ISSN: 2005-4238 IJAST 116
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The Role of (𝜶, 𝜷)- Level set on Complex Intuitionistic Fuzzy Soft Lattice Ordered Groups
S. Rajareegaa, J.Vimalab, D.Preethic
𝑎,𝑏,∗,𝑐 Department of Mathematics, Alagappa University, Karaikudi, Tamilnadu, India.
∗ Corresponding Author
Abstract:In this paper the notion of Complex Intuitionistic Fuzzy Soft Lattice Ordered Group is introduced.Moreover,the properties of (𝛼, 𝛽)- level sets of complex intuitionistic fuzzy soft lattice ordered group are presented and discussed.
Keywords: Fuzzy lattice ordered group, soft group, Soft lattice, Complex fuzzy sets, Complex intuitionistic fuzzy soft sets, (𝛼, 𝛽)- Level sets
MSC (2010): 06D72
1 Introduction
Fuzzy set theory [15], Intuitionistic fuzzy set theory [3], soft set theory [9] and Intuitionisticfuzzy soft set theory [8] are the general mathematical tools used to handle the uncertainties,vagueness and imprecision. But all of these theories posed a difficulty to handle the periodicity problems and which are impossible to address in one dimensional grade of membership. Toexceed these Ramot [11] introduced the new concept of complex fuzzy set.
Alkouri A.U.M [1]established the notion of complex intuitionistic fuzzy sets and Kumar T. [7]
identifiedthe distance measures and entropies on complex intuitionistic fuzzy soft sets.
SelvachandrenG. [10] found the notion of complex intuitionistic fuzzy soft group.
In [2], Aktas H. apply the notion of soft sets to group theory. Serife [13] introduced the connection between soft set theory and lattice theory. SatyaSaibaba [12] and Vimala J.
[14]initiate the concept of fuzzy lattice ordered group in a different manner.
In this paper we introduced the notion of complex intuitionistic fuzzy soft latticeordered group (𝒞ℐℱ𝒮ℒ − 𝒢). The properties of (𝛼, 𝛽)- level sets of 𝒞ℐℱ𝒮ℒ − 𝒢are presented and also identified some results of(𝛼, 𝛽)- level sets of 𝒞ℐℱ𝒮ℒ − 𝒢based on soft group and soft lattice structures.
2 Preliminaries
In this section, we summarize some of the important perliminaries pertaining to the development of the work.
Definition 2.1 [15] The fuzzy set X over a set H is a set 𝑋 = {< ℎ, 𝜇(ℎ) > |ℎ ∈ 𝐻} where
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𝜇: 𝐻 ⟶ [0,1] which is called the membership function of X and 𝜇(ℎ) is called the membership value of H in X.
Definition 2.2 [5,6] A Lattice ordered group is a system G = (𝐺, +, ≤) where (1) (G, +) is a group
(2) (G, ≤) is a lattice
(3) 𝑥 ≤ 𝑦 ⇒ 𝑎 + 𝑥 + 𝑦 ≤ 𝑎 + 𝑦 + 𝑏 for all 𝑎, 𝑏, 𝑥, 𝑦 ∈ 𝐺.
Definition 2.3 [12] A fuzzy subset 𝜆 of a lattice-ordered group G is said to be a fuzzy lattice- ordered subgroup or briefly, fuzzy l-subgroup if
(1) 𝜆(𝑥𝑦) ≥ 𝜆(𝑥) ∧ 𝜆(𝑦) (2) 𝜆(𝑥−1) ≥ 𝜆(𝑥)
(3) 𝜆(𝑥 ∨ 𝑦) ≥ 𝜆(𝑥) ∧ 𝜆(𝑦)
(4) 𝜆(𝑥 ∧ 𝑦) ≥ 𝜆 𝑥 ∧ 𝜆 𝑦 forall 𝑥, 𝑦 ∈ 𝐺.
Definition 2.4 [2] Let (F,A) be a soft set over G. Then (𝐹, 𝐴) is said to be a soft group over the group G if and only if 𝐹(𝑥) < 𝐺 for all 𝑥 ∈ 𝐴.
Definition 2.5 [2] Let (𝐹, 𝐴) and (𝐺, 𝐵) be two soft groups over the group G. Then (𝐺, 𝐵) is a soft subgroup of (𝐹, 𝐴), written (𝐺, 𝐵) < (𝐹, 𝐴) if
(1) 𝐵 ⊂ 𝐴
(2) 𝐺(𝑥) < 𝐹(𝑥) for all 𝑥 ∈ 𝐵.
