = ((⋃_{x = yz} {f_{S} (y) \cap f_{S} (z) \cap N}) \cap N) \cup M \\ \subseteq ((⋃_{x = yz} {f_{S} (yz) \cup M}) \cap N) \cup M \\ = (⋃_{x = yz} (f_{S} (yz) \cup M)) \cap (N \cup M) \\ = (f_{S} (x) \cup M) \cap (N \cup M) \\ = (f_{S} (x) \cap N) \cup M \end{matrix}$

and $\begin{matrix} ((g_{S} \tilde{\circ} g_{S}) (x) \cup M) \cap N \\ = ((⋂_{x = yz} {g_{S} (x) \cup g_{S} (y)}) \cup M) \cap N \\ = ((⋂_{x = yz} {g_{S} (x) \cup g_{S} (y) \cup M}) \cup M) \cap N \\ \supseteq ((⋂_{x = yz} {g_{S} (yz) \cap N}) \cup M) \cap N \\ = (⋂_{x = yz} {g_{S} (yz) \cap N}) \cup (M \cap N) \\ = (g_{S} (x) \cap N) \cup (M \cap N) \\ = (g_{S} (x) \cup M) \cap N \end{matrix}$

Conversely, let x, y ∈ S then by hypothesis, f_S (xy) $\cup M \supseteq (f_{S} (xy) \cup M) \cap N = (f_{S} (xy) \cap N) \cup M \supseteq ((f_{S} \tilde{\circ} f_{S}) (xy) \cap N) \cup M \supseteq (f_{S} (x) \cap f_{S} (y) \cap N) \cup M \supseteq f_{S} (x) \cap f_{S} (y) \cap N$ .

Also g_S (xy) ∩ N ⊆ (g_S (xy) ∩ N) ∪ M ⊆ (g_S (xy) $\cup M) \cap N \subseteq ((g_{S} \tilde{\circ} g_{S}) (xy) \cup M) \cap N \subseteq (g_{S} (x) \cup g_{S} (y) \cup M) \cap N \subseteq g_{S} (x) \cup g_{S} (y) \cup M .$ Hence 〈 (f_S, g_S); S〉 is (M, N)-DFS AG groupoid.

Definition 4.10. Let S be an AG-groupoid and 〈 (f_A, g_A); A〉 be a DFS-set of S over U. Then 〈 (f_A, g_A); A〉 is called (M, N)-double-framed soft left (resp. right) ideal (briefly, (M, N)-DFS left (resp. right)) ideal over U if f_A (ab) ∪ M ⊇ f_A (b) ∩ N (resp. f_A (ab) ∪ M ⊇ f_A (a) ∩ N) and g_A (ab) ∩ N ⊆ g_A (b) ∪ M (resp. g_A (ab) ∩ N ⊆ g_A (a) ∪ M) for all a, b ∈ A. A DFS-set 〈 (f_A, g_A); A〉 of S over U is called a (M, N)-double-framed soft two sided ideal (briefly, (M, N)-DFS two-sided ideal) of S over U if it is both a (M, N)-DFS left and a (M, N)-DFS right ideal of S over U.

Example 4.11. Consider the set S = {1, 2, 3, 4} with the following multiplication table $\begin{array}{l} . & 1 & 2 & 3 & 4 \\ 1 & 4 & 4 & 4 & 4 \\ 2 & 3 & 1 & 3 & 1 \\ 3 & 4 & 1 & 4 & 4 \\ 4 & 4 & 4 & 4 & 4 \end{array}$ Consider the DFS-set 〈 (f_S, g_S); S〉 of S over $U = ℤ$ with $M = 64 ℤ,$ $N = 128 ℤ$ defined by $f_{S} (1) = 16 ℤ,$ $f_{S} (2) = 8 ℤ,$ $f_{S} (3) = 4 ℤ,$ $f_{S} (4) = 2 ℤ$ and $g_{S} (1) = 128 ℤ,$ $g_{S} (2) = 32 ℤ,$ $g_{S} (3) = 64 ℤ,$ $g_{S} (4) = 256 ℤ$ is (M, N)-DFS right ideal of S.

Theorem 4.12. Every (M, N)-DFS left (resp. right) ideal is (M, N)-DFS AG groupoid.

Proof. Straightforward.

Theorem 4.13. Let A be a nonempty subset of an AG-groupoid S. Then 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 an (M, N)-DFS left (resp. right) ideal of S over U.

Proof. Straightforward.

Theorem 4.14. LA DFS set 〈 (f_S, g_S); S〉 of an AG-groupoid S over U is (M, N)-DFS left ideal of S if and only if X_S◊ 〈 (f_S, g_S); S 〉 ⊑ _[M,N] 〈 (f_S, g_S); S 〉.

Proof. Let 〈 (f_S, g_S); S〉 be an (M, N)-DFS left ideal of S. Let a ∈ S. If a is not product of any x, y ∈ S then $((χ_{S} \tilde{\circ} f_{S}) (a) \cap N) \cup M \subseteq (f_{S} (a) \cap N) \cup M$ and $((χ_{S}^{c} \tilde{\circ} g_{S}) (a) \cup M) \cap N \supseteq (g_{S} (a) \cup M) \cap N$

If there exist x, y ∈ S such that a = xy then $\begin{matrix} ((χ_{S} \tilde{\circ} f_{S}) (a) \cap N) \cup M \\ = ((⋃_{a = xy} {χ_{S} (x) \cap f_{S} (y)}) \cap N) \cup M \\ = ((⋃_{a = xy} {U \cap f_{S} (y) \cap N}) \cap N) \cup M \\ = ((⋃_{a = xy} {f_{S} (y) \cap N}) \cap N) \cup M \\ \subseteq ((⋃_{a = xy} {f_{S} (xy) \cup M}) \cap N) \cup M \\ = ((f_{S} (a) \cup M) \cap N) \cup M \\ = ((f_{S} (a) \cup M) \cap (M \cup N)) \\ = (f_{S} (a) \cup M) \cap N \\ = (f_{S} (a) \cap N) \cup (M \cap N) \\ = (f_{S} (a) \cap N) \cup M \end{matrix}$

and $\begin{matrix} ((χ_{S}^{c} \tilde{\circ} g_{S}) (a) \cup M) \cap N \\ = ((⋂_{a = xy} {χ_{S}^{c} (x) \cup g_{S} (y)}) \cup M) \cap N \\ = ((⋂_{a = xy} {\emptyset \cup g_{S} (y) \cup M}) \cup M) \cap N \\ = ((⋂_{a = xy} {g_{S} (y) \cup M}) \cup M) \cap N \\ \supseteq ((⋂_{a = xy} {g_{S} (xy) \cap N}) \cup M) \cap N \\ = ((g_{S} (a) \cap N) \cup M) \cap N \\ = (g_{S} (a) \cap N) \cap (M \cap N) \\ = (g_{S} (a) \cap N) \cap M \end{matrix}$

Hence X_S ◊ 〈 (f_S, g_S); S 〉 ⊑ _[M,N] 〈 (f_S, g_S); S 〉.

