Implementation of anti-lattice ordered fuzzy soft groups and its matrix operations in deciding process

Abstract

The concept of lattice ordered fuzzy soft groups(l- FSGs) was instigated by J. Vimala and J. Arockia Reeta. In this present work, we define anti- lattice ordered fuzzy soft groups (anti- l-FSGs) and extend anti- l-FSGs matrix with appropriate examples. Also we derive the properties relevant to anti- l-FSGs matrices with its distinctive operations. Furthermore, we have applied the bounded sum and bounded difference operations in real life deciding process to find the best solution.

Keywords

De Morgan’s law

1 Introduction

In the mid 1930s, Garrett Birkhoff [3] exhibited the significance of lattice theory. Followed that, Gratzer [6] discussed in depth the lattice theory foundation, and presented special topics and applications of lattice theory. The notion of fuzzy set [17, 18] was conferred by Zadeh. In 1999, Molodtsov firstly proposed the theory of soft sets [8]. By embedding the definition of Molodtsov’s soft set theory, Maji et al. investigated fuzzy soft set theory [7] with its operations and found many interesting applications of fuzzy soft set theory [9]. The concept of fuzzy soft groups(FSGs) was originated by Abdulkadir Aygunoglu and Halish Aygun [2]. In 2016, Vimala et al. bestowed l-FSGs [10 , 15] and its algebraic properties [13]. Then duality principle [1, 12] of l-FSGs has been studied in detail. In 2010, Cagman and Enginoglu [5] defined soft matrices on soft set theory and redefined the operations of soft sets theoretically. In 2011, Yong Yang and Chenli Ji [16] presented fuzzy soft matrices and discussed their rudimentary properties. In 2012, Cagman and Enginoglu [4] studied fuzzy soft matrices and constructed a practical fuzzy soft max- min decision method to solve problem successfully. He defined four special products of fuzzy soft matrices and its several operations.

This paper is organized in the following manner: In Section 2, the basic definitions related to fuzzy soft set, fuzzy soft groups and their properties are given. In Section 3, the concept of anti-l-FSGs matrices is defined and its pertinent properties are obtained with its unique operations such as bounded sum, bounded difference, complement and algebraic product. In Section 4, bounded sum and bounded difference operations have been applied in real life problems.

Throughout this paper, we use I for a unit closed interval [0,1], FSG for fuzzy soft group and l-FSG for lattice ordered fuzzy soft group.

2 Preliminaries

Definition 2.1. [17] Let X be a non-empty set, then a fuzzy set μ over X is a function from X into I = [0, 1].ie., μ: X → I.

Definition 2.2. [8] Let X be an initial universe set and E a set of parameters with respect to X. Let P(X) denote the power set of X and A ⊆ E. A pair (F, A) is called a soft set over X, where F is a mapping given by F:A →P(X).

A soft set over X is a parameterized family of subsets of the universe X.

Definition 2.3. [7] Let I^X denote the set of all fuzzy sets on X and A ⊂ E. A pair (f, A) is called a fuzzy soft set over X, where f is a mapping from A into I^X. That is, for each a ∈ A, f (a) = f_a: X → I, is a fuzzy set on X.

Definition 2.4. [2] Let X be a group and (f, A) be a fuzzy soft set over X. Then (f, A) is said to be a FSG over X iff for each a ∈ A and x, y∈ X,

f_a (x. y)≥ min {f_a (x), f_a (y)}

f_a (x⁻¹) ≥ f_a (x).

That is, for each a ∈ A, f_a is a fuzzy subgroup.

Definition 2.5. [7] For two fuzzy soft sets (f, A) and (g, B) over a common universe X, we say that (f, A) is a fuzzy soft subset of (g, B) and write (f, A) ⊑ (g, B) if

A ⊂ B, and

For each a ∈ A, f_a ≤ g_a,that is, f_a is fuzzy subset of g_a.

Note that for all a ∈ A, f_a and g_a are identical approximations.

Definition 2.6. [7] Union of two fuzzy soft sets (f, A) and (g, B) over a common universe X is the fuzzy soft set (h, C), where C = A ∪ B and $h (c) = {\begin{matrix} f_{c} & if c \in A - B \\ g_{c} & if c \in B - A \\ f_{c} \lor g_{c} & if c \in A \cap B \end{matrix}$ , for all c ∈ C.

We write (f, A) ⊔ (g, B)=(h, C).

Definition 2.7. [7] Intersection of two fuzzy soft sets (f, A) and (g, B) over a common universe X is the fuzzy soft set (h, C), where C = A ∩ B and h_c = f_c ∧ g_c, for all c ∈ C.

We write (f, A) ⊓ (g, B)=(h, C).

Definition 2.8. [4] Let Γ_A ∈ FS (U). Then a fuzzy relation form of Γ_A is defined by R_A = {(μ_{R
_A} (u, x)/(u, x)): (u, x) ∈ U × E} where the membership function of μ_{R
_A} is written by μ_{R
_A}: U × E → [0, 1], μ_{R
_A} (u, x) = μ_{γ_A(x)} (u). If U = {u₁, u₂ … …. u_m}, E = {x₁, x₂, … ….. x_n} and A ⊆ E, then the R_A can be presented by a table as in the following form

R _R	x ₁	x _n
u ₁	μR_A (u₁, x₁)	μR_A (u₁, x_n)
u ₂	μR_A (u₂, x₁)	μR_A (u₂, x_n)
·	·
·	·
·	·
u _m	μR_A (u_m, x₁)	μR_A (u _m , x _n )

If a_ij = μ_{R
_A} (u_i, x_j), we can define a matrix

[a_{ij}] = [\begin{matrix} x_{11} & x_{12} & . . . . . & x_{1 n} \\ x_{21} & x_{22} & . . . . . & x_{2 n} \\ . & . & . . . . . & . \\ . & . & . . . . . & . \\ . & . & . . . . . & . \\ x_{m 1} & x_{m 2} & . . . . . & x_{mn} \end{matrix}]

which is called an m × n fuzzy soft matrix of the fuzzy soft set Γ_A over U.