Definition 2.6 [13] Let (𝐹, 𝐴) be a soft set over the lattice L. Then (𝐹, 𝐴) is called a soft lattice over L if 𝐹(𝑥) is a sublattice of L for all 𝑥 ∈ 𝐴
Definition 2.7 Let (𝐹, 𝐴) and (𝐺, 𝐵) be two soft lattices over the lattice L. Then (𝐺, 𝐵) is a soft sublattice of (𝐹, 𝐴), written (𝐺, 𝐵) ⊆ (𝐹, 𝐴) if
(1) 𝐵 ⊂ 𝐴
(2) 𝐺(𝑥) 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑙𝑎𝑡𝑡𝑖𝑐𝑒 𝐹(𝑥) for all 𝑥 ∈ 𝐵.
Definition 2.8 [11] A complex fuzzy set A defined on a universe of discourse U is characterized by a membership function 𝜇𝐴(𝑥) that assigns a complex-valued grade of membership in A to any element 𝑥 ∈ 𝑈. By definition, all values of 𝜇𝐴(𝑥) lie within the unit circle in the complex plane and are expressed by 𝜇𝐴(𝑥) = 𝑟𝐴(𝑥)𝑒𝑖𝑤𝐴(𝑥), where 𝑖 = −1, 𝑟𝐴(𝑥) and 𝑤𝐴(𝑥) are both real-valued, 𝑟𝐴(𝑥) ∈ (0,2𝜋]. A complex fuzzy set A is thus of the form
𝐴 = {< 𝑥, 𝜇𝐴(𝑥) >: 𝑥 ∈ 𝑈} = {< 𝑥, 𝑟𝐴(𝑥)𝑒𝑖𝑤𝐴(𝑥)>: 𝑥 ∈ 𝑈}
Definition 2.9 [1] A complex intuitionistic fuzzy set S, defined on a universe of discourse U, is characterized by membership and non-membership functions 𝜇𝑆(𝑥) and 𝜈𝑆(𝑥) respectively, that assign any element 𝑥 ∈ 𝑈 a complex-valued grade of both membership and non-membership in S. By definition, the values of 𝜇𝑆(𝑥), 𝜈𝑆(𝑥), and their sum may receive all lying within the unit circle in the complex plane, and are on the form 𝜇𝑆(𝑥) = 𝑟𝑆(𝑥)𝑒𝑖𝑤𝜇 𝑆(𝑥) for membership function, 𝜈𝑆(𝑥) = 𝑘𝑆(𝑥)𝑒𝑖𝑤𝜈 𝑆(𝑥) for non-membership function, where 𝑖 = −1 each of
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𝑟𝑆(𝑥)and 𝑘𝑆(𝑥) are real valued and both belong to the interval [0.1] such that 0 ≤ 𝑟𝑆(𝑥) + 𝑘𝑆(𝑥) ≤ 1 also, 𝑤𝜇𝑆(𝑥) and 𝑤𝜈𝑆(𝑥) are real valued. We represent the Complex intuitionistic fuzzy set S as, 𝑆 = {< 𝑥, 𝜇𝑆(𝑥) = 𝑎, 𝜈𝑆(𝑥) = 𝑎′ >: 𝑥 ∈ 𝑈}
where 𝜇: 𝑈 ⟶ {𝑎|𝑎 ∈ 𝐶, |𝑎| ≤ 1}, 𝜈: 𝑈 ⟶ {𝑎′|𝑎′ ∈ 𝐶, |𝑎′| ≤ 1} and |𝜇𝑆(𝑥) + 𝜈𝑆(𝑥)| ≤ 1.
Definition 2.10 [7] Let E be a set of parameters, CIFS(U) denote the collection of all complex intuitionistic fuzzy sets on U, and 𝐹 be a function from E to CIFS(U). Then the set of ordered pairs {(𝜀, 𝐹 (𝜀)): 𝜀 ∈ 𝐸, 𝐹 (𝜀) ∈ 𝐶𝐼𝐹𝑆(𝑈)}, denoted by (𝐹 , 𝐸), is called a complex intuitionistic fuzzy soft set (CIFSS) on U.