Conversely, let x, y ∈ S and a = xy. Then $\begin{matrix} f_{S} (xy) \cup M \\ \supseteq (f_{S} (xy) \cup M) \cap N \\ = (f_{S} (a) \cap N) \cup M \\ \supseteq ((χ_{S} \tilde{\circ} f_{S}) (a) \cap N) \cup M \\ = ((⋃_{a = xy} {χ_{S} (x) \cap f_{S} (y)}) \cap N) \cup M \\ \supseteq ((χ_{S} (x) \cap f_{S} (y)) \cap N) \cup M \\ = (f_{S} (y) \cap N) \cup M \supseteq f_{S} (y) \cap N \end{matrix}$ and $\begin{matrix} g_{S} (xy) \cap N \\ \subseteq (g_{S} (xy) \cap N) \cup M \\ = (g_{S} (a) \cap N) \cup M \\ \subseteq ((χ_{S}^{c} \tilde{\circ} g_{S}) (a) \cap N) \cup M \\ = ((⋂_{a = xy} {χ_{S}^{c} (x) \cup g_{S} (y)}) \cap N) \cup M \\ \subseteq ((χ_{S}^{c} (x) \cup g_{S} (y)) \cap N) \cup M \\ = (g_{S} (y) \cap N) \cup M = (g_{S} (y) \cup M) \cap (N \cup M) \\ = (g_{S} (y) \cup M) \cap N \subseteq g_{S} (y) \cup M \end{matrix}$

Hence 〈 (f_S, g_S); S〉 is (M, N)-DFS left ideal of S.

Theorem 4.15. RA DFS set 〈 (f_S, g_S); S〉 of an AG-groupoid S over U is (M, N)-DFS right ideal of S if and only if 〈 (f_S, g_S); S〉 ◊ X_S ⊑ _[M,N] 〈 (f_S, g_S); S 〉.

Proof. Same as Theorem 4.14.

If an AG groupoid contains left identity then every (M, N)-DFS right ideal of it becomes (M, N)-DFS left ideal. We prove it.

Theorem 4.16. If S is an AG-groupoid with left identity e then every (M, N)-DFS right ideal of S is (M, N)-DFS left ideal.

Proof. Let S be an AG-groupoid with left identity e and 〈 (f_S, g_S); S〉 is (M, N)-DFS right ideal of S. Then $\begin{matrix} f_{S} (xy) \cup M & = f_{S} ((ex) y) \cup M \\ = f_{S} ((yx) e) \cup M \\ = (f_{S} ((yx) e) \cup M) \cup M \\ \supseteq (f_{S} (yx) \cap N) \cup M \\ = (f_{S} (yx) \cup M) \cap N \\ \supseteq (f_{S} (y) \cap N) \cap N \\ = f_{S} (y) \cap N \end{matrix}$ and $\begin{matrix} g_{S} (xy) \cap N & = g_{S} ((ex) y) \cap N \\ = g_{S} ((yx) e) \cap N \\ = (g_{S} ((yx) e) \cap N) \cap N \\ \subseteq (g_{S} (yx) \cup M) \cap N \\ = (g_{S} (yx) \cap N) \cup M \\ \subseteq (g_{S} (y) \cup M) \cup M \\ = g_{S} (y) \cup M \end{matrix}$

Hence 〈 (f_S, g_S); S〉 is (M, N)-DFS left ideal.

Similar to Theorem 4.7, one can easily prove the following result.

Proposition 4.17. If 〈 (f_S, g_S); S〉 and 〈 (h_S, k_S); S〉 are (M, N)-DFS left (resp. right) ideals then their intersection 〈 (f_S, g_S); S〉 ⊓ 〈 (h_S, k_S); S 〉 is (M, N)-DFS left (resp. right) ideals.

Lemma 4.18. If S is an AG-groupoid with left identity e, then X_S ◊ X_S = _[M,N]X_S.

Proof. Let x be any element of S then x = ex, where e is the left identity of S, then we have $\begin{matrix} ((χ_{S} \tilde{\circ} χ_{S}) (x) \cap N) \cup M \\ = ((⋃_{x = pq} {χ_{S} (p) \cap χ_{S} (q)}) \cap N) \cup M \\ \supseteq ({χ_{S} (e) \cap χ_{S} (x)} \cap N) \cup M \\ = (χ_{S} (x) \cap N) \cup M \end{matrix}$

and $\begin{matrix} ((χ_{S}^{c} \tilde{\circ} χ_{S}^{c}) (x) \cup M) \cap N \\ = ((⋂_{x = pq} {χ_{S}^{c} (p) \cup χ_{S}^{c} (q)}) \cup M) \cap N \\ \subseteq ({χ_{S}^{c} (e) \cup χ_{S}^{c} (x)} \cup M) \cap N \\ = (χ_{S}^{c} (x) \cup M) \cap N \end{matrix}$

Hence, $X_{S} ⊑_{[M, N]} X_{S} \tilde{\circ} X_{S} .$ The other inclusion $X_{S} \tilde{\circ} X_{S} . ⊑_{[M, N]} X_{S}$ is obvious. Therefore $X_{S} \tilde{\circ} X_{S} =_{[M, N]} X_{S} .$

Proposition 4.19. If S is an AG-groupoid with left identity e, then for (M, N)-DFS right ideal 〈 (f_S, g_S); S〉 we have 〈 (f_S, g_S); S 〉 ◊ X_S = _[M,N] 〈 (f_S, g_S); S 〉.

