Lattice ordered group has been developed by J.Vimala. After that J. Arockia Reeta and J. Vimala introduced Lattice ordered fuzzy soft groups(-FSG) and obtained real life applications. In this present work, we have attained some results on -FSGs under the operations of bounded sum and bounded difference

Dr. J. Arockia Reeta

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 Pandian Journal of Mathematical Sciences A Peer Reviewed Journal 5 Volume 2, Issue 1, July 2023      E-ISSN: 2583-8822   Be-Ecofriendly         Save Trees Save Life  Some results on 𝒍-fuzzy soft groups under bounded sum and bounded difference operations  Dr. J. Arockia Reeta, Guest Faculty, Department of Mathematics,  Bharathidasan University, Trichy, Tamilnadu, India.  ORCiD: https://orcid.org/0009-0001-1995-8775   Abstract   Lattice ordered group has been developed by J.Vimala.  After that J. Arockia Reeta and J. Vimala introduced Lattice ordered fuzzy soft groups(𝑙-FSG)  and obtained real life applications. In this present work, we have attained some results on 𝑙-FSGs under the operations of bounded sum and bounded difference.   Keywords: Fuzzy soft groups, 𝑙-FSG, operations of bounded sum and bounded difference in 𝑙-FSG.  1. Introduction The notion of lattice theory was proposed by Gratzer [6]. Fuzzy set theory has been propounded by Zadeh [10] and developed by Zimmerman [11].  Soft set theory has been originated by Molodsov [8]. Fuzzy soft set [7] has been developed and used in recent years to solve real life problems [5]. Fuzzy soft group was defined by Abdulkadir Aygunoglu and Halis Aygun [1]. Lattice ordered group has been investigated by J.Vimala [9]. A lattice structure on fuzzy soft group was constructed and some of its properties were investigated by J. Arockia reeta et al [2]. After that, anti-lattice ordered fuzzy soft groups [3] and its matrix operations have been studied and implemented in deciding process. Further, Algebraic relations over l-Fuzzy soft groups has been examined [4]. In this present work, we have attained some results on l-FSGs under the operations of bounded sum and bounded difference.  