Ratio Mathematica Volume 47, 2023 On Neutrosophic filter of BL-algebras A. Ibrahim* S. Karunya Helen Gunaseeli† Abstract In this paper, we introduce the notion of neutrosophic filter in BL-algebras and investigate several key characteristics with illustrations. Additionally, we obtain few conditions, order properties of neutrosophic filter of BL-algebras. Moreover, we prove that intersection of two neutrosophic filters of a BL- algebra is also a neutrosophic filter. Keywords: BL-algebra; filter; neutrosophic set; neutrosophic filter. 2020 AMS subject classification: 03G25,03E72,03F55,06F35 ‡ . 1. Introduction In 1965, L. A. Zadeh [2, 3, 14] was the first to introduce the notion of fuzzy sets to describe vagueness mathematically. He rectified those problems by designating every feasible individual in the universe a number resembling its grade of membership in the fuzzy set. In lattice implication algebras, Xu and Qin[12] first proferred the concept of filters. An important aspect of researching various logical algebras is filter theory. In the argument of the completeness of various logical algebra, filters are crucial. Several academicians have investigated the filter theory of different logical algebras. In 1998, the ideology of neutrosophy was introduced by Smarandache [1,9]. This became a new idea of philosophy to characterize the neutralites. The main idea of neutrosophy is *Assistant Professor, P.G. and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University, Trichirappalli, India; dribra@hhrc.ac.in, dribrahimaadhil@gmail.com. †Research Scholar, P.G. and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University, Trichirappalli, India; karjes821@gmail.com. ‡Received on August 11, 2022. Accepted on May 17, 2023. Published on June 30, 2023. DOI:10.23755/rm.v39i0.816. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 141 A. Ibrahim and S. Karunya Helen Gunaseeli that behind every concept there also exists an indeterminant degree in addition to truth and falsity. Neutrosophy, a discipline of philosophy that has just recently been recognised as a science, examines the genesis, character, and range of neutralities as