Proof. Let a is any element of S then a = ea = (ee) a = (ae) e, where e is the left identity of S, we have $\begin{matrix} ((f_{S} \tilde{\circ} χ_{S}) (a) \cap N) \cup M \\ = ((⋃_{a = xy} {f_{S} (x) \cap χ_{S} (y)}) \cap N) \cup M \\ \supseteq ((f_{S} (ae) \cap χ_{S} (e)) \cap N) \cup M \\ = (f_{S} (ae) \cap N) \cup M \\ = (f_{S} (ae) \cup M) \cap N \\ = ((f_{S} (ae) \cup M) \cup M) \cap N \\ \supseteq ((f_{S} (a) \cap N) \cup M) \cap N \\ = (f_{S} (a) \cap N) \cup M \end{matrix}$

and $\begin{matrix} ((g_{S} \tilde{\circ} χ_{S}^{c}) (a) \cup M) \cap N \\ = ((⋂_{a = xy} {g_{S} (x) \cup χ_{S}^{c} (y)}) \cup M) \cap N \\ \subseteq ((g_{S} (ae) \cup χ_{S}^{c} (e)) \cup M) \cap N \\ = (g_{S} (ae) \cup M) \cap N \\ = (g_{S} (ae) \cap N) \cup M \\ = ((g_{S} (ae) \cap N) \cap N) \cup M \\ \subseteq ((g_{S} (a) \cup M) \cap N) \cup M \\ = (g_{S} (a) \cup M) \cap N \end{matrix}$

Therefore 〈 (f_S, g_S); S〉 ⊑ _[M,N] 〈 (f_S, g_S); S 〉 ◊ X_S. By Theorem 4.15, we have the required result.

It is proved earlier in Lemma 3.6. that the set of all double framed soft sets of S over U denoted by DFS (U) is (M, N)-DFS AG groupoid. An element 〈 (f_S, g_S); S〉 of DFS (U) is said to be idempotent if 〈 (f_S, g_S); S 〉 ◊ 〈 (f_S, g_S); S 〉 = _[M,N] 〈 (f_S, g_S); S 〉.

Proposition 4.20. An idempotent (M, N)-DFS left ideal of an AG groupoid S is (M, N)-DFS right ideal.

Proof. Let 〈 (f_S, g_S); S〉 be a (M, N)-DFS left ideal of S over U which is idempotent. Since 〈 (f_S, g_S); S〉 is idempotent then we can write $(f_{S} \cap N) \cup M \tilde{\subseteq} ((f_{S} \tilde{\circ} f_{S}) \cap N) \cup M$ and $(g_{S} \cup M) \cap N \tilde{\supseteq} ((g_{S} \tilde{\circ} g_{S}) \cup M) \cap N$ then by Lemma 3.9 and Theorem 4.14, $\begin{matrix} ((f_{S} \tilde{\circ} χ_{S}) \cap N) \cup M \\ \tilde{\subseteq} ((f_{S} \tilde{\circ} f_{S}) \tilde{\circ} χ_{S} \cap N) \cup M \\ = ((χ_{S} \tilde{\circ} f_{S}) \tilde{\circ} f_{S} \cap N) \cup M \\ \tilde{\subseteq} ((f_{S} \tilde{\circ} f_{S}) \cap N) \cup M \\ \tilde{\subseteq} (f_{S} \cap N) \cup M \end{matrix}$

and $\begin{matrix} ((g_{S} \tilde{\circ} χ_{S}^{c}) \cup M) \cap N \\ \tilde{\supseteq} ((g_{S} \tilde{\circ} g_{S}) \tilde{\circ} χ_{S}^{c} \cup M) \cap N \\ = ((χ_{S}^{c} \tilde{\circ} g_{S}) \tilde{\circ} g_{S} \cup M) \cap N \\ \tilde{\supseteq} ((g_{S} \tilde{\circ} g_{S}) \cup M) \cap N \\ \tilde{\supseteq} (g_{S} \cup M) \cap N \end{matrix}$

Therefore, 〈 (f_S, g_S); S〉 is (M, N)-DFS right ideal of S over U and hence an (M , N)-DFS ideal.

5. Regular AG-groupoids

In this section, we discuss properties of (M, N)-DFS ideals in regular AG-groupoids and characterize regular AG-groupoids in terms of these ideals.

An AG-groupoid is said to be regular if for each a ∈ S, there exist an element x ∈ S such that a = (ax) a.

Proposition 5.1. An (M, N)-DFS right ideal of a regular AG-groupoid S over U is an (M, N)-DFS left ideal of S.

Proof. Let 〈 (f_S, g_S); S〉 be an (M, N)-DFS right ideal of a regular AG-groupoid S. Then for any a ∈ S, there exist an element x ∈ S such that a = (ax) a. Then $\begin{matrix} f_{S} (ab) \cup M & = f_{S} ((ax) a) b) \cup M \\ = f_{S} ((ba) (ax)) \cup M \\ \supseteq (f_{S} (ba) (ax) \cup M) \cup M \\ \supseteq (f_{S} (ba) \cap N) \cup M \\ = (f_{S} (ba) \cup M) \cap N \\ \supseteq f_{S} (b) \cap N \end{matrix}$ and $\begin{matrix} g_{S} (ab) \cap N & = g_{S} ((ax) a) b) \cap N \\ = g_{S} ((ba) (ax)) \cap N \\ \subseteq (g_{S} (ba) (ax) \cap N) \cap N \\ \subseteq (g_{S} (ba) \cup M) \cap N \\ = (g_{S} (ba) \cap N) \cup M \\ \subseteq g_{S} (b) \cup M \end{matrix}$

Hence 〈 (f_S, g_S); S〉 is (M, N)-DFS left ideal of S.

Proposition 5.2. An (M, N)-DFS right ideal of a regular AG-groupoid S over U is idempotent.