Proposition 2.9. [4] Let [a_ij], [b_ij] ∈ FSM_m×n. Then the fuzzy soft matrix [c_ij] is called

union of [a_ij] and [b_ij], denoted [a_ij] ∪ [b_ij], if c_ij = max {a_ij, b_ij} for all i and j.

intersection of [a_ij] and [b_ij], denoted [a_ij] ∩ [b_ij], if c_ij = min {a_ij, b_ij} for all i and j.

complement of [a_ij], denoted by [a_ij] ^o, if c_ij = 1 - a_ij for all i and j.

Proposition 2.10. [4] Let [a_ij], [b_ij], [c_ij] ∈ FSM_m×n. Then

[a_ij] ∩ [b_ij] = [b_ij] ∩ [a_ij]

[a_ij] ∪ [b_ij] = [b_ij] ∪ [a_ij]

([a_ij] ∩ [b_ij]) ∩ [c_ij] = [a_ij] ∩ ([b_ij] ∩ [c_ij])

([a_ij] ∪ [b_ij]) ∪ [c_ij] = [a_ij] ∪ ([b_ij] ∪ [c_ij])

[a_ij] ∩ ([b_ij] ∪ [c_ij]) = ([a_ij] ∩ [b_ij]) ∪ ([a_ij] ∩ [c_ij])

[a_ij] ∪ ([b_ij] ∩ [c_ij]) = ([a_ij] ∪ [b_ij]) ∩ ([a_ij] ∪ [c_ij])

Definition 2.11. [18] The bounded difference of two fuzzy sets $\tilde{A}$ and $\tilde{B}$ is denoted by $\tilde{C} = \tilde{A} ⊖ \tilde{B} = {(x, μ_{\tilde{A} ⊖ \tilde{B}} (x)); x \in X}$ , where $μ_{\tilde{A} ⊖ \tilde{B}} (x) = Max {0, μ_{\tilde{A}} (x) + μ_{\tilde{B}} (x) - 1}$ .

Definition 2.12. [18] The bounded sum of two fuzzy sets $\tilde{A}$ and $\tilde{B}$ is denoted by $\tilde{C} = \tilde{A} \oplus \tilde{B} = {(x, μ_{\tilde{A} \oplus \tilde{B}} (x)); x \in X}$ , where $μ_{\tilde{A} \oplus \tilde{B}} (x) = Min {1, μ_{\tilde{A}} (x) + μ_{\tilde{B}} (x)}$ .

Definition 2.13. [18] The algebraic product of two fuzzy sets $\tilde{C} = \tilde{A} . \tilde{B}$ is defined as $\tilde{C} = {(x, μ_{\tilde{A}} (x) . μ_{\tilde{B}} (x)) / x \in X}$ .

3 Anti-lattice ordered fuzzy soft groups and its matrix operations

[1] Throughout this section, Let X be a group and P(X) be the power set of X. If the set of parameters E is also a lattice with respect to certain binary operations or partial order, then a non-empty subset A of E also inherits the partial order from the set E and we use ∨ for maximum and ∧ for minimum.

Definition 3.1. [1] Let X be a group and a FSG (f, A) over X is said to be an l-FSG over X if for the mapping f: A → I^X, a ≤ b implies f_a ⊆ f_b, for all a, b ∈ A.

Definition 3.2. Let X be a group and a FSG (f, A) over X is said to be an anti - l-FSG over X if for the mapping f: A → I^X, a ≤ b implies f_a ⊇ f_b, for all a, b ∈ A.

Definition 3.3. Let X be a group and (f, A), (f, B), (f, C)…. be anti - l-FSGs over X. Anti - l-FSG matrix can be defined as $[P_{ij}]_{m \times n} = {[\begin{matrix} μ_{f_{A}} (x) \\ μ_{f_{B}} (x) \\ μ_{f_{C}} (x) \\ . \\ . \\ . \\ μ_{f_{M}} (x) \end{matrix}]}_{m \times n}$ $indent = {[\begin{matrix} μ_{f_{a_{1}}} (x) & μ_{f_{a_{2}}} (x) & μ_{f_{a_{3}}} (x) & . . . . . & μ_{f_{a_{n}}} (x) \\ μ_{f_{b_{1}}} (x) & μ_{f_{b_{2}}} (x) & μ_{f_{b_{3}}} (x) & . . . . . & μ_{f_{b_{n}}} (x) \\ μ_{f_{c_{1}}} (x) & μ_{f_{c_{2}}} (x) & μ_{f_{c_{3}}} (x) & . . . . . & μ_{f_{c_{n}}} (x) \\ . \\ . \\ . \\ μ_{f_{m_{1}}} (x) & μ_{f_{m_{2}}} (x) & μ_{f_{m_{3}}} (x) & . . . . . & μ_{f_{m_{n}}} (x) \end{matrix}]}_{m \times n}$ for each a_i ∈ A, b_i ∈ B, c_i ∈ C….. and A, B, C … M ⊂ E.

Definition 3.4. Complement of an anti - l-FSGs matrix over X is defined as $[P_{ij}]_{m \times n}^{c} = [1 - P_{ij}]_{m \times n}$ .

Remark 3.5. Complement of an anti - l-FSGs matrix is not an l-FSG matrix over X.

Proof. Let X be a group and (f, A) be an anti-l- FSGs over X. Then we have for all a_i, a_j ∈ A, a_i ≤ a_j $\begin{matrix} \Rightarrow & f_{a_{i}} \supseteq f_{a_{j}} \\ \Rightarrow & μ_{f_{a_{i}}} \geq μ_{f_{a_{j}}} \\ \Rightarrow & - μ_{f_{a_{i}}} \leq - μ_{f_{a_{j}}} \\ \Rightarrow & 1 - μ_{f_{a_{i}}} \leq 1 - μ_{f_{a_{j}}} \\ \Rightarrow & μ_{f_{a_{i}}}^{c} \leq μ_{f_{a_{j}}}^{c} \end{matrix}$

Since (f, A) is an anti-l- FSG, f_{a
_i} ⊈ f_{a
_j}

Hence complement of an anti - l-FSG is not an l-FSG which implies complement of an anti - l-FSGs matrix does not form an l-FSGs matrix over X. □