2.  Preliminaries Definition 2.1[9] Let X be a non-empty set, then a fuzzy set   over X is a function from X into [0,1]=I . ie., X:  I.   Definition 2.2 [5] 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 [5] Let XI  denote the set of all fuzzy sets on X and A  E. A pair ),( Af  is called a fuzzy soft set over X, where f  is a mapping from A into XI . That is, for each Aa, ,:=)( IXfaf a   is a fuzzy set on X.  Definition 2.4 [10] The bounded sum of two fuzzy sets A  and B  is denoted by C = A ⊕ B ={(𝑥, 𝜇𝐴 ⊕𝐵 (𝑥));𝑥 ∈ 𝑋}, where 𝜇𝐴 ⊕𝐵 (𝑥) = 𝑚𝑖𝑛{1, 𝜇𝐴 (𝑥) + 𝜇𝐵 (𝑥)}.  Pandian Journal of Mathematical Sciences A Peer Reviewed Journal 6 Volume 2, Issue 1, July 2023      E-ISSN: 2583-8822   Be-Ecofriendly         Save Trees Save Life Definition 2.5 [10] The bounded difference of two fuzzy sets A  and B  is denoted by C =A ⊖ B = {(x,μA ⊖B (x)); x ∈ X}, where μA ⊖B (x) = Max{0, μA (x) + μB (x) − 1}.   Definition 2.6 [9]The complement of a fuzzy soft set (f, A) is denoted by (f, A)c  and is defined by (f, A)c = (f c , ¬A), where f c : ¬A → P(U) is a mapping given by f c (σ) = (f(¬σ))c for all σ ∈ ¬A.  Definition 2.7[5] Let X be a group and (f, A) be a FSG over X. A FSG (f, A) is said to be an l - FSG over X. If for the mapping f: A → IX , a ≤ b implies fa ⊆ fb , for all a, b ∈ A.   Example 2.8 Let ℕ be the set of all natural numbers and (ℕ,≤) be a lattice. Let ℝ be the set of all real numbers. Define f:ℕ → Iℝ by f(n) = fn :ℝ → I for each n ∈ ℕ, where 𝑓 𝑥 =  1 −1𝑛 + 1,   𝑖𝑓 𝑥 = 𝑘5𝑛 , ∃ 𝑘𝜖𝑍0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒  Then the pair ((𝑓,ℕ),∨,∧, ⊆) forms an 𝑙 - FSG over ℝ. 3. Some results on 𝒍-fuzzy soft groups under bounded sum and bounded difference operations In this section, we develop some results on 𝑙-FSGs under the operations of bounded sum and bounded difference. 