Proof. Let S be regular AG-groupoid and 〈 (f_S, g_S); S〉 be an (M, N)-DFS right ideal of S. Then for any a ∈ S, we have $\begin{matrix} ((f_{S} \tilde{\circ} f_{S}) (a) \cap N) \cup M \\ \subseteq & ((f_{S} \tilde{\circ} χ_{S}) (a) \cap N) \cup M \\ = & (((f_{S} \tilde{\circ} χ_{S}) (a) \cap N) \cap N) \cup M \\ = & (((f_{S} \tilde{\circ} χ_{S}) (a) \cap N) \cup M) \cap (N \cup M) \\ \subseteq & ((f_{S} (a) \cap N) \cup M) \cap N = (f_{S} (a) \cap N) \cup M \end{matrix}$ and $\begin{matrix} ((g_{S} \tilde{\circ} g_{S}) (a) \cup M) \cap N \\ \supseteq & ((g_{S} \tilde{\circ} χ_{S}^{c}) (a) \cup M) \cap N \\ = & (((g_{S} \tilde{\circ} χ_{S}^{c}) (a) \cup M) \cup M) \cap N \\ = & (((g_{S} \tilde{\circ} χ_{S}^{c}) (a) \cup M) \cap N) \cup (M \cap N) \\ \supseteq & ((g_{S} (a) \cup M) \cap N) \cup M = (g_{S} (a) \cup M) \cap N \end{matrix}$ Hence 〈 (f_S, g_S); S〉 ◊ 〈 (f_S, g_S) ; S 〉 ⊑ _[M,N] 〈 (f_S, g_S) ; S〉. Next we prove the other inclusion. Since S is regular AG-groupoid so for any a ∈ S, there exist x ∈ S such that a = (ax) a, then $\begin{matrix} ((f_{S} \tilde{\circ} f_{S}) (a) \cap N) \cup M \\ = ((⋃_{a = xy} {f_{S} (x) \cap f_{S} (y)}) \cap N) \cup M \\ \supseteq ((f_{S} (ax) \cap f_{S} (a)) \cap N) \cup M \\ \supseteq (f_{S} (a) \cap N) \cup M \end{matrix}$ and $\begin{matrix} ((g_{S} \tilde{\circ} g_{S}) (a) \cup M) \cap N \\ = ((⋂_{a = xy} {g_{S} (x) \cup g_{S} (y)}) \cup M) \cap N \\ \subseteq ((g_{S} (ax) \cup g_{S} (a)) \cup M) \cap N \\ \subseteq (g_{S} (a) \cup M) \cap N \end{matrix}$ Hence 〈 (f_S, g_S); S〉 ⊑ _[M,N] 〈 (f_S, g_S); S 〉 ◊ 〈 (f_S, g_S); S〉 and thus 〈 (f_S, g_S); S〉 is idempotent.

Proposition 5.3. If 〈 (f_S, g_S); S〉 is an (M, N)-DFS right ideal of a regular AG-groupoid S with left identity e then 〈 (f_S, g_S); S 〉 (ab) = _[M,N] 〈 (f_S, g_S); S 〉 (ba) for all a, b ∈ S.

Proof. Let 〈 (f_S, g_S); S〉 be an (M, N)-DFS right ideal of a regular AG-groupoid S with left identity e. Let a, b ∈ S. Due to the regularity of S, there exist elements x, y ∈ S such that a = (ax) a and b = (by) b. Then ab = ((ax) a) ((by) b) = (ba) ((by) (ax)). Since 〈 (f_S, g_S); S〉 is (M, N)-DFS right ideal of S, thus $\begin{matrix} (f_{S} (ab) \cap N) \cup M \\ = (f_{S} ((ba) ((by) (ax))) \cap N) \cup M \\ = (f_{S} ((ba) ((by) (ax)) \cup M) \cap N \\ = ((f_{S} ((ba) ((by) (ax)) \cup M) \cup M) \cap N \\ \supseteq ((f_{S} (ba) \cap N) \cup M) \cap N \\ = (f_{S} (ba) \cap N) \cup M \end{matrix}$ and $\begin{matrix} (g_{S} (ab) \cup M) \cap N \\ = (g_{S} ((ba) ((by) (ax))) \cup M) \cap N \\ = (g_{S} ((ba) ((by) (ax)) \cap N) \cup M \\ = ((g_{S} ((ba) ((by) (ax)) \cap N) \cap N) \cup M \\ \subseteq ((g_{S} (ba) \cup M) \cap N) \cup M \\ = (g_{S} (ba) \cup M) \cap N \end{matrix}$

Hence 〈 (f_S, g_S); S〉 (ba) ⊑ _[M,N] 〈 (f_S, g_S); S 〉 (ab). Similarly the inclusion 〈 (f_S, g_S); S 〉 (ab) ⊑_[M,N] 〈 (f_S, g_S); S 〉 (ba) can be proved. Thus 〈 (f_S, g_S); S 〉 (ab) = _[M,N] 〈 (f_S, g_S); S 〉 (ba).

Theorem 5.4. If S a regular AG-groupoid then for every (M, N)-DFS right ideal 〈 (f_S, g_S); S〉 and every (M, N)-DFS left ideal 〈 (h_S, k_S); S〉, the relation 〈 (f_S, g_S); S〉 ◊ 〈 (h_S, k_S); S 〉 = _[M,N] 〈 (f_S, g_S); S 〉 ⊓ 〈 (h_S, k_S); S 〉 holds.

Proof. Since 〈 (f_S, g_S); S〉 is (M, N)-DFS right ideal and 〈 (h_S, k_S); S〉 is (M, N)-DFS left ideal of S over U then for any a ∈ S, $\begin{matrix} ((f_{S} \tilde{\circ} h_{S}) (a) \cap N) \cup M \\ \subseteq ((f_{S} \tilde{\circ} χ_{S}) (a) \cap N) \cup M \\ = (((f_{S} \tilde{\circ} χ_{S}) (a) \cap N) \cap N) \cup M \\ = (((f_{S} \tilde{\circ} χ_{S}) (a) \cap N) \cup M) \cap N \\ \subseteq ((f_{S} (a) \cap N) \cup M) \cap N \\ = (f_{S} (a) \cap N) \cup M \end{matrix}$ and $\begin{matrix} ((g_{S} \tilde{\circ} k_{S}) (a) \cup M) \cap N \\ \supseteq ((g_{S} \tilde{\circ} χ_{S}^{c}) (a) \cup M) \cap N \\ = (((g_{S} \tilde{\circ} χ_{S}^{c}) (a) \cup M) \cup M) \cap N \\ = (((g_{S} \tilde{\circ} χ_{S}^{c}) (a) \cup M) \cap N) \cup M \\ \supseteq ((g_{S} (a) \cup M) \cap N) \cup M \\ = (g_{S} (a) \cup M) \cap N \end{matrix}$ Similarly we can show that $((f_{S} \tilde{\circ} h_{S}) (a) \cap N) \cup M \subseteq (h_{S} (a) \cap N) \cup M$ and $((g_{S} \tilde{\circ} k_{S}) (a) \cup M) \cap N \supseteq (k_{S} (a) \cup M) \cap N .$ Hence it is established that