Example 3.6. Let N be the set of all natural numbers, $ℤ$ be the set of all integers and X be a group, Define f: N → I^X by f (n) = f_n: X → I, for each n ∈ N where $f_{n} (x) = {\begin{matrix} \frac{1}{n} & if x = k 3^{n}, \exists k \in ℤ \\ 0 & otherwise \end{matrix}$ Let A ⊂ N and A={2, 5, 7}. Then (f, A) = {f₂ (x), f₅ (x), f₇ (x)} Where $f_{2} (x) (simply f_{2}) = \frac{k 3^{2}}{0.5}$ $= {\frac{0}{0.5}, \frac{\pm 3^{2}}{0.5}, \frac{\pm 2 . 3^{2}}{0.5}, \frac{\pm 3 . 3^{2}}{0.5}, \frac{\pm 4 . 3^{2}}{0.5} . . . . . . \frac{\pm 3^{3} . 3^{2}}{0.5} . . . .}$ and $f_{5} = \frac{k 3^{5}}{0.2} = {\frac{0}{0.2}, \frac{\pm 3^{5}}{0.2}, \frac{\pm 2 . 3^{5}}{0.2}, \frac{\pm 3 . 3^{5}}{0.2}, \frac{\pm 4 . 3^{5}}{0.2} . . . . . . . . . . . . . . .}$ Similarly, for each 2≤5 implies f₂ ⊇ f₅. Hence (f, A) is an anti -l- FSGs over X. Similarly consider another anti l- FSGs (f, B) and (f, C) where B={1, 3, 5} and C={2, 8, 9}, then (f, B) = {f₁, f₃, f₅} and (f, C) = {f₂, f₈, f₉}. Now $(f, A) = {\frac{k 3^{2}}{0.5}, \frac{k 3^{5}}{0.2}, \frac{k 3^{7}}{0.14}}$ , $(f, B) = {\frac{k 3^{1}}{1}, \frac{k 3^{3}}{0.3}, \frac{k 3^{5}}{0.2}}$ , $(f, C) = {\frac{k 3^{2}}{0.5}, \frac{k 3^{8}}{0.12}, \frac{k 3^{9}}{0.11}}$ Consider the anti -l- FSGs matrix $[P_{ij}]_{3 \times 3} = [\begin{matrix} μ_{f_{A}} (x) \\ μ_{f_{B}} (x) \\ μ_{f_{C}} (x) \end{matrix}] = {[\begin{matrix} 0.5 & 0.2 & 0.14 \\ 1 & 0.3 & 0.2 \\ 0.5 & 0.12 & 0.11 \end{matrix}]}_{3 \times 3}$ Complement of [P_ij] _3×3 is $[P_{ij}]_{3 \times 3}^{c} = {[\begin{matrix} 0.5 / k 3^{2} & 0.8 / k 3^{5} & 0.86 / k 3^{7} \\ 0 / k 3^{1} & 0.7 / k 3^{3} & 0.8 / k 3^{5} \\ 0.5 / k 3^{2} & 0.88 / k 3^{8} & 0.89 / k 3^{9} \end{matrix}]}_{3 \times 3}$ which is not an l-FSGs matrix over X since k3² ⊈ k3⁵ ⊈ k3⁷, k3¹ ⊈ k3³ ⊈ k3⁵ and k3² ⊈ k3⁸ ⊈ k3⁹.

Definition 3.7. Bounded sum of two anti - l-FSGs matrices [P_ij] _(m×n) and [Q_ij] _(m×n) over X is defined as [P_ij] _m×n ⊕ [Q_ij] _m×n = [P_ij ⊕ Q_ij] _m×n = [Min {P_ij + Q_ij, 1}] _m×n.

Proposition 3.8. Bounded sum of two anti - l-FSGs matrices is also an anti - l-FSG matrix over X.

Proof. Let (f, A) and (f, B) be two anti-l-FSGs over X. Then f_a (x. y) ≥ Min {f_a (x), f_a (y)}, f_a (x⁻¹) ≥ f_a (x) andf_b (x. y) ≥ Min {f_b (x), f_b (y)}, f_b (x⁻¹) ≥ f_b (x) ∀x, y ∈ X, a ∈ A and b ∈ B. $\begin{matrix} (f_{a} \oplus f_{b}) (x . y) & = & f_{a} (x . y) \oplus f_{b} (x . y) \\ \geq & Min {f_{a} (x), f_{a} (y)} \oplus \\ Min {f_{b} (x), f_{b} (y)} \\ = & (f_{a} (x) \land f_{a} (y)) \oplus (f_{b} (x) \land f_{b} (y)) \\ = & (f_{a} (x) \oplus f_{b} (x)) \land (f_{a} (y) \oplus f_{b} (y)) \\ = & (f_{a} \oplus f_{b}) (x) \land (f_{a} \oplus f_{b}) (y) and \\ (f_{a} \oplus f_{b}) (x^{- 1}) & \geq & (f_{a} \oplus f_{b}) (x) . \end{matrix}$ Hence bounded sum of two FSGs is also a FSG over X. Consider two anti -l- FSGs matrices [P_ij] _1×n and [Q_ij] _1×n where [P_ij] _1×n = [f_A (x)] _1×n and [Q_ij] _1×n = [f_B (x)] _1×n [P_ij] _1×n ⊕ [Q_ij] _1×n $= {[\begin{matrix} μ_{f_{a_{1}}} (x) & μ_{f_{a_{2}}} (x) & μ_{f_{a_{3}}} (x) & . . . . . & μ_{f_{a_{n}}} (x) \end{matrix}]}_{1 \times n} \oplus$ ${[\begin{matrix} μ_{f_{b_{1}}} (x) & μ_{f_{b_{2}}} (x) & μ_{f_{b_{3}}} (x) & . . . . . & μ_{f_{b_{n}}} (x) \end{matrix}]}_{1 \times n}$ , for a₁, a₂, ….., a_n ∈ A and b₁, b₂, ….., b_n ∈ B. ⇒ [P_ij] _1×n ⊕ [Q_ij] _1×n = ${[\begin{matrix} μ_{f_{a_{1}}} (x) \oplus μ_{f_{b_{1}}} (x) & . . . . . & μ_{f_{a_{n}}} (x) \oplus μ_{f_{b_{n}}} (x) \end{matrix}]}_{1 \times n}$ where (f, A) and (f, B) are anti- l-FSGs over X. Then for each a₁, a₂ ∈ A, a₁ ≤ a₂ implies f_{a
₁} ⊇ f_{a
₂} and for each b₁, b₂ ∈ B, b₁ ≤ b₂ implies f_{b
₁} ⊇ f_{b
₂}. $\begin{matrix} \Rightarrow & μ_{f_{a_{1}} (x)} \geq μ_{f_{a_{2}} (x)} and μ_{f_{b_{1}} (x)} \geq μ_{f_{b_{2}} (x)} \\ \Rightarrow & μ_{f_{a_{1}} (x)} + μ_{f_{b_{1}} (x)} \geq μ_{f_{a_{2}} (x)} + μ_{f_{b_{2}} (X)} \\ \Rightarrow & Min {1, μ_{f_{a_{1}} (x)} + μ_{f_{b_{1}} (X)}} \geq \\ Min {1, μ_{f_{a_{2}} (x)} + μ_{f_{b_{2}} (x)}} \\ \Rightarrow & f_{a_{1}} (x) \oplus f_{b_{1}} (x) \supseteq f_{a_{2}} (x) \oplus f_{b_{2}} (X) \end{matrix}$