1Throughout this work, 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. Proposition 4.1 Let (𝑓, 𝐴) be an 𝑙- FSG over a group X. Then we have the following results for all 𝑎1, 𝑎2, 𝑎3 ∈ 𝐴 and 𝑎1 ≤ 𝑎2 ≤ 𝑎3,   1.  𝑓𝑎1 ∨ (𝑓𝑎2 ⊕ 𝑓𝑎3 ) = (𝑓𝑎1 ∨ 𝑓𝑎2 ) ⊕ (𝑓𝑎1 ∨ 𝑓𝑎3 ),  2.  𝑓𝑎1 ∧ (𝑓𝑎2 ⊕ 𝑓𝑎3 ) = 𝑓𝑎1 ,  3.  (𝑓𝑎1 ⊖ 𝑓𝑎2 ) ∨ 𝑓𝑎3 = 𝑓𝑎3 ,  4.  𝑓𝑎1 ⊕ (𝑓𝑎2 ∨ 𝑓𝑎3 ) = 𝑓𝑎1 ⊕ 𝑓𝑎3 ,  5.  𝑓𝑎1 ⊕ (𝑓𝑎2 ∧ 𝑓𝑎3 ) = 𝑓𝑎1 ⊕ 𝑓𝑎2 .  Proof:  1.  Let X be a group and (𝒇, 𝑨) be an 𝒍- FSG over X. Then we have for all 𝒂𝟏, 𝒂𝟐, 𝒂𝟑 ∈ 𝑨, 𝑎1 ≤ 𝑎2 ≤ 𝑎3 ⇒ 𝑓𝑎1 ⊆ 𝑓𝑎2 ⊆ 𝑓𝑎3 . Then, we get  𝑓𝑎1 ∨ 𝑓𝑎2 = 𝑓𝑎2 ,    𝑓𝑎1 ∨ 𝑓𝑎3 = 𝑓𝑎3 ,  𝑓𝑎1 ⊆ 𝑓𝑎2 ⊕ 𝑓𝑎3 ⇒ 𝑓𝑎1 ∨  𝑓𝑎2 ⊕ 𝑓𝑎3 = 𝑓𝑎2 ⊕ 𝑓𝑎3   and  (𝑓𝑎1 ∨ 𝑓𝑎2 ) ⊕ (𝑓𝑎1 ∨ 𝑓𝑎3 ) = 𝑓𝑎2 ⊕ 𝑓𝑎3 . Therefore    𝑓𝑎1 ∨  𝑓𝑎2 ⊕ 𝑓𝑎3 =  𝑓𝑎1 ∨ 𝑓𝑎2 ⊕  𝑓𝑎1 ∨ 𝑓𝑎3 . 2. 𝒇𝒂𝟐 ⊕ 𝒇𝒂𝟑 = {(𝒙,𝟏 ∧ [𝝁𝒇𝒂𝟐(𝒙) + 𝝁𝒇𝒂𝟑(𝒙)])/𝒙 ∈ 𝑿} We know that 𝑎1 ≤ 𝑎2 ≤ 𝑎3 ⇒ 𝜇𝑓𝑎1 (𝑥) ≤ 𝜇𝑓𝑎2 (𝑥) ≤ 𝜇𝑓𝑎3 (𝑥), for each 𝑥 ∈ 𝑋. ⇒ 𝜇𝑓𝑎1 𝑥 < 𝜇𝑓𝑎2 𝑥 + 𝜇𝑓𝑎3 𝑥  and 𝜇𝑓𝑎1 (𝑥) ≤ 1, for each 𝑥 ∈ 𝑋.  Pandian Journal of Mathematical Sciences A Peer Reviewed Journal 7 Volume 2, Issue 1, July 2023      E-ISSN: 2583-8822   Be-Ecofriendly         Save Trees Save Life ⇒ 𝜇𝑓𝑎1 (𝑥) < 1 ∧ (𝜇𝑓𝑎2 (𝑥) + 𝜇𝑓𝑎3 (𝑥)), for each 𝑥 ∈ 𝑋. ⇒ 𝜇𝑓𝑎1 (𝑥) ∧ [1 ∧ (𝜇𝑓𝑎2 (𝑥) + 𝜇𝑓𝑎3 (𝑥))] = 𝜇𝑓𝑎1 (𝑥), for each 𝑥 ∈ 𝑋. Then {(𝑥, 𝜇𝑓𝑎1 (𝑥))/𝑥 ∈ 𝑋} ∧ {(𝑥, 1 ∧ [𝜇𝑓𝑎2 (𝑥) + 𝜇𝑓𝑎3 (𝑥)])/𝑥 ∈ 𝑋} = {(𝑥, 𝜇𝑓𝑎1 )/𝑥 ∈ 𝑋}. Hence we get 𝑓𝑎1 ∧ (𝑓𝑎2 ⊕ 𝑓𝑎3 ) = 𝑓𝑎1 . 3. 𝒇𝒂𝟏 ⊖ 𝒇𝒂𝟐 =  (𝒙, 𝟎 ∨ [𝝁𝒇𝒂𝟏(𝒙) + 𝝁𝒇𝒂𝟐(𝒙) − 𝟏])/𝒙 ∈ 𝑿 . We know that 𝑎1 ≤ 𝑎2 ≤ 𝑎3 ⇒ 𝜇𝑓𝑎1 (𝑥) ≤ 𝜇𝑓𝑎2 (𝑥) ≤ 𝜇𝑓𝑎3 (𝑥), for each 𝑥 ∈ 𝑋. ⇒ 1 + 𝜇𝑓𝑎3 (𝑥) ≥ 𝜇𝑓𝑎2 (𝑥) + 𝜇𝑓𝑎1 (𝑥), for each 𝑥 ∈ 𝑋. ⇒ 𝜇𝑓𝑎1 (𝑥) + 𝜇𝑓𝑎2 (𝑥) − 1 ≤   𝜇𝑓𝑎3 (𝑥), for each 𝑥 ∈ 𝑋. ⇒ 0 ∨ (𝜇𝑓𝑎1 (𝑥) + 𝜇𝑓𝑎2 (𝑥) − 1) ∨ 𝜇𝑓𝑎3 (𝑥) = 𝜇𝑓𝑎3 (𝑥), for each 𝑥 ∈ 𝑋. Therefore  {𝑥, 0 ∨ [𝜇𝑓𝑎1 (𝑥) + 𝜇𝑓𝑎2 (𝑥) − 1])/𝑥 ∈ 𝑋} ∨ {(𝑥, 𝜇𝑓𝑎3 (𝑥)/𝑥 ∈ 𝑋} = {(𝑥, 𝜇𝑓𝑎3 (𝑥))/𝑥 ∈ 𝑋}. Then we get (𝑓𝑎1 ⊖ 𝑓𝑎2 ) ∨ 𝑓𝑎3 = 𝑓𝑎3 . (iv) and (v) Proof is obvious.  