well as how they interact with various ideational spectra. Hajek[6] introduced the concept of BL-algebras (Basic Logic), is a type of logical algebra. In this paper, the concept of neutrosophic filters of BL-algebras is discussed. In section 2, few fundamental definitions and results are explained. In section 3, we introduce the notion of neutrosophic filters of BL-algebras along with some of its related features. 2. Preliminaries In this section, we recall few fundamental definitions and their characteristics that are useful for developing the primary findings. Definition 2.1[6,11] A BL-algebra is an algebra (ℬ, ∨, ∧,∘, →, 0,1) of type (2, 2, 2, 2, 0, 0) such that the following are satisfied for all 𝑗1, 𝑘1, 𝑙1 ∈ ℬ, (i) (ℬ, ∨, ∧, 0,1) is a bounded lattice, (ii) (ℬ, ∘, 1) is a commutative monoid, (iii) ′ ∘ ′ and ′ → ′form an adjoint pair, that is, 𝑙1 ≤ 𝑗1→ 𝑘1if and only if 𝑗1 ∘ 𝑙1 ≤ 𝑘1for all 𝑗1, 𝑘1, 𝑙1 ∈ ℬ, (iv) 𝑗1∧ 𝑘1= 𝑗1 ∘ (𝑗1 → 𝑘1), (v) (𝑗1 → 𝑘1) ∨ (𝑘1 → 𝑗1) = 1. Proposition 2.2[7,13] The following axioms are satisfied in a BL- algebra ℬ for all 𝑗1, 𝑘1, 𝑙1∈ ℬ, (i) 𝑘1 → (𝑗1 → 𝑙1) = 𝑗1→ (𝑘1→ 𝑙1) = (𝑗1 ∘ 𝑘1) → 𝑙1, (ii) 1 → 𝑗1 = 𝑗1, (iii) 𝑗1 ≤ 𝑘1 if and only if 𝑗1 → 𝑘1 = 1, (iv) 𝑗1∨𝑘1 = ((𝑗1→ 𝑘1) → 𝑘1) ∧ ((𝑘1 → 𝑗1) → 𝑗1), (v) 𝑗1 ≤ 𝑘1 implies 𝑘1 → 𝑙1 ≤ 𝑗1 → 𝑙1, (vi) 𝑗1 ≤ 𝑘1 implies 𝑙1 → 𝑗1≤ 𝑙1 → 𝑘1, (vii) 𝑗1 → 𝑘1 ≤ (𝑙1→ 𝑗1) → (𝑙1→ 𝑘1), (viii) 𝑗1 → 𝑘1≤ (𝑘1 → 𝑙1) → (𝑘1→ 𝑙1), (ix) 𝑗1 ≤ (𝑗1→ 𝑘1) → 𝑘1, (x) 𝑗1 ∘ (𝑗1 → 𝑘1) = 𝑗1∧𝑘1, (xi) 𝑗1 ∘ 𝑘1 ≤ 𝑗1∧𝑘1 (xii) 𝑗1→ 𝑘1 ≤ (𝑗1 ∘ 𝑙1) → (𝑘1 ∘ 𝑙1), (xiii) 𝑗1 ∘ (𝑘1 → 𝑙1) ≤ 𝑘1→ (𝑗1 ∘ 𝑙1), (xiv) (𝑗1 → 𝑘1) ∘ (𝑘1 → 𝑙1) ≤ 𝑗1 →𝑙1, (xv) (𝑗1 ∘ 𝑗1 ∗ ) = 0. 142 On Neutrosophic filter of BL-algebras Note. In the above sequence, ℬ is used to intend the BL- algebras and the operations ′ ∨ ′, ′ ∧′, ′ ∘′ have preference on the way to the operations ′ → ′. Note. In a BL- algebra ℬ,′ ∗ ′is a complement defined as 𝑗1 ∗ = 𝑗1 → 0 for all 𝑗1∈ ℬ. Definition 2.3[15] A non-empty subset 𝐹 of a BL- algebra ℬ is a filter of ℬ if the following axioms hold for all 𝑗1, 𝑘1∈ ℬ, (i) If 𝑗1, 𝑘1∈𝐹, then 𝑗1 ∘ 𝑘1∈𝐹, (ii) If 𝑗1∈𝐹 and 𝑗1≤ 𝑘1, then 𝑘1∈𝐹. Proposition 2.4[15] A nonempty subset 𝐹 of a BL- algebra ℬ is a filter of ℬ if and