Hence bounded sum of two m × n anti - l-FSGs matrices is also an m × n anti - l-FSGs matrix over X. □

Example 3.9. From the example 3.6, we have $[P_{ij}]_{3 \times 3} = {[\begin{matrix} 0.5 & 0.2 & 0.14 \\ 1 & 0.3 & 0.2 \\ 0.5 & 0.12 & 0.11 \end{matrix}]}_{3 \times 3}$ Consider the another anti - l-FSGs $(f, R) = {\frac{k 3^{1}}{1}, \frac{k 3^{4}}{0.25}, \frac{k 3^{5}}{0.2}}$ , $(f, S) = {\frac{k 3^{2}}{0.5}, \frac{k 3^{3}}{0.3}, \frac{k 3^{6}}{0.16}}$ , $(f, T) = {\frac{k 3^{2}}{0.5}, \frac{k 3^{5}}{0.2}, \frac{k 3^{9}}{0.11}}$ where R={1, 4, 5}, S={2, 3,6} and T={2, 5, 9} Then $[Q_{ij}]_{3 \times 3} = [\begin{matrix} μ_{f_{R}} (x) \\ μ_{f_{S}} (x) \\ μ_{f_{T}} (x) \end{matrix}] = {[\begin{matrix} 1 & 0.25 & 0.2 \\ 0.5 & 0.3 & 0.16 \\ 0.5 & 0.2 & 0.11 \end{matrix}]}_{3 \times 3}$ The bounded sum of [P_ij] _3×3 and [Q_ij] _3×3 is [P_ij] _3×3 ⊕ [Q_ij] _3×3 $= {[\begin{matrix} 0.5 & 0.2 & 0.14 \\ 1 & 0.3 & 0.2 \\ 0.5 & 0.12 & 0.11 \end{matrix}]}_{3 \times 3} \oplus {[\begin{matrix} 1 & 0.25 & 0.2 \\ 0.5 & 0.3 & 0.16 \\ 0.5 & 0.2 & 0.11 \end{matrix}]}_{3 \times 3}$ $= {[\begin{matrix} 1 & 0.45 & 0.34 \\ 1 & 0.6 & 0.36 \\ 1 & 0.32 & 0.22 \end{matrix}]}_{3 \times 3}$ Then [P_ij] _3×3 ⊕ [Q_ij] _3×3= ${[\begin{matrix} 1 / k 3^{1} & 0.45 / k 3^{5} & 0.34 / k 3^{7} \\ 1 / k 3^{1} & 0.6 / k 3^{3} & 0.36 / k 3^{6} \\ 1 / k 3^{2} & 0.32 / k 3^{8} & 0.22 / k 3^{9} \end{matrix}]}_{3 \times 3}$ is also an anti - l-FSGs matrix over X.

Definition 3.10. Bounded difference of two anti - l-FSGs matrices [P_ij] _(m×n) and [Q_ij] _(m×n) over X is defined as [P_ij] _m×n ⊖ [Q_ij] _m×n = [P_ij ⊖ Q_ij] _m×n = [Max {P_ij + Q_ij − 1, 0}] _m×n.

Proposition 3.11. Bounded difference of two anti - l-FSGs matrices is also an anti - l-FSGs matrix over X.