Definition 2.11 [7] Suppose that (𝐹 , 𝐸) and (𝐺 , 𝐸) are two complex intuitionistic fuzzy soft sets over a universal set U. Then (𝐹 , 𝐸) ⊆ (𝐺 , 𝐸) if and only if 𝜇𝐹 (𝜀)(𝑥) ≤ 𝜇𝐺 (𝜀)(𝑥)and 𝜈𝐹 (𝜀)(𝑥) ≥ 𝜈𝐺 (𝜀)(𝑥) , ∀𝑥 ∈ 𝑈, 𝜀 ∈ 𝐸
Definition 2.12 [10] Let (𝐹 , 𝐸) ∈ 𝐶𝐼𝐹𝑆𝑆(𝑈) and 𝛼, 𝛽 ∈ 𝒪1. The (𝛼, 𝛽)- level set of (𝐹 , 𝐸), denoted as (𝐹 , 𝐸)(𝛼,𝛽 ), is a soft set on U defined below:
(𝐹 , 𝐸)(𝛼 ,𝛽)= {(𝑎, 𝐹 (𝛼,𝛽)(𝑎))| 𝑎 ∈ 𝐸, 𝐹 (𝛼 ,𝛽)(𝑎) ∈ 𝑃(𝑈)},
where 𝐹 (𝛼,𝛽 )(𝑎) = {𝑥 ∈ 𝑈 |𝜇𝐹 (𝑎)(𝑥) ≥ 𝛼, 𝜈𝐹 (𝑎)(𝑥) ≤ 𝛽} for all 𝑎 ∈ 𝐸 and the symbol 𝒪1 will denote {𝑥 ∈ ℂ||𝑥| ≤ 1}
3 Complex Intuitionistic Fuzzy Soft Lattice Ordered Group
In this section, we introduce the notion of Complex Intuitionistic Fuzzy Soft Lattice Ordered Group which is associated with complex intuitionistic fuzzy soft sets and lattice ordered groups.
Throughout this paper G denotes the lattice ordered group and 𝑖)𝜇 ≥ 𝜈, if both 𝑟 ≥ 𝜏 and 𝑤 ≥ 𝜓 and
𝑖𝑖)𝜇 ≤ 𝜈, if both 𝑟 ≤ 𝜏 and 𝑤 ≤ 𝜓, where 𝜇 = 𝑟𝑒𝑖𝑤 and 𝜈 = 𝜏𝑒𝑖𝜓, with 𝑟, 𝜏 ∈ [0,1] and 𝑤, 𝜓 ∈ (0,2𝜋] [10].
Definition 3.1 Let G be a lattice ordered group and Let 𝒩 = {< 𝑥, 𝜇𝒩(𝑥), 𝜈𝒩(𝑥) > |𝑥 ∈ 𝐺} be a CIFS on G. Then 𝒩 is said to be a complex intuitionistic fuzzy lattice ordered subgroup
(𝒞ℐℱℒ − 𝑆𝑢𝑏𝑔𝑟𝑜𝑢𝑝)of G, if the following conditions holds for all x, y ∈ 𝐺:
1. 𝜇𝒩(𝑥𝑦) ≥ 𝜇𝒩(𝑥) ∧ 𝜇𝒩(𝑦) 2. 𝜈𝒩(𝑥𝑦) ≤ 𝜈𝒩(𝑥) ∨ 𝜈𝒩(𝑦) 3. 𝜇𝒩(𝑥−1) ≥ 𝜇𝒩(𝑥)
4. 𝜈𝒩(𝑥−1) ≤ 𝜈𝒩(𝑥)
5. 𝜇𝒩(𝑥 ∨ 𝑦) ≥ 𝜇𝒩(𝑥) ∧ 𝜇𝒩(𝑦) 6. 𝜇𝒩(𝑥 ∧ 𝑦) ≥ 𝜇𝒩(𝑥) ∧ 𝜇𝒩(𝑦) 7. 𝜈𝒩(𝑥 ∨ 𝑦) ≤ 𝜈𝒩(𝑥) ∨ 𝜈𝒩(𝑦) 8. 𝜈𝒩(𝑥 ∧ 𝑦) ≤ 𝜈𝒩(𝑥) ∨ 𝜈𝒩(𝑦).
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Definition 3.2 Let (𝐹 , 𝐸) ∈ 𝐶𝐼𝐹𝑆𝑆(𝐺). Then (𝐹 , 𝐸) is said to be a complex intuitionistic fuzzy soft lattice ordered group (𝒞ℐℱ𝒮ℒ − 𝒢) on G if 𝐹 (𝑎) is a complex intuitionistic fuzzy lattice ordered subgroup of G for all 𝑎 ∈ 𝜌(𝐹 , 𝐸), Where 𝜌(𝐹 , 𝐸)= 𝑎 ∈ 𝐸 𝐹 𝑎 is non null}
is the Support set of 𝐹 , 𝐸 by [10].
Example 3.3 Consider the 𝑙 − 𝑔𝑟𝑜𝑢𝑝𝐺 = (ℤ, +,∧,∨), and the parameters 𝐸 = {𝑎, 𝑏}.