only if the following are satisfied for all 𝑗1, 𝑘1∈ ℬ, (i) 1∈𝐹, (ii) 𝑗1, 𝑗1 → 𝑘1∈𝐹 implies 𝑘1∈𝐹. A filter 𝐹 of a BL-algebra ℬ is proper if 𝐹≠ ℬ. Definition 2.5[8,9] Let𝑋 be a set.A neutrosophic subset 𝑅 of 𝑋 is a triple (𝑇𝑅 , 𝐼𝑅 , 𝐹𝑅 ) where 𝑇𝑅 : 𝑋→[0,1] is truth membership function, 𝐼𝑅 : 𝑋→[0,1] is indeterminacy function and 𝐹𝑅 : 𝑋 → [0,1] is false membership function and 0 ≤ 𝑇𝑅 (𝑗1) + 𝐼𝑅 (𝑗1) + 𝐹𝑅 (𝑗1) ≤ 3 for all 𝑗1 ∈ 𝑋. Hence, for each 𝑗1 ∈ 𝑋, 𝑇𝑅 (𝑗1), 𝐼𝑅 (𝑗1) and 𝐹𝑅 (𝑗1) are all standard real numbers in [0,1] . Note. The values of 𝑇𝑅 (𝑗1), 𝐼𝑅 (𝑗1) and 𝐹𝑅 (𝑗1) have no limitations and we have the obvious condition 0 ≤ 𝑇𝑅 (𝑗1) + 𝐼𝑅 (𝑗1) + 𝐹𝑅 (𝑗1) ≤ 3. Definition 2.6[4, 9] Let 𝑅 and 𝑆 be two neutrosophic sets on 𝑋. Define 𝑅 ≤ 𝑆 if and only if 𝑇𝑅 (𝑗1) ≤ 𝑇𝑆 (𝑗1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑆 (𝑗1), 𝐹𝑅 (𝑗1) ≥ 𝐹𝑆 (𝑗1) for all 𝑗1 ∈ 𝑋. Definition 2.7[5,9] Let 𝑅 and S be two neutrosophic sets on X. Define 𝑅 ∧ 𝑆 = (𝑇𝑅 ∧ 𝑇𝑆,𝐼𝑅 ∨ 𝐼𝑆 , 𝐹𝑅 ∨ 𝐹𝑆); 𝑅 ∨ 𝑆 = (𝑇𝑅 ∨ 𝑇𝑆,𝐼𝑅 ∧ 𝐼 𝑆 , 𝐹𝑅 ∧ 𝐹𝑆) Where ′ ∧ ′ is the minimum and ′ ∨ ′ is the maximum between real numbers. Definition 2.8[10] Let 𝑅 be a neutrosophic set in 𝑋 and 𝛼, 𝛽, 𝛾 ∈ [0,1] with 0 ≤ 𝛼 + 𝛽 + 𝛾 ≤ 3 and (𝛼, 𝛽, 𝛾 ) – level set of 𝑅 denoted by 𝑅(𝛼,𝛽,𝛾 ) is defined as 𝑅(𝛼,𝛽,𝛾 ) = {𝑗1 ∈ 𝑋/ 𝑇𝑅 (𝑗1) ≥ 𝛼, 𝐼𝑅 (𝑗1) ≤ β and 𝐹𝑅 (𝑗1) ≤ γ}. 143 A. Ibrahim and S. Karunya Helen Gunaseeli 3. Properties of neutrosophic filter In this section, we introduce the definition of neutrosophic filter of BL- algebra and obtain some relevant properties with illustrations. Definition 3.1 A neutrosophic set 𝑅 of algebra ℬ is called a neutrosophic filter if it satisfies the subsequent conditions: (i) 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅 (1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (1) and 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (1), (ii) min{𝑇𝑅 (𝑗1 → 𝑘1), 𝑇𝑅 (𝑗1)}≤ 𝑇𝑅 (𝑘1),min{𝐼𝑅 (𝑗1 → 𝑘1), 𝐼𝑅 (𝑗1)} ≥ 𝐼𝑅 (𝑘1) and min{𝐹𝑅 ( 𝑗1 → 𝑘1), 𝐹𝑅 (𝑗1)} ≥ 𝐹𝑅 (𝑘1)}for all 𝑗1, 𝑘1 ∈ ℬ. Example 3.2 Let ℬ = {0, 𝑢, 𝑣, 1}. The binary operations ′ ∘ ′ and ′ → ′are given by the subsequent tables (3.1) and (3.2). Table3.1: ′ ° ′ Operation Table 3.2: ′ → ′Operation Then, (ℬ, ∨, ∧, ∘, →, 