〈 (f_S, g_S); S 〉 ◊ 〈 (h_S, k_S); S 〉 ⊑ _[M,N] 〈 (f_S, g_S); S 〉 ⊓ 〈 (h_S, k_S); S 〉. Next, since S is regular so for a ∈ S, there exist x ∈ S such that a = (ax) a. Thus, $\begin{matrix} ((f_{S} \tilde{\circ} h_{S}) (a) \cap N) \cup M \\ = ((⋃_{a = xy} {f_{S} (x) \cap h_{S} (y)}) \cap N) \cup M \\ \supseteq ((f_{S} (ax) \cap h_{S} (a)) \cap N) \cup M \\ = (f_{S} (ax) \cup M) \cap ((h_{S} (a) \cap N) \cup M) \\ = ((f_{S} (ax) \cup M) \cup M) \cap ((h_{S} (a) \cap N) \cup M) \\ \supseteq ((f_{S} (a) \cap N) \cup M) \cap ((h_{S} (a) \cap N) \cup M) \\ = ((f_{S} (a) \cap h_{S} (a)) \cap N) \cup M \end{matrix}$ and $\begin{matrix} ((g_{S} \tilde{\circ} k_{S}) (a) \cup M) \cap N \\ = ((⋂_{a = xy} {g_{S} (x) \cup k_{S} (y)}) \cup M) \cap N \\ \subseteq ((g_{S} (ax) \cup k_{S} (a)) \cup M) \cap N \\ = (g_{S} (ax) \cap N) \cup ((k_{S} (a) \cup M) \cap N) \\ = ((g_{S} (ax) \cap N) \cap N) \cup ((k_{S} (a) \cup M) \cap N) \\ \subseteq ((g_{S} (a) \cup M) \cap N) \cup ((k_{S} (a) \cup M) \cap N) \\ = ((g_{S} (a) \cup k_{S} (a)) \cup M) \cap N \end{matrix}$ Thus 〈 (f_S, g_S); S〉 ⊓ 〈 (h_S, k_S); S 〉 ⊑_[M,N]〈 (f_S, g_S); S 〉 ◊ 〈 (h_S, k_S); S 〉 and hence the required result follows.

6. A decision making scheme based on DFS-sets

Suppose a set of objects is under consideration. The properties of the objects are often called parameters. There could be situations where the parameters may be multi-valued. For the sake of argument, lets take a collection of books titled, ” Calculus & Analytic Geometry”. The properties of these books are the calculus contents and the geometry contents. The parameter, the calculus contents, is intended to be two-valued. i.e. Calculus contents means considering single variable calculus and several-variable calculus. Other parameter, the geometry contents of the book mean 2-dimensional geometry and 3-dimensional geometry. Situations, as exemplified above, where the parameters involved are two-valued, the concept of DFS-sets may be useful.

The concept of DFS-sets can be naturally extended to n-framed soft sets where the parameters are n-valued. We present a decision making scheme based on DFS-sets in the following.

According to the definition of a DFS-set, we can construct tabular form for every DFS-set for a finite universal set U and finite set of parameters E. Therefore let U = {u₁, u₂,…, u_m} and E = {e₁, e₂,…, e_n} and 〈 (α_A, β_A); A〉 be a DFS-set of A over U, then its tabular representation is given in Table 1 where,

Table 1

〈 (α_A, β_A); A〉 (α_A (e₁), β_A (e₁)) (α_A (e₂), β_A (e₂)) ⋯ (α_A (e_n), β_A (e_n))

u ₁ (a₁₁, b₁₁) (a₁₂, b₁₂) ⋯ (a_1n, b_1n)

u ₂ (a₂₁, b₂₁) (a₂₂, b₂₂) ⋯ (a_2n, b_2n)

⋮ ⋮ ⋮ ⋱ ⋮

u _m (a_m1, b_m1) (a_m2, b_m2) ⋯ (a_mn, b_mn)

〈 (α_A, β_A); A〉	(α_A (e₁), β_A (e₁))	(α_A (e₂), β_A (e₂))	⋯	(α_A (e_n), β_A (e_n))
u ₁	(a₁₁, b₁₁)	(a₁₂, b₁₂)	⋯	(a_1n, b_1n)
u ₂	(a₂₁, b₂₁)	(a₂₂, b₂₂)	⋯	(a_2n, b_2n)
⋮	⋮	⋮	⋱	⋮
u _m	(a_m1, b_m1)	(a_m2, b_m2)	⋯	(a_mn, b_mn)

$a_{ij} = {\begin{matrix} 1 if u_{i} \in α_{A} (e_{j}) \\ 0 if u_{i} \notin α_{A} (e_{j}) \end{matrix}$ and $b_{ij} = {\begin{matrix} [c] c 1 if u_{i} \in β_{A} (e_{j}) \\ 0 if u_{i} \notin β (e_{j}) \end{matrix} .$

Example 6.1. Assume that U = {n₁. n₂, n₃, n₄, n₅} is the set of five newspapers under consideration and let A = {e₁, e₂, e₃} = {coverage, publication, quality} is the set of parameters. The parameters involved here are two-valued. i.e. e₁ stands for coverage which means national news and international news, e₂ stands for publication which means online version and print version, e₃ stands for quality which means unbiased news and true news. Suppose Mr. X wants to subscribe monthly package of a newspaper. According to the data collected, the DFS-set 〈 (α_A, β_A); A〉 can be viewed as the collection of the following approximations:

α_A (e₁) = { n₁, n₂, n₃, n₅ }, β_A (e₁) ={ n₁, n₂, n₃ }

α_A (e₂) = { n₁, n₂, n₃ }, β_A (e₂) ={ n₃, n₄, n₅ }

α_A (e₃) = { n₁, n₅ }, β_A (e₃) = { n₁, n₂ }.