Proof. Let (f, A) and (f, B) be two anti-l-FSGs over X. Then f_a (x. y) ≥ Min {f_a (x), f_a (y)}, f_a (x⁻¹) ≥ f_a (x) andf_b (x. y) ≥ Min {f_b (x), f_b (y)}, f_b (x⁻¹) ≥ f_b (x) ∀x, y ∈X, a ∈ A and b ∈ B. $\begin{matrix} (f_{a} ⊖ f_{b}) (x . y) & = & f_{a} (x . y) ⊖ f_{b} (x . y) \\ \geq & Min {f_{a} (x), f_{a} (y)} ⊖ \\ Min {f_{b} (x), f_{b} (y)} \end{matrix}$ $\begin{matrix} = & (f_{a} (x) \land f_{a} (y)) ⊖ (f_{b} (x) \land f_{b} (y)) \\ = & (f_{a} (x) ⊖ f_{b} (x)) \land (f_{a} (y) ⊖ f_{b} (y)) \\ ∥ & = & (f_{a} ⊖ f_{b}) (x) \land (f_{a} ⊖ f_{b}) (y) and \\ (f_{a} ⊖ f_{b}) (x^{- 1}) & \geq & (f_{a} ⊖ f_{b}) (x) . \end{matrix}$ Hence bounded difference of two FSGs is also a FSGs over X. Consider two anti -l- FSGs matrices [P_ij] _1×n and [Q_ij] _1×n where [P_ij] _1×n = [f_A (x)] _1×n and [Q_ij] _1×n = [f_B (x)] _1×n [P_ij] _1×n ⊖ [Q_ij] _1×n $= {[\begin{matrix} μ_{f_{a_{1}}} (x) & μ_{f_{a_{2}}} (x) & μ_{f_{a_{3}}} (x) & . . . . . & μ_{f_{a_{n}}} (x) \end{matrix}]}_{1 \times n} ⊖$ ${[\begin{matrix} μ_{f_{b_{1}}} (x) & μ_{f_{b_{2}}} (x) & μ_{f_{b_{3}}} (x) & . . . . . & μ_{f_{b_{n}}} (x) \end{matrix}]}_{1 \times n}$ , for a₁, a₂, ….., a_n ∈ A and b₁, b₂, ….., b_n ∈ B. $\Rightarrow [P_{ij}]_{1 \times n} ⊖ [Q_{ij}]_{1 \times n} = μ_{f_{a_{1}}} (x) ⊖ {[\begin{matrix} μ_{f_{b_{1}}} (x) & . . . . . & μ_{f_{a_{n}}} (x) ⊖ μ_{f_{b_{n}}} (x) \end{matrix}]}_{1 \times n}$ where (f, A) and (f, B) are anti- l-FSGs over X. Then for each a₁, a₂ ∈ A, a₁ ≤ a₂ implies f_{a
₁} ⊇ f_{a
₂} and for each b₁, b₂ ∈ B, b₁ ≤ b₂ implies f_{b
₁} ⊇ f_{b
₂}. $\begin{matrix} \Rightarrow & μ_{f_{a_{1}}} \geq μ_{f_{a_{2}}} and μ_{f_{b_{1}}} \geq μ_{f_{b_{2}}} \\ \Rightarrow & μ_{f_{a_{1}}} + μ_{f_{b_{1}}} \geq μ_{f_{a_{2}}} + μ_{f_{b_{2}}} \\ \Rightarrow & Max {0, μ_{f_{a_{1}}} + μ_{f_{b_{1}}} - 1} \\ \geq & Max {0, μ_{f_{a_{2}}} + μ_{f_{b_{2}}} - 1} \\ \Rightarrow & f_{a_{1}} (x) ⊖ f_{b_{1}} (x) \supseteq f_{a_{2}} (x) ⊖ f_{b_{2}} (x) \end{matrix}$ Hence bounded difference of two m × n anti - l-FSGs matrices is also an m × n anti - l-FSGs matrix over X. □

Example 3.12. From the example 3.9, the bounded difference of [P_ij] _3×3 and [Q_ij] _3×3 is [P_ij] _3×3 ⊖ [Q_ij] _3×3 $= {[\begin{matrix} 0.5 & 0.2 & 0.14 \\ 1 & 0.3 & 0.2 \\ 0.5 & 0.12 & 0.11 \end{matrix}]}_{3 \times 3} ⊖ {[\begin{matrix} 1 & 0.25 & 0.2 \\ 0.5 & 0.3 & 0.16 \\ 0.5 & 0.2 & 0.11 \end{matrix}]}_{3 \times 3}$ $= {[\begin{matrix} 0.5 / k 3^{2} & 0 & 0 \\ 0.5 / k 3^{2} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]}_{3 \times 3}$ is also an anti - l-FSGs matrix over X.

Proposition 3.13. Let [P_ij] _m×n and [Q_ij] _m×n be two anti -l- FSGs matrices over X. Then

$[P_{ij}]_{m \times n}^{c^{c}} = [P_{ij}]_{m \times n}$ ,

$[P_{ij}]_{m \times n} \oplus [P_{ij}]_{m \times n}^{c} = [1]_{m \times n}$ ,

$[P_{ij}]_{m \times n} ⊖ [P_{ij}]_{m \times n}^{c} = [0]_{m \times n}$ ,

[P_ij] _m×n ⊕ [Q_ij] _m×n = [Q_ij] _m×n ⊕ [P_ij] _m×n,

[P_ij] _m×n ⊖ [Q_ij] _m×n = [Q_ij] _m×n ⊖ [P_ij] _m×n.

Proof. Let [P_ij] _m×n and [Q_ij] _m×n be two anti -l- FSGs matrices over X.

$[P_{ij}]_{m \times n}^{c^{c}}$ $\begin{matrix} = & [1 - P_{ij}]_{m \times n}^{c} \\ = & [1 - {1 - P_{ij}}]_{m \times n} \\ = & [P_{ij}]_{m \times n} \end{matrix}$

$[P_{ij}]_{m \times n} \oplus [P_{ij}]_{m \times n}^{c}$ $\begin{matrix} = & [Min {P_{ij} + P_{ij}^{c}, 1}]_{m \times n} \\ = & [Min {P_{ij} + 1 - P_{ij}, 1}]_{m \times n} \\ = & [1]_{m \times n} \end{matrix}$

$[P_{ij}]_{m \times n} ⊖ [P_{ij}]_{m \times n}^{c}$ $\begin{matrix} = & [Max {P_{ij} + P_{ij}^{c} - 1, 0}]_{m \times n} \\ = & [Max {P_{ij} + 1 - P_{ij} - 1, 0}]_{m \times n} \\ = & [0]_{m \times n} \end{matrix}$

[P_ij] _m×n ⊕ [Q_ij] _m×n $\begin{matrix} = & [Min {P_{ij} + Q_{ij}, 1}]_{m \times n} \\ = & [Min {Q_{ij} + P_{ij}, 1}]_{m \times n} \\ = & [Q_{ij}]_{m \times n} \oplus [P_{ij}]_{m \times n} \end{matrix}$

[P_ij] _m×n ⊖ [Q_ij] _m×n $\begin{matrix} = & [Max {P_{ij} + Q_{ij} - 1, 0}]_{m \times n} \\ = & [Max {Q_{ij} + P_{ij} - 1, 0}]_{m \times n} \\ = & [Q_{ij}]_{m \times n} ⊖ [P_{ij}]_{m \times n} \end{matrix}$ □

Proposition 3.14. (De Morgan’s Law) Let [P_ij] _m×n and [Q_ij] _m×n be two anti -l- FSGs matrices over X. Then

$[P_{ij} \oplus Q_{ij}]_{m \times n}^{c} = [P_{ij}]_{m \times n}^{c} ⊖ [Q_{ij}]_{m \times n}^{c}$ ,

$[P_{ij} ⊖ Q_{ij}]_{m \times n}^{c} = [P_{ij}]_{m \times n}^{c} \oplus [Q_{ij}]_{m \times n}^{c}$ .