Next Consider the two CIFSS of G, which are defined as follows:
(i) (𝐹 , 𝐸) = {< 𝑥, 𝜇𝐹 (𝑎)(𝑥), 𝜈𝐹 (𝑎)(𝑥) >, < 𝑥, 𝜇𝐹 (𝑏)(𝑥), 𝜈𝐹 (𝑏)(𝑥) >: 𝑥 ∈ 𝐺}, where
𝜇𝐹 𝑎 𝑥 =
0.7𝑒𝑖2𝜋(0.5), if x = 2k; k ∈ ℤ/{0}
0.9𝑒𝑖2𝜋(0.7), x = 0 0.3𝑒𝑖2𝜋(0.4), otherwise
,
𝜈𝐹 (𝑎)(𝑥) =
0.3𝑒𝑖2𝜋(0.5), if x = 2k; k ∈ ℤ/{0}
0.1𝑒𝑖2𝜋(0.3), x = 0 0.7𝑒𝑖2𝜋(0.6), otherwise
and
𝜇𝐹 𝑏 𝑥 =
0.5𝑒𝑖2𝜋 0.3 , if x = 2k; k ∈ ℤ/{0}
0.8𝑒𝑖2𝜋 0.9 , x = 0 0.3𝑒𝑖2𝜋 0.3 , otherwise
,
𝜈𝐹 (𝑏)(𝑥) =
0.5𝑒𝑖2𝜋 0.7 , if x = 2k; k ∈ ℤ/{0}
0.2𝑒𝑖2𝜋 0.1 , x = 0 0.7𝑒𝑖2𝜋 0.7 , otherwise
(ii) (𝐺 , 𝐸) = {< 𝑥, 𝜇𝐺 (𝑎)(𝑥), 𝜈𝐺 (𝑎)(𝑥) >, < 𝑥, 𝜇𝐺 (𝑏)(𝑥), 𝜈𝐺 (𝑏)(𝑥) >: 𝑥 ∈ 𝐺}, Where
𝜇𝐺 𝑎 𝑥 =
0.7𝑒𝑖2𝜋 0.9 , if x = 2k; k ∈ ℤ/{0}
0.8𝑒𝑖2𝜋 0.8 , x = 0 0.6𝑒𝑖2𝜋 0.7 , otherwise
,
𝜈𝐺 (𝑎)(𝑥) =
0.3𝑒𝑖2𝜋 0.1 , if x = 2k; k ∈ ℤ/{0}
0.2𝑒𝑖2𝜋 0.2 , x = 0 0.4𝑒𝑖2𝜋 0.3 , otherwise
and 𝐺 (𝑏) = 𝐹 (𝑏)
From the above 𝐹 , 𝐸 ∈ 𝒞ℐℱ𝒮ℒ − 𝒢(𝐺), Whereas 𝐺 , 𝐸 ∉ 𝒞ℐℱ𝒮ℒ − 𝒢(𝐺)
Proposition 3.4 Let 𝐹 , 𝐸 ∈ 𝒞ℐℱ𝒮ℒ − 𝒢 over G. (𝐹 , 𝐸)(𝛼1,𝛽1) and (𝐹 , 𝐸)(𝛼2,𝛽2) are (𝛼1, 𝛽1) and (𝛼2, 𝛽2) - level soft sets of (𝐹 , 𝐸), respectively, where 𝛼1, 𝛼2, 𝛽1, 𝛽2 ∈ 𝒪1. If 𝛼1 ≥ 𝛼2 and
𝛽2 ≥ 𝛽1, then we have (𝐹 , 𝐸)(𝛼1,𝛽1) ⊆ (𝐹 , 𝐸)(𝛼2,𝛽2) Proof. By Definition 2.12,
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(𝐹 , 𝐸)(𝛼1,𝛽1) = {(𝑎, 𝐹 (𝛼1,𝛽1)(𝑎))| 𝑎 ∈ 𝐸, 𝐹 (𝛼1,𝛽1)(𝑎) ∈ 𝑃(𝐺)}, where(𝑎) = {𝑥 ∈ 𝐺 |𝜇𝐹 (𝑎)(𝑥) ≥ 𝛼1 𝑎𝑛𝑑 𝜈𝐹 (𝑎)(𝑥) ≤ 𝛽1}
∀𝑎 ∈ 𝐸, 𝛼1, 𝛽1 ∈ 𝒪1 and
(𝐹 , 𝐸)(𝛼2,𝛽2)= {(𝑎, 𝐹 (𝛼2,𝛽2)(𝑎))| 𝑎 ∈ 𝐸, 𝐹 (𝛼2,𝛽2)(𝑎) ∈ 𝑃(𝐺)},
where𝐹 (𝛼2,𝛽2)(𝑎) = {𝑥 ∈ 𝐺 |𝜇𝐹 (𝑎)(𝑥) ≥ 𝛼2 𝑎𝑛𝑑 𝜈𝐹 (𝑎)(𝑥) ≤ 𝛽2} ∀𝑎 ∈ 𝐸, 𝛼2, 𝛽2 ∈ 𝒪1.