Example 4.1 From the example 2.8, consider the 𝑙-FSG (𝑓, 𝐴) = {𝑓2, 𝑓5 , 𝑓7, 𝑓8, 𝑓10}   𝑖𝑒. , (𝑓, 𝐴) =  k5−20.7,k5−50.8,k5−70.88,k5−80.89,k5−100.9 . Clearly it shows that  𝑓2 ⊆ 𝑓5 ⊆ 𝑓7 ⊆ 𝑓8 ⊆ 𝑓10 . 𝑓2 ∨ (𝑓5 ⊕ 𝑓7) =  𝑘5−20.7∨  𝑘5−50.8⊕𝑘5−70.88  .=  𝑘5−20.7∨𝑘5−51 .=  𝑘5−51 . (1) (𝑓2 ∨ 𝑓5) ⊕ (𝑓2 ∨ 𝑓7) =   k5−20.7∨k5−50.8 ⊕  k5−20.7∨k5−70.88  .=  k5−50.8 ⊕k5−70.88 .=   k5−51 . (2)  From (1) and (2), we get 𝑓2 ∨ (𝑓5 ⊕ 𝑓7) = (𝑓2 ∨ 𝑓5) ⊕ (𝑓2 ∨ 𝑓7). Proposition 4.2 Let (𝑓, 𝐴) be an 𝑙- FSG over a group X. Then       1.  (𝑓𝑎1 ⊕ 𝑓𝑎2 )𝑐 = 𝑓𝑎1𝑐 ⊖ 𝑓𝑎2𝑐 ,      2.  (𝑓𝑎1 ⊖ 𝑓𝑎2 )𝑐 = 𝑓𝑎1𝑐 ⊕ 𝑓𝑎2𝑐 ,      3.  (𝑓𝑎1𝑐 ⊕ 𝑓𝑎2𝑐 )𝑐 = 𝑓𝑎1 ⊖ 𝑓𝑎2 ,      4.  (𝑓𝑎1𝑐 ⊖ 𝑓𝑎2𝑐 )𝑐 = 𝑓𝑎1 ⊕ 𝑓𝑎2 ,  Proof: 1. 𝑓𝑎1 ⊕ 𝑓𝑎2 = {(𝑥, min{1, 𝜇𝑓𝑎1 (𝑥) + 𝜇𝑓𝑎2 (𝑥))/𝑥 ∈ 𝑋}. Then (𝑓𝑎1 ⊕ 𝑓𝑎2 )𝑐 = {(𝑥, 1 − {1 ∧ {𝜇𝑓𝑎1 (𝑥) + 𝜇𝑓𝑎2 (𝑥)}})/𝑥 ∈ 𝑋}.              = {(𝑥, {1 − 1} ∨ {1 − 𝜇𝑓𝑎1 (𝑥) − 𝜇𝑓𝑎2 (𝑥)})/𝑥 ∈ 𝑋}.   = {(𝑥, 0 ∨ {1 − 𝜇𝑓𝑎1 (𝑥) + 1 − 𝜇𝑓𝑎2 (𝑥) − 1})/𝑥 ∈ 𝑋}.  Pandian Journal of Mathematical Sciences A Peer Reviewed Journal 8 Volume 2, Issue 1, July 2023      E-ISSN: 2583-8822   Be-Ecofriendly         Save Trees Save Life   = 𝑓𝑎1𝑐 ⊖ 𝑓𝑎2𝑐 .  2.  The proof is similar to (1).   3.  𝑓𝑎1𝑐 ⊕ 𝑓𝑎2𝑐 = {(𝑥, 𝑚𝑖𝑛{1,1 − 𝜇𝑓𝑎1 (𝑥) + 1 − 𝜇𝑓𝑎2 (𝑥)})/𝑥 ∈ 𝑋}  (𝑓𝑎1𝑐 ⊕ 𝑓𝑎2𝑐 )𝑐 = {(𝑥, 1 − {1 ∧ {1 − 𝜇𝑓𝑎1 (𝑥) + 1 − 𝜇𝑓𝑎2 (𝑥)}})/𝑥 ∈ 𝑋}    = {(𝑥, {1 − 1} ∨ {1 − 1 + 𝜇𝑓𝑎1 (𝑥) − 1 + 𝜇𝑓𝑎2 (𝑥)})/𝑥 ∈ 𝑋}    = {(𝑥, 0 ∨ {𝜇𝑓𝑎1 (𝑥) + 𝜇𝑓𝑎2 (𝑥) − 1})/𝑥 ∈ 𝑋}    = 𝑓𝑎1 ⊖ 𝑓𝑎2   4.  The proof is similar to (3). Proposition 4.3 Let (𝑓, 𝐴) and (f, B) be an 𝑙- FSG over a group X. Then  (1)((𝑓, 𝐴) ∪ (𝑓, 𝐵))𝑐 = (𝑓𝑐 , ¬𝐴) ∪ (𝑓𝑐 , ¬𝐵) = (𝑓, 𝐴 ∪ 𝐵)𝑐 . (2)((𝑓, 𝐴) ∩ (𝑓, 𝐵))𝑐 = (𝑓𝑐 , ¬𝐴) ∩ (𝑓𝑐 , ¬𝐵) = (𝑓, 𝐴 ∩ 𝐵)𝑐 .   