Proof. Let [P_ij] _m×n and [Q_ij] _m×n be two -l- FSGs matrices over X. Then (i) $\begin{matrix} [P_{ij} \oplus Q_{ij}]_{m \times n} & = & [P_{ij}]_{m \times n} \oplus [Q_{ij}]_{m \times n} \\ = & [Min {P_{ij} + Q_{ij}, 1}]_{m \times n} \end{matrix}$ $[P_{ij} \oplus Q_{ij}]_{m \times n}^{c}$ $\begin{matrix} = & [Min {P_{ij} + Q_{ij}, 1}]_{m \times n}^{c} \\ = & [1 - Min {P_{ij} + Q_{ij}, 1}]_{m \times n} \\ = & [Max {1 - {P_{ij} + Q_{ij}}, 1 - 1]_{m \times n} \\ = & [Max {1 - {P_{ij} + Q_{ij}}, 0]_{m \times n} \\ = & [Max {1 - P_{ij} + 1 - Q_{ij} - 1}, 0]_{m \times n} \\ = & [Max {P_{ij}^{c} + Q_{ij}^{c} - 1}, 0]_{m \times n} \\ = & [P_{ij}]_{m \times n}^{c} ⊖ [Q_{ij}]_{m \times n}^{c} \end{matrix}$

Hence $[P_{ij} \oplus Q_{ij}]_{m \times n}^{c} = [P_{ij}]_{m \times n}^{c} ⊖ [Q_{ij}]_{m \times n}^{c}$

(ii) [P_ij ⊖ Q_ij] _m×n $\begin{matrix} = & [P_{ij}]_{m \times n} ⊖ [Q_{ij}]_{m \times n} \\ = & [Max {P_{ij} + Q_{ij} - 1, 0}]_{m \times n} \end{matrix}$

$[P_{ij} ⊖ Q_{ij}]_{m \times n}^{c}$ $\begin{matrix} = & [Max {P_{ij} + Q_{ij} - 1, 0}]_{m \times n}^{c} \\ = & [1 - Max {P_{ij} + Q_{ij} - 1, 0}]_{m \times n} \\ = & [Min {1 - {P_{ij} + Q_{ij} - 1}, 1 - 0}]_{m \times n} \\ = & [Min {1 - P_{ij} + 1 - Q_{ij}}, 0]_{m \times n} \\ = & [Min {P_{ij}^{c} + Q_{ij}^{c}}, 1]_{m \times n} \\ = & [P_{ij}]_{m \times n}^{c} \oplus [Q_{ij}]_{m \times n}^{c} \end{matrix}$

Hence $[P_{ij} ⊖ Q_{ij}]_{m \times n}^{c} = [P_{ij}]_{m \times n}^{c} \oplus [Q_{ij}]_{m \times n}^{c}$ .□

Proposition 3.15. Let [P_ij] _m×n, [Q_ij] _m×n and [R_ij] _m×n be three anti -l- FSG matrices over X. Then

[P_ij] _m×n ⊕ {[Q_ij] _m×n ∨ [R_ij] _m×n} = [P_ij ⊕ Q_ij] _m×n ∨ [P_ij ⊕ R_ij] _m×n,

[P_ij] _m×n ⊕ {[Q_ij] _m×n ∧ [R_ij] _m×n} = [P_ij ⊕ Q_ij] _m×n ∧ [P_ij ⊕ R_ij] _m×n,

[P_ij] _m×n ⊖ {[Q_ij] _m×n ∨ [R_ij] _m×n} = [P_ij ⊖ Q_ij] _m×n ∨ [P_ij ⊖ R_ij] _m×n,

[P_ij] _m×n ⊖ {[Q_ij] _m×n ∧ [R_ij] _m×n = [P_ij ⊖ Q_ij] _m×n ∧ [P_ij ⊖ R_ij] _m×n.

Proof. (i) [P_ij] _m×n ⊕ {[Q_ij] _m×n ∨ [R_ij] _m×n} $\begin{matrix} = & [P_{ij}]_{m \times n} \oplus [Q_{ij} \lor R_{ij}]_{m \times n} \\ = & [Min {P_{ij} + (Q_{ij} \lor R_{ij}), 1}]_{m \times n} \\ = & [Min {(P_{ij} + Q_{ij}) \lor (P_{ij} + R_{ij})}, 1]_{m \times n} \\ = & [Min {P_{ij} + Q_{ij}, 1}]_{m \times n} \lor \\ [Min {P_{ij} + R_{ij}, 1}]_{m \times n} \\ = & {[P_{ij}]_{m \times n} \oplus [Q_{ij}]_{m \times n}} \lor \\ {[P_{ij}]_{m \times n} \oplus [R_{ij}]_{m \times n}} \\ = & [P_{ij} \oplus Q_{ij}]_{m \times n} \lor [P_{ij} \oplus R_{ij}]_{m \times n} \end{matrix}$

Hence [P_ij] _m×n ⊕ {[Q_ij] _m×n ∨ [R_ij] _m×n} = [P_ij ⊕ Q_ij] _m×n ∨ [P_ij ⊕ R_ij] _m×n

Similarly we get (ii) [P_ij] _m×n ⊕ {[Q_ij] _m×n ∧ [R_ij] _m×n} = [P_ij ⊕ Q_ij] _m×n ∧ [P_ij ⊕ R_ij] _m×n