Obviously 𝐸 ⊆ 𝐸. We have to prove that ∀𝑎 ∈ 𝐸, 𝐹 (𝛼1,𝛽1)(𝑎) ⊆ 𝐹 (𝛼2,𝛽2)(𝑎).
Since 𝛼1 ≥ 𝛼2 and 𝛽2 ≥ 𝛽1, and let 𝛼𝑗 = 𝑟𝛼𝑗𝑒𝑖𝑤𝛼 𝑗 𝑎𝑛𝑑 𝛽𝑗 = 𝑟𝛽𝑗𝑒𝑖𝑤𝛽 𝑗, 𝑤ℎ𝑒𝑟𝑒 𝑟𝛼𝑗, 𝑟𝛽𝑗 ∈ [0,1] 𝑎𝑛𝑑 𝑤𝛼𝑗, 𝑤𝛽𝑗 ∈ (0,2𝜋], 𝑗 = 1,2,
then ∀𝑎 ∈ 𝐸, 𝑥 ∈ 𝐺 𝑟𝐹 𝑎 𝑥 ≥ 𝑟𝛼2, 𝑤𝐹 𝑎 𝑥 ≥ 𝑤𝛼2 and
𝜏𝐹 𝑎 𝑥 ≤ 𝑟𝛽2, 𝜓𝐹 𝑎 (𝑥) ≤ 𝑤𝛽2} ⊇ {𝑥 ∈ 𝐺 |𝑟𝐹 (𝑎)(𝑥) ≥ 𝑟𝛼1, 𝑤𝐹 (𝑎)(𝑥) ≥ 𝑤𝛼1and 𝜏𝐹 (𝑎)(𝑥) ≤ 𝑟𝛽1, 𝜓𝐹 (𝑎)(𝑥) ≤ 𝑤𝛽1}
{𝑥 ∈ 𝐺 |𝜇𝐹 (𝑎)(𝑥) ≥ 𝛼2 𝑎𝑛𝑑 𝜈𝐹 (𝑎)(𝑥) ≤ 𝛽2} ⊇ {𝑥 ∈ 𝐺 |𝜇𝐹 (𝑎)(𝑥) ≥ 𝛼1 𝑎𝑛𝑑 𝜈𝐹 (𝑎)(𝑥) ≤ 𝛽1}thus ∀𝑎 ∈ 𝐸, 𝐹 (𝛼2,𝛽2)(𝑎) ⊇ 𝐹 (𝛼1,𝛽1)(𝑎).
Hence (𝐹 , 𝐸)(𝛼1,𝛽1) ⊆ (𝐹 , 𝐸)(𝛼2,𝛽2).
Proposition 3.5 Let (𝐹 1, 𝐸) and 𝐹 2, 𝐸 ∈ 𝒞ℐℱ𝒮ℒ − 𝒢 over G. (𝐹 1, 𝐸)(𝛼,𝛽) and (𝐹 2, 𝐸)(𝛼,𝛽) are (𝛼, 𝛽)- level soft sets of (𝐹 1, 𝐸) and (𝐹 2, 𝐸), respectively, where 𝛼, 𝛽 ∈ 𝒪1. If (𝐹 1, 𝐸) ⊆ (𝐹 2, 𝐸), then we have (𝐹 1, 𝐸)(𝛼 ,𝛽 ) ⊆ (𝐹 2, 𝐸)(𝛼,𝛽)
Proof. By Defnition 2.12,
(𝐹 1, 𝐸)(𝛼,𝛽 ) = {(𝑎, 𝐹 1(𝛼 ,𝛽 )(𝑎))| 𝑎 ∈ 𝐸, 𝐹 1(𝛼 ,𝛽 )(𝑎) ∈ 𝑃(𝐺)},
where 𝐹 1(𝛼 ,𝛽 )(𝑎) = {𝑥 ∈ 𝐺 |𝜇𝐹 1(𝑎)(𝑥) ≥ 𝛼 𝑎𝑛𝑑 𝜈𝐹 1(𝑎)(𝑥) ≤ 𝛽} and (𝐹 2, 𝐸)(𝛼,𝛽) = {(𝑎, 𝐹 2(𝛼 ,𝛽 )(𝑎))| 𝑎 ∈ 𝐸, 𝐹 2(𝛼 ,𝛽 )(𝑎) ∈ 𝑃(𝐺)},
where 𝐹 2(𝛼 ,𝛽 )(𝑎) = {𝑥 ∈ 𝐺 |𝜇𝐹 2(𝑎)(𝑥) ≥ 𝛼 𝑎𝑛𝑑 𝜈𝐹 2(𝑎)(𝑥) ≤ 𝛽} , ∀𝑎 ∈ 𝐸, 𝛼, 𝛽 ∈ 𝒪1. Obviously, 𝐸 ⊆ 𝐸. We have to prove that 𝐹 1(𝛼 ,𝛽 )(𝑎)⊆ 𝐹 2(𝛼 ,𝛽 )(𝑎)