Proof:(1) Let (𝑓, 𝐴) = {𝑓𝑎1 , 𝑓𝑎2 , 𝑓𝑎3 , 𝑓𝑎4 , 𝑓𝑎5 } and (𝑓, 𝐵) = {𝑓𝑏1 ,𝑓𝑏2 , 𝑓𝑏3 , 𝑓𝑏4 ,𝑓𝑏5 } . Therefore  (𝑓, 𝐴) ∪ (𝑓, 𝐵) = {𝑓𝑎1 , 𝑓𝑎2 , 𝑓𝑎3 , 𝑓𝑎4 , 𝑓𝑎5 , 𝑓𝑏1 , 𝑓𝑏2 ,𝑓𝑏3 ,𝑓𝑏4 , 𝑓𝑏5 }. ( 𝑓, 𝐴 ∪  𝑓, 𝐵 )𝑐 =  𝑓¬𝑎1 , 𝑓¬𝑎2 , 𝑓¬𝑎3 , 𝑓¬𝑎4 , 𝑓¬𝑎5 , 𝑓¬𝑏1 , 𝑓¬𝑏2 ,𝑓¬𝑏3 ,𝑓¬𝑏4 , 𝑓¬𝑏5 .    =  𝑓¬𝑎1 , 𝑓¬𝑎2 , 𝑓¬𝑎3 , 𝑓¬𝑎4 , 𝑓¬𝑎5 ∪  𝑓¬𝑏1 ,𝑓¬𝑏2 ,𝑓¬𝑏3 ,𝑓¬𝑏4 , 𝑓¬𝑏5 ,    =  𝑓𝑐 , ¬𝐴 ∪  𝑓𝑐 , ¬𝐵 ,    = (𝑓𝑐 , ¬𝐴 ∪ ¬𝐵),    = (𝑓, 𝐴 ∪ 𝐵)𝑐 . Hence ((𝑓, 𝐴) ∪ (𝑓, 𝐵))𝑐 = (𝑓𝑐 , ¬𝐴) ∪ (𝑓𝑐 , ¬𝐵) = (𝑓, 𝐴 ∪ 𝐵)𝑐 . Proof of (2) is similar to (1). Proposition 4.4 Let (𝑓, 𝐴) and (f, B) be an 𝑙- FSG over a group X. Then  (1)(𝑓, 𝐴 ∪ 𝐵)𝑐 = (𝑓, 𝐴)𝑐 ∪ (𝑓, 𝐵)𝑐    (2)(𝑓, 𝐴 ∩ 𝐵)𝑐 = (𝑓, 𝐴)𝑐 ∩ (𝑓, 𝐵)𝑐 . Proof : (1) (𝑓, 𝐴 ∪ 𝐵)𝑐 = (𝑓𝑐 , ¬(𝐴 ∪ 𝐵)),     = (𝑓𝑐 , ¬𝐴 ∪ ¬𝐵)),     = (𝑓𝑐 , ¬𝐴) ∪ (𝑓𝑐 , ¬𝐵)), = (𝑓, 𝐴)𝑐 ∪ (𝑓, 𝐵)𝑐 .  Proof of (2) is similar to (1). Proposition 4.5 Let (𝑓, 𝐴) be an 𝑙- FSG over a group X. Then it holds the following results: ∀ 𝑎1, 𝑎2, 𝑎3 ∈ 𝐴,   1.  𝑓𝑎1 ⊕ (𝑓𝑎2 ⊖ 𝑓𝑎1 ) = 𝑓𝑎1 , when 𝜇𝑓𝑎1 , 𝜇𝑓𝑎2 ≤ 0.5,  2.  𝑓𝑎1 ⊖ (𝑓𝑎2 ⊕ 𝑓𝑎1 ) = 𝑓𝑎1 , when 𝜇𝑓𝑎1 , 𝜇𝑓𝑎2 ≥ 0.5.  Proof:  Let X be a group and (𝑓, 𝐴) be an 𝑙- FSG over X. Then we have for all 𝑎1, 𝑎2, 𝑎3 ∈ 𝐴, 𝑎1 ≤ 𝑎2 ≤ 𝑎3 ⇒ 𝑓𝑎1 ⊆ 𝑓𝑎2 ⊆ 𝑓𝑎3 .  𝑓𝑎2 ⊖ 𝑓𝑎1 =  (𝑥, 0 ∨ [𝜇𝑓𝑎2 (𝑥) + 𝜇𝑓𝑎1 (𝑥) − 1])/𝑥 ∈ 𝑋 = {(𝑥, 0)/𝑥 ∈ 𝑋}, Then 𝑓𝑎1 ⊕ (𝑓𝑎2 ⊖ 𝑓𝑎1 ) = {(𝑥, min{1, 𝜇𝑓𝑎1 (𝑥) + 0/𝑥 ∈ 𝑋}},  Pandian Journal of Mathematical Sciences A Peer Reviewed Journal 9 Volume 2, Issue 1, July 2023      E-ISSN: 2583-8822   Be-Ecofriendly         Save Trees Save Life = {𝑥, 𝜇𝑓𝑎1 (𝑥)},     = 𝑓𝑎1 (𝑥). Proof of (2) is similar to (1). 5. Conclusion In this present work, we have obtained some results on 𝑙-FSGs under the operations of bounded sum and bounded difference. In future, application of  𝑙-FSGs will be established and analysed.  