The tabular representation of the DFS-set 〈 (α_A, β_A); A〉 is given in Table 2.

Table 2

〈 (α_A, β_A); A〉	(α_A (e₁), β_A (e₁))	(α_A (e₂), β_A (e₂))	(α_A (e₃), β_A (e₃))
n ₁	(1, 1)	(1, 0)	(1, 1)
n ₂	(1, 1)	(1, 0)	(0, 1)
n ₃	(1, 1)	(1, 1)	(0, 0)
n ₄	(0, 0)	(0, 1)	(0, 0)
n ₅	(1, 0)	(0, 1)	(1, 0)

6.1. Choice values of an object

The choice values of an object u_i ∈ U are $c_{i}^{(1)}$ and $c_{i}^{(2)}$ given by, $c_{i}^{(1)} = \sum_{j} a_{ij} and c_{i}^{(2)} = \sum_{j} b_{ij}$ where a_ij and b_ij are the entries in the tabular form of a DFS-set as given in Table 1.

6.2. DFS-set based decision making

In this section, we will present an algorithm for decision making problems based on DFS-sets.

The following algorithm may be followed by Mr. X to subscribe a monthly package of the newspaper.

Algorithm.

1. Input the DFS-set.

2. Input the set A of choice parameters of Mr. X which is a subset of E.

3. Compute the choice values $c_{i}^{(1)}$ and $c_{i}^{(2)}$ for each u_i ∈ U and find $c_{i} = c_{i}^{(1)} + c_{i}^{(2)} .$

4. Find k for which c_k = max c_i.

Then u_k is the optimal choice object. If there the more than one values of k then any of them could be chosen by Mr. X whichever he likes.

Now we use the algorithm to solve a real-world problem.

Example 6.2. Reconsider Example 6.1 and let us apply the algorithm, we get Table 3 in which choice values are shown.

Table 3

〈 (α_A, β_A); A〉 (α_A (e₁), β_A (e₁)) (α_A (e₂), β_A (e₂)) (α_A (e₃), β_A (e₃)) $c_{i}^{(1)}$ $c_{i}^{(2)}$

n ₁ (1, 1) (1, 0) (1, 1) 3 2

n ₂ (1, 1) (1, 0) (0, 1) 2 2

n ₃ (1, 1) (1, 1) (0, 0) 2 2

n ₄ (0, 0) (0, 1) (0, 0) 0 1

n ₅ (1, 0) (0, 1) (1, 0) 2 1

〈 (α_A, β_A); A〉	(α_A (e₁), β_A (e₁))	(α_A (e₂), β_A (e₂))	(α_A (e₃), β_A (e₃))	$c_{i}^{(1)}$	$c_{i}^{(2)}$
n ₁	(1, 1)	(1, 0)	(1, 1)	3	2
n ₂	(1, 1)	(1, 0)	(0, 1)	2	2
n ₃	(1, 1)	(1, 1)	(0, 0)	2	2
n ₄	(0, 0)	(0, 1)	(0, 0)	0	1
n ₅	(1, 0)	(0, 1)	(1, 0)	2	1

Here c₁ = 3 +2 = 5, c₂ = 2 +2 = 4, c₃ = 2 +2 = 4, c₄ = 0 +1 = 1, c₅ = 2 +1 = 3. Hence c₁ = max c_i and so Mr. X should subscribe newspaper n₁.

6.3. Weighted table of a DFS-set

Extending the idea of weighted soft sets given by Liu [19], we define weighted DFS-sets. We define a two-valued weight denoted by $w_{i} = (w_{i}^{(1)}, w_{i}^{(2)})$ on each parameter e_i with the property that $Σ w_{i}^{(j)} = 1$ . The entries of the weighted table of DFS-set 〈 (α_A, β_A); A〉 are defined as $(x_{ij}, y_{ij}) = (w_{j}^{(1)} \times a_{ij}, w_{j}^{(2)} \times b_{ij})$ where (a_ij, b_ij) are the entries in the tabular presentation of the DFS-set 〈 (α_A, β_A); A 〉.

6.4. Weighted choice values of an object

The weighted choice values of an object u_i ∈ U are $c_{i}^{(1)}$ and $c_{i}^{(2)}$ given by, $c_{i}^{(1)} = \sum_{j} x_{ij} and c_{i}^{(2)} = \sum_{j} y_{ij}$ where $x_{ij} = w_{j}^{(1)} \times a_{ij}$ and $y_{ij} = w_{j}^{(2)} \times b_{ij}$ and (a_ij, b_ij) are the entries in the tabular presentation of the DFS-set 〈 (α_A, β_A); A〉.

6.5. Weighted DFS set based decision making

In this section, we will present an algorithm for decision making problems based on DFS-sets.

After imposing weights on each parameter, Mr. X should follow the following algorithm to subscribe a monthly package of the newspaper.

1. Input the DFS-set.

2. Input the set A of choice parameters of Mr. X which is a subset of E.

3. Construct the weighted DFS-set.

4. Compute the choice values $c_{i}^{(1)}$ and $c_{i}^{(2)}$ for each u_i ∈ U and find $c_{i} = c_{i}^{(1)} + c_{i}^{(2)} .$

5. Find k for which c_k = max c_i.

Then u_k is the optimal choice object. If there the more than one values of k then any of them could be chosen by Mr. X whichever he likes.

Now we use the algorithm to solve a real-world problem.

Example 6.3. Reconsider Example 6.1 with the tabular representation given in Table 4. Suppose Mr. X puts weights w₁, w₂ and w₃ on each of the parameters e₁, e₂ and e₃ respectively. The weights are given below;

Table 4

〈 (α_A, β_A); A〉 (α_A (e₁), β_A (e₁)) (α_A (e₂), β_A (e₂)) (α_A (e₃), β_A (e₃))

n ₁ (1, 1) (1, 0) (1, 1)

n ₂ (1, 1) (1, 0) (0, 1)

n ₃ (1, 1) (1, 1) (0, 0)

n ₄ (0, 0) (0, 1) (0, 0)

n ₅ (1, 0) (0, 1) (1, 0)

〈 (α_A, β_A); A〉	(α_A (e₁), β_A (e₁))	(α_A (e₂), β_A (e₂))	(α_A (e₃), β_A (e₃))
n ₁	(1, 1)	(1, 0)	(1, 1)
n ₂	(1, 1)	(1, 0)	(0, 1)
n ₃	(1, 1)	(1, 1)	(0, 0)
n ₄	(0, 0)	(0, 1)	(0, 0)
n ₅	(1, 0)	(0, 1)	(1, 0)

w_{1} = (0.3, 0.7), w_{2} = (0.4, 0.6), w_{3} = (0.5, 0.5)

The tabular presentation of weighted DFS-set is given in Table 5. Hence c₁ = 1.2 + 1.2 = 2.4, c₂ = 0.7 + 1.2 = 1.9, c₃ = 0.7 + 1.3 = 2, c₄ = 0 +0.6 = 0.6, c₅ = 0.8 + 0.6 = 1.4 and so Mr. X chooses the newspaper n₁.