0, 1) is a BL- algebra. Define a neutrosophic set 𝑅 of ℬ as follows: 𝑅 = {(1, [0.9,0.2,0.1]), (𝑢, [0.5, 0.3, 0]), (𝑣, [0.5,0.3,0]), (0, [0.9,0.2,0.1])}. It is evident that 𝑅 is a neutrosophic filter of ℬ and assure the conditions (i) and (ii) of the definition 3.1. Example 3.3 Let ℬ = {0, 𝑎, 𝑏, 1}. The binary operations are given by the tables (3.3) and (3.4). Let 𝑆 of ℬ be a neutrosophic set as follows: 𝑆 = {(1, [0.5,0.3,0.2]), (𝑎, [0.9,0.2,0.1]), (𝑏, [0.9,0.2,0.1]), (0, [0.9,0.2,0.1])} Here, 𝑆 is not a neutrosophic filter of ℬ. Since 𝑇𝑆 (1) = 0.5 ≱ 0.9 = min{𝑇𝑆 (𝑎 ∘ 1), 𝑇𝑆 (𝑎)}, 𝑇𝑆 (1) = 0.5 ≱ 0.9 =min{𝑇𝑆 (𝑏 ∘ 1), 𝑇𝑆 (𝑏)}. ∘ 0 𝑢 𝑣 1 0 0 0 0 0 𝑢 0 0 𝑢 𝑣 𝑣 0 𝑢 𝑣 𝑣 1 0 𝑢 𝑣 1 → 0 𝑢 𝑣 1 0 1 1 1 1 𝑢 𝑢 1 1 1 𝑣 0 𝑢 1 1 1 0 𝑢 𝑣 1 144 On Neutrosophic filter of BL-algebras Table 3.3∶ ′ ° ′ Operation Table 3.4:′ ⟶′Operation Proposition 3.4 Let 𝑅 be a neutrosophic filter in a BL- algebra ℬ.If 𝑗1 ≤ 𝑘1then 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅 (𝑘1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (𝑘1), 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1)for all 𝑗1, 𝑘1 ∈ ℬ. Proof: Let 𝑅 be a neutrosophic filter of a BL-algebraℬ. If 𝑗1 ≤ 𝑘1, then 𝑗1 ⟶ 𝑘1 = 1for all 𝑗1, 𝑘1 ∈ ℬ [From(iii) of the proposition 2.2] Then, from (i) and(ii) of the definition 3.1, 𝑇𝑅 (𝑗1) = min{𝑇𝑅( 1 ) , 𝑇𝑅 (𝑗1)} = min{𝑇𝑅(𝑗1 → 𝑘1) , 𝑇𝑅 (𝑗1)} ≤ 𝑇𝑅 (𝑘1), 𝐼𝑅 (𝑗1) = min{𝐼𝑅 ( 1 ), 𝐼𝑅 (𝑗1) } = min{𝐼𝑅 ( 𝑗1 → 𝑘1 ), 𝐼𝑅 (𝑗1) } ≥ 𝐼𝑅 (𝑘1), 𝐹𝑅 (𝑗1) = min{ 𝐹𝑅 ( 1 ), 𝐹𝑅 (𝑗1)} = min{𝐹𝑅 (𝑗1 → 𝑘1), 𝐹𝑅 (𝑗1)} ≥ 𝐹𝑅 (𝑘1) for all 𝑗1, 𝑘1 ∈ ℬ. ∎ Proposition 3.5 Let 𝑅 be a neutrosophic filter of a BL-algebra ℬ. If 𝑗1 ≤ 𝑘1then 𝑇𝑅 (𝑗1)is order preserving and 𝐼𝑅 (𝑗1), 𝐹𝑅 (𝑗1)are order reversing. Proof: Let 𝑅 be a neutrosophic filter of a BL-algebra ℬ. To prove: If 𝑗1 ≤ 𝑘1then 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅(𝑘1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (𝑘1), 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1) for all𝑗1, 𝑘1 ∈ ℬ. Then, from the proposition 3.4, the proof is straight forward. Thus 𝑇𝑅 (𝑗1)is order preserving and 𝐼𝑅 (𝑗1), 𝐹𝑅 (𝑗1)are order reversing. ∎ → 0 𝑎 𝑏 1 0 1 1 1 1 𝑎 𝑎 1 1 1 𝑏 0 𝑎 1 1 1 0 𝑎 𝑏 1 ∘ 0 𝑎 𝑏 1 0 0 0 0 𝑏 𝑎 0 0 𝑎 𝑏 𝑏 0 𝑎 𝑏 𝑏 1 0 𝑎 𝑏 1 145 A. Ibrahim and S. Karunya Helen Gunaseeli Proposition 3.6 Let 𝑅 be a neutrosophic set of a BL–algebra ℬ. 