Since (𝐹 1, 𝐸) ⊆ (𝐹 2, 𝐸), By Definition 2.11., ∀𝑥 ∈ 𝐺, 𝑎 ∈ 𝐸
𝑟𝐹 1 𝑎 𝑥 ≤ 𝑟𝐹 2 𝑎 𝑥 , 𝑤𝐹 1 𝑎 𝑥 ≤ 𝑤𝐹 2 𝑎 𝑥 𝑎𝑛𝑑 𝜏𝐹 1 𝑎 𝑥 ≥ 𝜏𝐹 2 𝑎 𝑥 , 𝜓𝐹 1 𝑎 𝑥 ≥ 𝜓𝐹 2 𝑎 𝑥 (1)
Assume that 𝑥 ∈ 𝐹 1(𝛼 ,𝛽 )(𝑎), then we have
𝑟𝛼 ≤ 𝑟𝐹 1 𝑎 𝑥 , 𝑤𝛼 ≤ 𝑤𝐹 1 𝑎 𝑥 𝑎𝑛𝑑 𝑟𝛽 ≥ 𝜏𝐹 1 𝑎 𝑥 , 𝑤𝛽 ≥ 𝜓𝐹 1 𝑎 𝑥 (2) 𝑤ℎ𝑒𝑟𝑒 𝛼 = 𝑟𝛼𝑒𝑖𝑤𝛼 𝑎𝑛𝑑 𝛽 = 𝑟𝛽𝑒𝑖𝑤𝛽, ∀𝑟𝛼, 𝑟𝛽 ∈ [0,1], 𝑤𝛼, 𝑤𝛽 ∈ (0,2𝜋]
From (1) and (2),
𝑟𝐹 2(𝑎)(𝑥) ≥ 𝑟𝛼, 𝑤𝐹 2(𝑎)(𝑥) ≥ 𝑤𝛼 𝑎𝑛𝑑 𝜏𝐹 2(𝑎)(𝑥) ≤ 𝑟𝛽, 𝜓𝐹 2(𝑎)(𝑥) ≤ 𝑤𝛽. Hence 𝑥 ∈ 𝐹 2(𝛼 ,𝛽 )(𝑎), ∀𝑎 ∈ 𝐸.
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Thus, we have 𝐹 1(𝛼 ,𝛽 )(𝑎) ⊆ 𝐹 2(𝛼 ,𝛽 )(𝑎), this implies (𝐹 1, 𝐸)(𝛼 ,𝛽) ⊆ (𝐹 2, 𝐸)(𝛼,𝛽)
Proposition 3.6 Let 𝐹 , 𝐸 ∈ 𝐶𝐼𝐹𝑆𝑆(𝐺) where (𝐹 , 𝐸) is non- null. Then the following are equivalent
(1) 𝐹 , 𝐸 ∈ 𝒞ℐℱ𝒮ℒ − 𝒢(𝐺).
(2) For all 𝑎 ∈ 𝐸 and for arbitrary 𝛼, 𝛽 ∈ 𝒪1 with 𝐹 (𝛼,𝛽)(𝑎) ≠ ∅, (𝐹 , 𝐸)(𝛼 ,𝛽) is both soft group as well as soft lattice.
Proof.(1) ⇒ (2) Let 𝐹 , 𝐸 ∈ 𝒞ℐℱ𝒮ℒ − 𝒢(𝐺) and 𝐹 , 𝐸) 𝛼,𝛽 = 𝑎, 𝐹 𝛼,𝛽 𝑎 𝑎 ∈
𝐸, 𝐹 𝛼,𝛽 𝑎 ∈ 𝑃 𝑈 , where 𝐹 (𝛼,𝛽)(𝑎) = {𝑥 ∈ 𝑈 |𝜇𝐹 (𝑎)(𝑥) ≥ 𝛼, 𝜈𝐹 (𝑎)(𝑥) ≤ 𝛽} for all 𝑎 ∈ 𝐸.