References  [1] Abdulkadhir Aygunoglu and H. Aygun, Introduction to fuzzy soft groups, Computer Mathematics and Applications, 58 (2009), 1279-1286. [2] J.Arockia Reeta, and J.Vimala, A study on distributive and modular lattice ordered fuzzy soft group and its duality,  Applied Mathematics Journal of Chinese University, 31(4) (2016), 491-502. [3] J. Arockia Reeta, J. Vimala,  Implementation of anti-lattice ordered fuzzy soft groups and its matrix operations in deciding process,Journal of Intelligent & fuzzy systems,(2018), 35(4): pp. 4857-4864. [4] J. Arockia Reeta, J. Vimala and AB saeid, Algebraic relations over l- fuzzy soft groups, Afrika Mathematika, 32(5)(2020), 1-15. [5] N.Cagman, F.Citak and S.Enginoglu, Fuzzy soft set theory and its applications, Iranian Journal of Fuzzy Systems, 8(3) (2011), 137-147.  [6] G. Gratzer, Lattice Theory: First Concepts and Distributive Lattices, University of Manitoba, W. H. Freeman and Company, San Francisco, 1971.  [7] P.K. Maji, P. Biswas and A.R. Roy, A fuzzy soft sets, Journal of Fuzzy Mathematics, 3 (2001), 589-602.  [8] D.A. Molodtsov, Soft set theory - First result, Computer Mathematics and Applications, 37 (1999), 19-31. [9] J. Vimala, Lattice Ordered Groups, International Journal of Scientific and Engg. Research, 5(9), (2014) 58-60. [10] L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  [11] H. J. Zimmermann, Fuzzy set theory and its Applications, Springer International Edition, Fourth Edition, New Delhi, 1934.  Author Contribution Statement: NIL. Author Acknowledgement: NIL. Author Declaration: I declare that there is no competing interest in the content and authorship of this scholarly work.   The content of the article is licensed under https://creativecommons.org/licenses/by/4.0/ International License.   

Some results on -fuzzy soft groups under bounded sum and bounded difference operations

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Some results on -fuzzy soft groups under bounded sum and bounded difference operations

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