𝑅 is a neutrosophic filter of ℬ if and only if 𝑗1 ⟶ (𝑘1 ⟶ 𝑙1) = 1 implies 𝑇𝑅 (𝑙1) ≥ min{ 𝑇𝑅 (𝑗1) , 𝑇𝑅 (𝑘1) }, 𝐼𝑅 (𝑙1)≤ min{ 𝐼𝑅 ( 𝑗1 ), 𝐼𝑅 (𝑘1)} and 𝐹𝑅 (𝑙1) ≤ min{ 𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1)} for all 𝑗1, 𝑘1, 𝑙1 ∈ ℬ. Proof: Let 𝑅 be a neutrosophic filter of a BL- algebra ℬ. Then, from (ii) of the definition 3.1, we have 𝑇𝑅 (𝑙1) ≥ min{𝑇𝑅 ( 𝑙1 ⟶ 𝑘1) , 𝑇𝑅 (𝑘1) }for all 𝑗1,𝑘1,𝑙1 ∈ ℬ. Now,𝑇𝑅 ( 𝑙1 ⟶ 𝑘1) ≥ min{ 𝑇𝑅 ( 𝑗1 →(𝑘1 → 𝑙1)) , 𝑇R(𝑗1) }. If 𝑗1 →(𝑘1 → 𝑙1) = 1, then we have 𝑇𝑅 (𝑙1 ⟶ 𝑘1) ≥min{𝑇𝑅 (1) , 𝑇𝑅 (𝑗1) } = 𝑇𝑅 (𝑗1). So, 𝑇𝑅 ( 𝑙1) ≥ min{𝑇𝑅 (𝑗1) , 𝑇𝑅 (𝑘1) }. Similarly, 𝐼𝑅 (𝑙1) ≤ min{ 𝐼𝑅 (𝑗1), 𝐼𝑅 (𝑘1)} and 𝐹𝑅 (𝑙1) ≤ min{𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1)}. Conversely, Let 𝑗1 ⟶ (𝑗1 ⟶ 1) = 1 for all 𝑗1 ∈ ℬ. Then,𝑇𝑅 ( 1 ) ≥ min{𝑇𝑅 (𝑗1) , 𝑇𝑅 (𝑗1) } = 𝑇𝑅 (𝑗1). On the other hand, from (𝑗1 → 𝑘1) → (𝑗1 → 𝑘1) = 1 implies 𝑇𝑅 (𝑘1) ≥ min{𝑇𝑅 ( 𝑗1 → 𝑘1),𝑇𝑅 (𝑗1)}, 𝐼𝑅 (𝑘1)≤ min{ 𝐼𝑅 ( 𝑗1 → 𝑘1 ), 𝐼𝑅 (𝑗1)} and 𝐹𝑅 (𝑘1) ≤ min{ 𝐹𝑅 (𝑗1 → 𝑘1), 𝐹𝑅 (𝑗1)}. Then, from the definition 3.1, 𝑅 is a neutrosophic filter of ℬ. ∎ Corollary 3.7 Let 𝑅 be a neutrosophic set of BL- algebra ℬ. 𝑅 is a neutrosophic filter of ℬ if and only if 𝑗1 ∘ 𝑘1 ≤ 𝑙1 or 𝑘1 ∘ 𝑙1 ≤ 𝑙1 implies 𝑇𝑅 (𝑙1) ≥min{ 𝑇𝑅 ( 𝑗1) , 𝑇𝑅 (𝑘1 ) }, 𝐼𝑅 (𝑙1)≤ min{𝐼𝑅 (𝑗1), 𝐼𝑅 (𝑘1 )} and 𝐹𝑅 (𝑙1) ≤min{ 𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1 )} for all 𝑗1, 𝑘1 , 𝑙1 ∈ ℬ. Proof: From (i) of the proposition 2.2 and the proposition 3.6 the proof is obvious. ∎ Proposition 3.8 Let 𝑅 be a neutrosophic set of a BL-algebra ℬ. 𝑅is a neutrosophic filter of ℬ if and only if (i) If 𝑗1 ≤ 𝑘1then 𝑇𝑅(𝑗1) ≤ 𝑇𝑅 (𝑘1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (𝑘1) and 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1) , (ii) 𝑇𝑅 (𝑗1 ∘ 𝑘1) ≥ min{𝑇𝑅 ( 𝑗1) , 𝑇𝑅 (𝑘1) },𝐼𝑅 (𝑗1 ∘ 𝑘1)≤ min{𝐼𝑅 ( 𝑗1), 𝐼𝑅 (𝑘1)} and 𝐹𝑅 (𝑗1 ∘ 𝑘1) ≤ min{ 𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1)}for all 𝑗1, 𝑘1 ∈ ℬ. Proof: Let 𝑅 be a neutrosophic filter of a BL-algebra ℬ. Then, from the proposition 3.4, we have 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅 (𝑘1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (𝑘1) and 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1) when 𝑗1 ≤ 𝑘1. 