Then for all 𝑎 ∈ 𝐸 and for arbitrary 𝛼, 𝛽 ∈ 𝒪1 with 𝐹 (𝛼,𝛽)(𝑎) ≠ ∅, let 𝑥, 𝑦 ∈ 𝐹 (𝛼,𝛽 )(𝑎).
Now, 𝜇𝐹 (𝑎)(𝑥𝑦−1) ≥ 𝜇𝐹 (𝑎)(𝑥) ∧ 𝜇𝐹 (𝑎)(𝑦) ≥ 𝛼 ∧ 𝛼 = 𝛼 and 𝜈𝐹 (𝑎)(𝑥𝑦−1) ≤ 𝜈𝐹 (𝑎)(𝑥) ∨ 𝜈𝐹 (𝑎)(𝑦) ≤ 𝛽
⇒ 𝑥𝑦−1 ∈ 𝐹 (𝛼,𝛽)(𝑎), ∴ (𝐹 , 𝐸)(𝛼 ,𝛽) is a soft group.
Similarly it is easy to verify 𝑥 ∧ 𝑦 & 𝑥 ∨ 𝑦 ∈ 𝐹 (𝛼,𝛽 )(𝑎), this implies (𝐹 , 𝐸)(𝛼,𝛽) is a soft lattice.
(2) ⇒ (1) Let us assume that (𝐹 , 𝐸)(𝛼,𝛽) is both soft group as well as soft lattice.
Let 𝑎 ∈ 𝐸 and 𝑥, 𝑦 ∈ 𝐺.Take 𝜇𝐹 (𝑎)(𝑥) = 𝛼1, 𝜇𝐹 (𝑎)(𝑦) = 𝛼2 and 𝜈𝐹 (𝑎)(𝑥) = 𝛽1, 𝜈𝐹 (𝑎)(𝑥) = 𝛽2, for all 𝛼1, 𝛼2, 𝛽1 𝑎𝑛𝑑 𝛽2 ∈ 𝒪1. Then 𝑥 ∈ 𝐹 (𝛼1,𝛽1)(𝑎), 𝑦 ∈ 𝐹 (𝛼2,𝛽2)(𝑎)
Let us assume that 𝛼1 < 𝛼2 and 𝛽1 > 𝛽2. Then by Proposition 3.5.
(𝐹 , 𝐸)(𝛼2,𝛽2)⊆ (𝐹 , 𝐸)(𝛼1,𝛽1), so 𝑦 ∈ 𝐹 (𝛼1,𝛽1)(𝑎). Thus 𝑥, 𝑦 ∈ 𝐹 (𝛼1,𝛽1)(𝑎).
Since (𝐹 , 𝐸)(𝛼,𝛽 ) is both soft group as well as soft lattice, by hypothesis 𝑥𝑦−1, 𝑥 ∧ 𝑦 & 𝑥 ∨ 𝑦 ∈ 𝐹 (𝛼,𝛽 )(𝑎), implies that
𝜇𝐹 (𝑎)(𝑥𝑦−1) ≥ 𝛼1 = 𝜇𝐹 (𝑎)(𝑥) ∧ 𝜇𝐹 (𝑎)(𝑦) and 𝜈𝐹 (𝑎)(𝑥𝑦−1) ≤ 𝛽1 = 𝜈𝐹 (𝑎)(𝑥) ∨ 𝜈𝐹 (𝑎)(𝑦) 𝜇𝐹 (𝑎)(𝑥 ∨ 𝑦) ≥ 𝛼1 = 𝜇𝐹 (𝑎)(𝑥) ∧ 𝜇𝐹 (𝑎)(𝑦) and 𝜈𝐹 (𝑎)(𝑥 ∨ 𝑦) ≤ 𝛽1 = 𝜈𝐹 (𝑎)(𝑥) ∨ 𝜈𝐹 (𝑎)(𝑦) 𝜇𝐹 (𝑎)(𝑥 ∧ 𝑦) ≥ 𝛼1 = 𝜇𝐹 (𝑎)(𝑥) ∧ 𝜇𝐹 (𝑎)(𝑦) and 𝜈𝐹 (𝑎)(𝑥 ∧ 𝑦) ≤ 𝛽1 = 𝜈𝐹 (𝑎)(𝑥) ∨ 𝜈𝐹 (𝑎)(𝑦) Therefore 𝐹 , 𝐸 ∈ 𝒞ℐℱ𝒮ℒ − 𝒢(𝐺)
Proposition 3.7 Let 𝐹 , 𝐸 ∈ 𝒞ℐℱ𝒮ℒ − 𝒢(𝐺) and (𝐹 , 𝐸)|𝑒 = {(𝑎, 𝐹 (𝑎)) | 𝑎 ∈ 𝐸, 𝐹 (𝑎) ∈ 𝜌(𝐺)} where 𝐹 (𝑎)|𝑒 = {𝑥 ∈ 𝐺 | 𝜇𝐹 (𝑎)(𝑥) = 𝜇𝐹 (𝑎)(𝑒), 𝜇𝐹 (𝑎)(𝑥) = 𝜈𝐹 (𝑎)(𝑒)} in which 𝑒 is the unit element of G. Then (𝐹 , 𝐸)|𝑒 is a soft group and soft lattice.