Table 5

〈 (α_A, β_A); A〉	(α_A (e₁), β_A (e₁))	(α_A (e₂), β_A (e₂))	(α_A (e₃), β_A (e₃))	$c_{i}^{(1)}$	$c_{i}^{(2)}$
n ₁	(0.3, 0.7)	(0.4, 0)	(0.5, 0.5)	1.2	1.2
n ₂	(0.3, 0.7)	(0.4, 0)	(0, 0.5)	0.7	1.2
n ₃	(0.3, 0.7)	(0.4, 0.6)	(0, 0)	0.7	1.3
n ₄	(0, 0)	(0, 0.6)	(0, 0)	0	0.6
n ₅	(0.3, 0)	(0, 0.6)	(0.5, 0)	0.8	0.6

Before putting weights, we have c₁ > c₂ = c₃ > c₅ > c₄. Suppose that Mr. X cannot subscribe n₁ due to some reason. Then his next options are to choose between news papers n₂, n₃. But since c₂ = c₃, he is in confusion which one to subscribe. So to decide which newspaper suits him, he puts weights on each parameter and after putting weights, c₁ > c₃ > c₂ > c₅ > c₄. It is now very clear that Mr. X should subscribe n₃. Hence after assigning weights which means that by taking into account the importance of each parameter, the algorithm results in more suitable decision.

7. Conclusion

In this paper we developed theory of (M, N)-DFS ideals of AG-groupoids and discussed some of their properties in regular AG-groupoids. Some future work is given in the following points.

1. We shall extend the results of this study define (M, N)-DFS bi-ideals, (M, N)-DFS interior ideals and (M, N)-DFS quasi ideals of AG-groupoids, pseudo-BCI algebras (see [43, 44]) and neutrosophic triplet groups (see [45, 46]).

2. We shall combine DFS-set with fuzzy set and rough set to form some hybrid soft sets and develop some decision making methods like some developed in [21 , 48].

3. Importantly, we will generalize DFS-set to n-framed soft set and develop decision making schemes.

Footnotes

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 Professor Xueling Ma for editing and communicating the paper.

References

Acar ,

Koyuncu and

Tanay , Soft sets and soft rings, Comput Math Appl 59 (2010), 3458–3463.

A.O.

Atagun and

Sezgin , Soft substructures of rings, fields and modules, Comput Math Appl 61 (2011), 592–601.

S.M.

Anvariyeh ,

Mirvakili ,

Kazanci and

Davvaz , Algebraic hyperstructures of soft sets associated to semi-hypergroups, Southeast Asian Bull of Math 35 (2011), 911–925.

Cagman and

Enginoglu , FP-soft set theory and its applications, Ann Fuzzy Math Inform 2 (2011), 219–226.

Cagman and

Enginoglu , Soft matrix theory and its decision making, Comput Math Appl 59 (2010), 3308–3314.

J.R.

Cho ,

Pusan and

Jezek , Kepka, Paramedial groupoids, Czechoslovak Mathematical Journal 49(124) (1996), 277–290.

Holgate , Groupoids satisfying a simple invertive law, Mathematics Students 61 (1992), 101–106.

Y.B.

Jun and

S.S.

Ahn , Double-framed soft sets with applications in BCK/BCI-algebras, J Appl Math (2012), 15.

Y.B.

Jun ,

K.J.

Lee and

Zhan , Soft p-ideals of soft BCIal-gebras, Comput Math Appl 58 (2009), 2060–2068.

10.

Kazanci ,

Yilmaz and

Yamak , Soft sets and soft BCHalgebras, Hacettepe Journal of Mathematics and Statistics 39(2) (2010), 205–217.

11.

M.A.

Kazim and

Naseeruddin , On almost semigroups, The Aligarh Bulletin of Mathematics 2 (1972), 1–7.

12.

Khan ,

Izhar and

A.S.

Sezer , Characterizations of intraregular Abel-Grassmann's groupoids using Double Framed Soft Ideals, Intl J Anal Appl 15(1) (2017), 62–74.

13.

Khan ,

Farooq and

Davvaz , On (M, N)-intersectional soft interior hyperideals of ordered semihy-pergroups, J Intell Fuzzy Sys 36(6) (2017), 3895–3904.

14.

Khan ,

Izhar and

M.M.

Khalaf , Double framed soft LAsemigroups, J Intell Fuzzy Sys 33 (2017), 3339–3353.

15.

Khan ,

Y.B.

Jun and

Mehmood , Generalized fuzzy interior ideals in Abel Grassmann's groupoids, Intl J Math Mathematical Sci (2010), Art. ID. 838392.

16.

Khan ,

Smarandache and

Anis , Theory of abel grassmann's groupoids, ISBN 978-1-59973-347-0, Educational Publisher Columbus (2015).

17.

Khan ,

Smarandache and

Aziz , Fuzzy abel grass-mann's groupoids, ISBN 978-1-59973-340-1, Educational Publisher Columbus (2015).

18.

D.V.

Kovkov ,

V.M.

Kolbanov and

D.A.

Molodtsov , Soft sets theory based optimization, J Computers and Sys Sciences Int 46(6) (2007), 872–880.

19.

T.Y.

Lin , A set theory for soft computing, a unified view of fuzzy sets via neighbourhoods, Proceedings of 1996 IEEE Intl Conf of Fuzzzy Systems, New Orleans, 1996, pp. 1140–1146.

20.