146 On Neutrosophic filter of BL-algebras Since 𝑗1 ∘ 𝑘1 ≤ 𝑗1 ∘ 𝑘1 and from the corollary 3.7, we have 𝑇𝑅 (𝑗1 ∘ 𝑘1) ≥ min{ 𝑇𝑅 ( 𝑗1), 𝑇𝑅 (𝑘1) }, 𝐼𝑅 (𝑗1 ∘ 𝑘1)≤ min{ 𝐼𝑅 ( 𝑗1), 𝐼𝑅 (𝑘1)} and 𝐹𝑅 (𝑗1 ∘ 𝑘1) ≤ min{ 𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1)}. Conversely, Let 𝑅 be a neutrosophic set and satisfies (i) and (ii). If 𝑗1 ∘ 𝑘1 ≤ 𝑙1 then from (i) and (ii) we get, 𝑇𝑅 (𝑙1) ≥ min{𝑇𝑅 ( 𝑗1) , 𝑇𝑅 (𝑘1) }, 𝐼𝑅 (𝑙1)≤ min{ 𝐼𝑅 ( 𝑗1), 𝐼𝑅 (𝑘1)} and 𝐹𝑅 (𝑙1) ≤ min{ 𝐹𝑅 (𝑗1), 𝐹𝑅 (𝑘1)}for all 𝑗1, 𝑘1, 𝑙1 ∈ ℬ. Then, from the corollary 3.7, we have 𝑅 is a neutrosophic filter. ∎ Proposition 3.9 Let 𝑅 be a neutrosophic set of a BL-algebra ℬ. If 𝑅 is a neutrosophic filter of ℬ, then it satisfies the following for all 𝑗1, 𝑘1, 𝑙1 ∈ ℬ. (i) If 𝑇𝑅 ( 𝑗1 ⟶ 𝑘1)= 𝑇𝑅 (1), then 𝑇𝑅 (𝑗1) ≤ 𝑇(𝑘1),𝐼𝑅 ( 𝑗1 ⟶ 𝑘1)= 𝐼 𝑅(1), then 𝐼𝑅 (𝑗1) ≥ 𝐼 𝑅(𝑘1),𝐹𝑅 ( 𝑗1 ⟶ 𝑘1)= 𝐹𝑅(1), then 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1). (ii) 𝑇𝑅 (𝑗1 ⟶ 𝑘1)≤ 𝑇𝑅 (𝑗1 ∘ 𝑙1 ⟶ 𝑘1 ∘ 𝑙1), 𝐼𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐼𝑅 (𝑗1 ∘ 𝑙1 ⟶ 𝑘1 ∘ 𝑙1) and𝐹𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐹𝑅 (𝑗1 ∘ 𝑙1 ⟶ 𝑘1 ∘ 𝑙1). (iii) 𝑇𝑅 ( 𝑗1 ⟶ 𝑘1) ≤ 𝑇𝑅 ((𝑘1 → 𝑙1) → (𝑗1 → 𝑙1)),𝐼𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐼𝑅 (( 𝑘1→ 𝑙1) →(𝑗1 → 𝑙1))and 𝐹𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐹𝑅 ((𝑘1→ 𝑙1) → ( 𝑗1→ 𝑙1)). (iv) 𝑇𝑅 ( 𝑗1 ⟶ 𝑘1) ≤ 𝑇𝑅 ((𝑙1 → 𝑗1) → (𝑙1 → 𝑘1)),𝐼𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐼𝑅 ((𝑙1→ 𝑗1) → (𝑙1→ 𝑘1)) and𝐹𝑅 ( 𝑗1 ⟶ 𝑘1) ≥ 𝐹𝑅 ((𝑙1 → 𝑗1) → (𝑙1 → 𝑘1)). Proof: (i) Let 𝑅 be a neutrosophic filter of a BL-algebra ℬ. Then from the definition 3.1, and since 𝑇𝑅 ( 𝑗1 ⟶ 𝑘1)= 𝑇𝑅 (1), we have 𝑇𝑅 (𝑘1) ≥ min{ 𝑇𝑅 (𝑗1) , 𝑇𝑅 (𝑗1 ⟶ 𝑘1) } = min{𝑇𝑅 (𝑗1) , 𝑇𝑅 (1) }= 𝑇𝑅 (𝑗1). Thus, 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅 (𝑘1). Similarly we get, 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (𝑘1), 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (𝑘1). From the proposition 2.2 and (i) of the proposition 3.8, we can prove (ii), (iii) and (iv) easily. ∎ Proposition 3.10 Let 𝑅 be a neutrosophic set of a BL- algebra ℬ. 