Proof. For each 𝑎 ∈ 𝐸 and for arbitrary 𝑥, 𝑦 ∈ 𝐹 (𝑎)|𝑒, we have 𝜇𝐹 (𝑎)(𝑥𝑦−1) ≥ 𝜇𝐹 (𝑎)(𝑥) ∧ 𝜇𝐹 (𝑎)(𝑦) = 𝜇𝐹 (𝑎)(𝑒) 𝜈𝐹 (𝑎)(𝑥𝑦−1) ≤ 𝜈𝐹 (𝑎)(𝑥) ∨ 𝜈𝐹 (𝑎)(𝑦) = 𝜈𝐹 (𝑎)(𝑒) Since 𝜇𝐹 (𝑎)(𝑥𝑦−1) ≤ 𝜇𝐹 (𝑎)(𝑒) and 𝜈𝐹 (𝑎)(𝑥𝑦−1) ≥ 𝜈𝐹 (𝑎)(𝑒)
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This implies 𝑥𝑦−1 ∈ 𝐹 (𝑎)|𝑒
Similarly we can easily show 𝑥 ∧ 𝑦 & 𝑥 ∨ 𝑦 ∈ 𝐹 (𝑎)|𝑒. Therefore (𝐹 , 𝐸)|𝑒 is a soft group and soft lattice.
Remark: In above Proposition 3.7., 𝐹 (𝑎)|𝑒 forms a lattice ordered subgroup of G.
Proposition 3.8 Let 𝐹 , 𝐸 ∈ 𝒞ℐℱ𝒮ℒ − 𝒢 over G. (𝐹 , 𝐸)(𝛼1,𝛽1) and (𝐹 , 𝐸)(𝛼2,𝛽2) are (𝛼1, 𝛽1) and (𝛼2, 𝛽2) - level soft sets of (𝐹 , 𝐸), respectively, where 𝛼1, 𝛼2, 𝛽1, 𝛽2 ∈ 𝒪1. If 𝛼1 ≥ 𝛼2 and
𝛽2 ≥ 𝛽1, then we have (𝐹 , 𝐸)(𝛼1,𝛽1) < (𝐹 , 𝐸)(𝛼2,𝛽2) and (𝐹 , 𝐸)(𝛼1,𝛽1) ⊆ (𝐹 , 𝐸)(𝛼2,𝛽2) that is (𝐹 , 𝐸)(𝛼1,𝛽1) is a soft sublattice and also soft subgroup of (𝐹 , 𝐸)(𝛼2,𝛽2)
Proof: The proof is obvious from Proposition 3.4. & 3.6.
4 Conclusion
In this paper, we introduced the notion ofComplex intuitionistic fuzzy soft lattice ordered group and also some of the properties of 𝛼, 𝛽 −level sets of 𝒞ℐℱ𝒮ℒ − 𝒢based on the soft group and soft lattice structures are derived. As a future work we planned to study the algebraic properties and the operations of complex intuitionisticfuzzy soft sets underthe lattice ordered group structure.
Acknowledgement
The article has been written with the joint financial support of RUSA-Phase 2.0 grant sanctioned vide letter No.F 24-51/2014-U, Policy (TN Multi-Gen), Dept. of Edn. Govt. of India, Dt. 09.10.2018, UGC-SAP (DRS-I) vide letter No.F.510/8/DRS-I/2016(SAP-I) Dt. 23.08.2016, DST- PURSE 2nd Phase programme vide letter No. SR/PURSE Phase 2/38 (G) Dt. 21.02.2017 and DST (FST - level I) 657876570 vide letter No.SR/FIST/MS-I/2018/17 Dt. 20.12.2018.
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