Ma and

Zhan , Applications of a new soft set to h-hemiregularhemirings via(M, N)-SI-h-ideals, J Intell Fuzzzy Syst 26 (2014), 2515–2525.

21.

Ma ,

Zhan ,

M.I.

Ali and

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

22.

Ma ,

Liu and

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

23.

P.K.

Maji ,

Biswas and

A.R.

Roy , An application of soft sets in a decision making problem, Comput Math Appl 44 (2002), 1077–1083.

24.

P.K.

Maji ,

Biswas and

A.R.

Roy , Soft set theory, Comput Math Appl 45 (2003), 555–562.

25.

Molodtsov , Soft set theory - First results, Comput Math Appl 37 (1999), 19–31.

26.

Mushtaq and

S.M.

Yusuf , On LA-semigroups, The Ali-garh Bulletin of Mathematics 8 (1978), 65–70.

27.

Naseeruddin , Some studies on almost semigroups and flocks, PhD Thesis, The Aligarh Muslim University India, 1970.

28.

P.V.

Protic and

Stevanovic , On Abel-Grassmann's groupoids (review), Proceeding of Mathematics Conference in Pristina, 1999, pp. 31–38.

29.

A.R.

Roy and

P.K.

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

30.

A.S.

Sezer , A new approach to LA-semigroup theory via the soft sets, J Intell and Fuzzy Sys 26 (2014), 2483–2495.

31.

A.S.

Sezer , Certain characterizations of LA-semigroups by soft sets, J Intell and Fuzzy Sys 27(2) (2014), 1035–1046.

32.

Xiao ,

Gong and

Zou , A combined forecasting approach based on fuzzy soft sets, J Comput Appl Math 228(1) (2009), 326–333.

33.

Yousafzai ,

Khan ,

Amjad and

Zeb , On fuzzy fully regular AG-groupoids, J Intell & Fuzzy Syst 26 (2014), 2973–2982.

34.

L.A.

Zadeh , Fuzzy sets, Information and Control 8 (1965), 338–353.

35.

Zhan ,

W.A.

Dudek and

Neggers , A new soft union set: Characterizations of hemirings, Int J Mach Learn & Cyber DOI 10.1007/s13042-015-0343-8.

36.

Zhan and

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

37.

Zhan and

J.C.R.

Alcantud , A novel type of soft rough covering and its application to multicriteria group decision making, Artificial Intelligence Review (2018). DOI: 10.1007/s10462-018-9617-3

38.

Zhan and

J.C.R.

Alcantud , A survey of parameter reduction of soft sets and corresponding algorithms, Artificial Intelligence Review (2018). DOI: 10.1007/s10462-017-9592-0

39.

Zhan ,

Liu and

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

40.

Zhan ,

M.I.

Ali and

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

41.

Zhan and

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

42.

Zhan and

Xu , Two types of coverings based multi granulation rough fuzzy sets and applications to decision making, Artificial Intelligence Review (2018).

43.

X.H.

Zhang , Fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras, J Intell Fuzzy Syst 33 (2017), 1767–1774.

44.

X.H.

Zhang ,

Park and

S.P.

Wu , Soft set theoretical approach to pseudo-BCI algebras, J Intell Fuzzy Syst 34 (2018), 559–568.

45.

X.H.

Zhang ,

C.X.

Bo ,

Smarandache and

Park , New operations of totally dependent-neutrosophic sets and totally dependent-neutrosophic soft sets, Symmetry 10(6) 2018.

46.

X.H.

Zhang ,

Smarandache and

X.L.

Liang , Neutrosophic duplet semi-group and cancellable neutrosophic triplet groups, Symmetry 9 (2017), doi: 10.3390/sym9110275

47.

Zhang and

Zhan , Novel classes of fuzzy soft ß-coverings-based fuzzy rough sets with applications to multicriteria fuzzy group decision making, Soft Computing (2018).

48.

Zhang and

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

49.

Zou and

Xiao , Data analysis approaches of soft sets under incomplete information, Knowledge-Based Systems 21(8) (2008), 941–945.

( M,N )-Double framed soft ideals of Abel Grassmann’s groupoids

Abstract

Keywords

1. Introduction

2. Preliminaries

4. (M, N)-double framed soft ideals

5. Regular AG-groupoids

6. A decision making scheme based on DFS-sets

Table 1 〈 (α A , β A ); A〉 (α A (e1), β A (e1)) (α A (e2), β A (e2)) ⋯ (α A (e n ), β A (e n )) u 1 (a11, b11) (a12, b12) ⋯ (a1n, b1n) u 2 (a21, b21) (a22, b22) ⋯ (a2n, b2n) ⋮ ⋮ ⋮ ⋱ ⋮ u m (am1, bm1) (am2, bm2) ⋯ (a mn , b mn )

6.2. DFS-set based decision making

Table 3 〈 (α A , β A ); A〉 (α A (e1), β A (e1)) (α A (e2), β A (e2)) (α A (e3), β A (e3)) c i ( 1 ) c i ( 2 ) n 1 (1, 1) (1, 0) (1, 1) 3 2 n 2 (1, 1) (1, 0) (0, 1) 2 2 n 3 (1, 1) (1, 1) (0, 0) 2 2 n 4 (0, 0) (0, 1) (0, 0) 0 1 n 5 (1, 0) (0, 1) (1, 0) 2 1

6.4. Weighted choice values of an object

6.5. Weighted DFS set based decision making

Table 4 〈 (α A , β A ); A〉 (α A (e1), β A (e1)) (α A (e2), β A (e2)) (α A (e3), β A (e3)) n 1 (1, 1) (1, 0) (1, 1) n 2 (1, 1) (1, 0) (0, 1) n 3 (1, 1) (1, 1) (0, 0) n 4 (0, 0) (0, 1) (0, 0) n 5 (1, 0) (0, 1) (1, 0)

Footnotes

Acknowledgements

References

Table 4

〈 (α_A, β_A); A〉 (α_A (e₁), β_A (e₁)) (α_A (e₂), β_A (e₂)) (α_A (e₃), β_A (e₃))

n ₁ (1, 1) (1, 0) (1, 1)

n ₂ (1, 1) (1, 0) (0, 1)

n ₃ (1, 1) (1, 1) (0, 0)

n ₄ (0, 0) (0, 1) (0, 0)

n ₅ (1, 0) (0, 1) (1, 0)