(iii) $\begin{array}{l} [_{P_{i j}] m \times n} θ {[_{Q_{i j}] m \times n} \lor [_{R_{i j}] m \times n}} \\ = [_{P_{i j}] m \times n} θ [_{Q_{i j} \lor R_{i j}] m \times n} \\ = [_{M a x {{(P_{i j} + Q_{i j}) \lor (P_{i j} + R_{i j})} - 1}, 0] m \times n} \\ = [_{M a x {P_{i j} + Q_{i j} - 1, 0}] m \times n} \lor [_{M a x {P_{i j} + R_{i j} - 1, 0}] m \times n} \\ = {[_{P_{i j}] m \times n} θ [_{Q_{i j}] m \times n}} \lor {[_{P_{i j}] m \times n} θ [_{R_{i j}] m \times n}} \\ = [_{P_{i j} θ Q_{i j}] m \times n} \lor [_{P_{i j} θ R_{i j}] m \times n} \end{array}$ Hence [P_ij] _m×n ⊖ {[Q_ij] _m×n ∨ [R_ij] _m×n} = [P_ij ⊖ Q_ij] _m×n ∨ [P_ij ⊖ R_ij] _m×n

Similarly we get (iv) [P_ij] _m×n ⊖ {[Q_ij] _m×n ∧ [R_ij] _m×n} = [P_ij ⊖ Q_ij] _m×n ∧ [P_ij ⊖ R_ij] _m×n □

Definition 3.16. Algebraic product of two anti - l-FSGs matrices [P_ij] _(m×n) and [Q_ij] _(m×n) over X is defined as [P_ij] • [Q_ij] = [P_ij. Q_ij] _m×n.

Proposition 3.17. Algebraic product of two anti- l-FSGs matrices is also an anti - l-FSGs matrix over X.

Example 3.18. From the example 3.9, Product of [P_ij] _3×3 and [Q_ij] _3×3 is [P_ij] _3×3 • [Q_ij] _3×3 $= {[\begin{matrix} 0.5 & 0.2 & 0.14 \\ 1 & 0.3 & 0.2 \\ 0.5 & 0.12 & 0.11 \end{matrix}]}_{3 \times 3} • {[\begin{matrix} 1 & 0.25 & 0.2 \\ 0.5 & 0.3 & 0.16 \\ 0.5 & 0.2 & 0.11 \end{matrix}]}_{3 \times 3}$ $= {[\begin{matrix} 0.5 / k 3^{2} & 0.05 / k 3^{5} & 0.028 / k 3^{7} \\ 0.5 / k 3^{2} & 0.09 / k 3^{3} & 0.032 / k 3^{6} \\ 0.25 / k 3^{2} & 0.024 / k 3^{8} & 0.0121 / k 3^{9} \end{matrix}]}_{3 \times 3}$ is also an anti - l-FSGs matrix over X.

Proposition 3.19. Let [P_ij] _m×n, [Q_ij] _m×n and [R_ij] _m×n be three anti -l- FSGs matrices over X. Then (i) [P_ij] _m×n • {[Q_ij] _m×n ∨ [R_ij] _{m × n}} = [P_ij. Q_ij] _m×n ∨ [P_ij. R_ij] _m×n, (ii) $\begin{array}{l} [_{P_{i j}] m \times n} • {[_{Q_{i j}] m \times n} \lor [_{R_{i j}] m \times n}} \\ = [_{P_{i j}] m \times n} • [_{Q_{i j} \lor R_{i j}] m \times n} \\ = [_{P_{i j}] m \times n} • [_{M a x {Q_{i j}, R_{i j}}] m \times n} \\ = [_{M a x {P_{i j}, Q_{i j}, P_{i j}, R_{i j}}] m \times n} \\ = [_{P_{i j}, Q_{i j}] m \times n} \lor [_{P_{i j}, R_{i j}] m \times n} \end{array}$ Hence [P_ij] _{m × n} • {[Q_ij] _m×n ∨ [R_ij] _m×n} = [P_ij. Q_ij] _m×n ∨ [P_ij. R_ij] _m×n. Similarly we get (ii) [P_ij] _m×n • {[Q_ij] _m×n ∧ [R_ij] _m×n} = [P_ij. Q_ij] _m×n ∧ [P_ij.R_ij] _m×n. □

4 Implementation of Anti-lattice Ordered Fuzzy Soft Groups and Its Matrix Operations in Deciding Process

Consider a set of people need to reach the place P8 from P1, they have the following six permissible routes {R1, R2, R3, R4, R5, R6} to arrive P8. According to [14], any routes with resting places(RP) form a group under the operation of ’arriving’. Consider the group X= {Rp, P1, P2, P3, P4, P5, P6, P7, P8} and the parameter E= {e₁= very minimum no. of people wanted(route), e₂= minimum no. people wanted, e₃= maximum no. of people wanted to go through P5, e₄= maximum no. of people wanted to go through P6, e₅= very maximum no. of people wanted, e₆= very very maximum no. of people wanted}. Then f_{e
₁} = {R1}, f_{e
₂} = {R2}, f_{e
₃} = {R3},f_{e
₄} = {R4}, f_{e
₅} = {R5}, f_{e
₆} = {R6}. Where,

R1 = {P1/0, P2/0.3,P3/0.2, P4/0.2, P5/ 0.1, P6/ 0.2, P7/0.5, P8/0.1, RP/1}

R2 = {P1/0, P2/0.3, P3/0.2, P4/0.2, P5/ 0.1, P6/ 0.2, P8/0.5, RP/1}

R3 = {P1/0, P2/0.3, P3/0.2, P4/0.2, P5/0.2, P8/0.5, RP/1}

R4 = {P1/0, P2/0.3, P3/0.2, P4/0.2, P6/0.2, P8/0.5, RP/1}

R5 = {P1/0, P2/0.3, P3/0.2, P4/0.2, P8/0.5, RP/1}

R6 = {P1/0, P2/0.3, P4/0.2, P8/0.5, RP/1}

Then (f, E) is defined as (f, E) = {R1, R2, R3, R4, R5, R6}. Here for each e₁ ∈ E, f_{e
₁} is a fuzzy subgroup over X and for each e₁, e₂ ∈ E, e₁ ≤ e₂ implies f_{e
₁} ⊇ f_{e
₂}. Hence (f, E) is an anti- l-FSG over X and we have the following lattice structure.

Fig. 1

Anti- l-FSG.

4.1 How to select the vehicle type

People can select the suitable vehicles according to the conditions of the roads by applying bounded sum and bounded difference operations in the following way.

Step: 1 Construct anti- l-FSG matrices [P_ij] and [Q_ij] by fixing the parameters ’VT1’ types of vehicles, ’VT2’ types of vehicles, ’VT3’ types of vehicles, ’VT4’ types of vehicles and ’VT5’ types of vehicles(Collecting data from two sources). $\begin{array}{l} [_{P_{i j}] 5 \times 6} = \\ \begin{matrix} V T 1 \\ V T 2 \\ V T 3 \\ V T 4 \\ V T 5 \end{matrix} (\begin{matrix} \begin{matrix} R 1 \\ 0.60 \end{matrix} & \begin{matrix} R 2 \\ 0.50 \end{matrix} & \begin{matrix} R 3 \\ 0.15 \end{matrix} & \begin{matrix} R 4 \\ 0.80 \end{matrix} & \begin{matrix} R 5 \\ 0.40 \end{matrix} & \begin{matrix} R 6 \\ 0.75 \end{matrix} \\ 0.70 & 0.55 & 0.20 & 0.65 & 0.75 & 0.85 \\ 0.40 & 0.30 & 0.58 & 0.79 & 0.74 & 0.64 \\ 0.55 & 0.62 & 0.725 & 0.64 & 0.54 & 0.83 \\ 0.65 & 0.72 & 0.75 & 0.64 & 0.49 & 0.63 \end{matrix}) \end{array}$ $\begin{array}{l} [_{Q_{i j}] 5 \times 6} = \\ \begin{matrix} V T 1 \\ V T 2 \\ V T 3 \\ V T 4 \\ V T 5 \end{matrix} (\begin{matrix} \begin{matrix} R 1 \\ 0.56 \end{matrix} & \begin{matrix} R 2 \\ 0.65 \end{matrix} & \begin{matrix} R 3 \\ 0.55 \end{matrix} & \begin{matrix} R 4 \\ 0.88 \end{matrix} & \begin{matrix} R 5 \\ 0.50 \end{matrix} & \begin{matrix} R 6 \\ 0.85 \end{matrix} \\ 0.77 & 0.65 & 0.30 & 0.765 & 0.65 & 0.985 \\ 0.54 & 0.50 & 0.68 & 0.79 & 0.84 & 0.74 \\ 0.60 & 0.72 & 0.85 & 0.74 & 0.64 & 0.73 \\ 0.55 & 0.62 & 0.85 & 0.764 & 0.649 & 0.67 \end{matrix}) \end{array}$

Step: 2

To compute bounded sum of two anti-l-FSG matrices. $\begin{array}{l} [_{P_{i j}] 5 \times 6} \oplus [_{Q_{i j}] 5 \times 6} = \\ \begin{matrix} V T 1 \\ V T 2 \\ V T 3 \\ V T 4 \\ V T 5 \end{matrix} (\begin{matrix} \begin{matrix} R 1 \\ 1 \end{matrix} & \begin{matrix} R 2 \\ 1 \end{matrix} & \begin{matrix} R 3 \\ 0.70 \end{matrix} & \begin{matrix} R 4 \\ 1 \end{matrix} & \begin{matrix} R 5 \\ 0.90 \end{matrix} & \begin{matrix} R 6 \\ 1 \end{matrix} \\ 1 & 1 & 0.50 & 1 & 1 & 1 \\ 0.94 & 0.80 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}) \end{array}$

From this bounded sum matrix,

vehicle type VT1 is not apt for R3 and R5 routes,

vehicle type VT2 is not appropriate for R3 route and

vehicle type VT3 is not fit for R1 and R2 routes.

Step:3

To compute bounded difference of two anti-l-FSG matrices. $\begin{array}{l} [_{P_{i j}] 5 \times 6} \oplus [_{Q_{i j}] 5 \times 6} = \\ \begin{matrix} V T 1 \\ V T 2 \\ V T 3 \\ V T 4 \\ V T 5 \end{matrix} (\begin{matrix} \begin{matrix} R 1 \\ 0.16 \end{matrix} & \begin{matrix} R 2 \\ 0.15 \end{matrix} & \begin{matrix} R 3 \\ 0 \end{matrix} & \begin{matrix} R 4 \\ 0.68 \end{matrix} & \begin{matrix} R 5 \\ 0 \end{matrix} & \begin{matrix} R 6 \\ 0.60 \end{matrix} \\ 0.47 & 0.20 & 0 & 0.42 & 0.40 & 0.84 \\ 0 & 0 & 0.26 & 0.58 & 0.58 & 0.38 \\ 0.15 & 0.34 & 0.58 & 0.38 & 0.18 & 0.56 \\ 0.20 & 0.34 & 0.60 & 0.40 & 0.14 & 0.30 \end{matrix}) \end{array}$

From this bounded difference matrix,

vehicle type VT1 is apt for R4 route,

vehicle type VT2 is appropriate for R1 and R6 routes,

vehicle type VT3 is apt for R5 route,

vehicle type VT4 is appropriate for R2 route and

vehicle type VT5 is apt for R2 and R3 routes.

Since the highest bounded difference value is 0.835, VT2 is the best appropriate vehicle type for the route R5. So people can elect VT2 vehicle type on the R5 for their comfortable sophisticated journey.

5 Conclusion

In this paper, the notion of anti-l-FSG and anti-l-FSG matrices are presented. The bounded sum of anti-l-FSG matrices, bounded difference of anti-l-FSG matrices, algebraic product of anti-l-FSG matrices and their related properties are discussed. Also De Morgan’s laws are proved with respect to the operations of bounded sum and bounded difference. Finally, bounded sum and bounded difference operations of anti-l-FSG matrices have been implemented victoriously in deciding process. In future work, one could study the different properties and operations of anti-l-FSGs matrices and could find their applications in wide area.

Footnotes

Acknowledgements

Authors are very grateful to the referees for their valuable comments and suggestions to improve this work. This work is original and this paper has not been submitted elsewhere.

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