𝑅is a neutrosophic filter of ℬ if and only if (i) 𝑇𝑅 (𝑗1) ≤ 𝑇𝑅 (1), 𝐼𝑅 (𝑗1) ≥ 𝐼𝑅 (1) and 𝐹𝑅 (𝑗1) ≥ 𝐹𝑅 (1), (ii) 𝑇𝑅 ( (𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≥ min{ 𝑇𝑅 ( 𝑗1) , 𝑇𝑅 (𝑘1) }, 𝐼𝑅 ((𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≤ min{𝐼𝑅 ( 𝑗1) , 𝐼𝑅 (𝑘1) } and 𝐹𝑅 ( (𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≤ min{ 𝐹𝑅 ( 𝑗1) , 𝐹𝑅 (𝑘1) }for all 𝑗1,𝑘1,𝑙1 ∈ ℬ. 147 A. Ibrahim and S. Karunya Helen Gunaseeli Proof: Let 𝑅 be a neutrosophic filter of BL-algebra ℬ. From the definition 3.1 (i) is straight forward. Since,𝑇𝑅 ((𝑗1 → (𝑘1 → 𝑙1)) → 𝑙1)≥ min{𝑇𝑅 ((𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1), 𝑇𝑅 (𝑘1)} (3.1) Now, we have, (𝑗1 →(𝑘1 → 𝑙1)) →(𝑘1 → 𝑙1) = 𝑗1∨(𝑘1 → 𝑙1) ≥ 𝑗1. 𝑇𝑅 ((𝑗1 →(𝑘1 → 𝑙1)) →(𝑘1 → 𝑙1))≥ 𝑇𝑅 (𝑗1) (3.2) [From the proposition 3.5] Using (3.2) in (3.1), we have 𝑇𝑅 ( (𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≥ min{𝑇𝑅 (𝑗1) , 𝑇𝑅 (𝑘1) } Similarly, we get 𝐼𝑅 ( (𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≤ min{ 𝐼𝑅 ( 𝑗1) , 𝐼𝑅 (𝑘1) } and 𝐹𝑅 ( (𝑗1 →(𝑘1 → 𝑙1)) → 𝑙1) ≤ min{ 𝐹𝑅 ( 𝑗1) , 𝐹𝑅 (𝑘1) }. Conversely, assume (i) and (ii) hold. Since 𝑇𝑅 (𝑘1) = 𝑇𝑅 (1 → 𝑘1) = 𝑇𝑅 ( (𝑗1 → 𝑘1) → (𝑗1 → 𝑘1) → 𝑘1) ≥ min{𝑇𝑅 ( 𝑗1 → 𝑘1) , 𝑇𝑅 (𝑘1) }. Similarly, 𝐼𝑅 (𝑘1)≤ min{𝐼𝑅 (𝑗1 → 𝑘1) , 𝐼𝑅 (𝑘1) }and 𝐹𝑅 (𝑘1) ≤ min{ 𝐹𝑅 ( 𝑗1 → 𝑘1) , 𝐹𝑅 (𝑘1) }. From (i), 𝑅 is a neutrosophic filter of ℬ. ∎ Proposition 3.11 The intersection of two neutrosophic filters of ℬ is also a neutrosophic filter of ℬ. Proof: Let 𝑅 and 𝑆 be two neutrosophic filters of ℬ, To Prove: 𝑅 ∩ 𝑆 is a neutrosophic filter of ℬ. We have 𝑇𝑅 (𝑘1) ≥ min{ 𝑇𝑅 (𝑙1) , 𝑇𝑅 (𝑗1) } and 𝑇𝑆(𝑘1 ) ≥ min{ 𝑇𝑆(𝑙1 ) , 𝑇𝑆(𝑗1) } for all 𝑗1, 𝑘1 , 𝑙1 ∈ ℬ. Since 𝑇𝑅∩𝑆(𝑘1 ) = min{𝑇𝑅 (𝑘1 ) , 𝑇𝑆(𝑘1 ) } ≥ min{min{ 𝑇𝑅 (𝑙1) , 𝑇𝑅 (𝑗1) }, min{ 𝑇𝑆(𝑙1 ) , 𝑇𝑆(𝑗1) }} = min{min{ 𝑇𝑅 (𝑙1) , 𝑇𝑆(𝑙1 ) }, min{ 𝑇𝑅 (𝑗1) , 𝑇𝑆(𝑗1) }} = min{ min{ 𝑇𝑅∩𝑆(𝑙1) ,𝑇𝑅∩𝑆(𝑗1) }}. Similarly,𝐼𝑅∩𝑆(𝑘1 ) = min{ min{ 𝐼𝑅∩𝑆(𝑙1) ,𝐼𝑅∩𝑆(𝑗1) }} and 𝐹𝑅∩𝑆(𝑘1 ) = min{ min{ 𝐹𝑅∩𝑆(𝑙1) ,𝐹𝑅∩𝑆(𝑗1) }}. Hence, 𝑇𝑅∩𝑆(𝑘1 ) ≥ min{𝑇𝑅∩𝑆(𝑙1) ,𝑇𝑅∩𝑆(𝑗1) }, 148 On Neutrosophic filter of BL-algebras 𝐼𝑅∩𝑆(𝑘1 ) ≤min{𝐼𝑅∩𝑆(𝑙1) ,𝐼𝑅∩𝑆(𝑗1) } and 𝐹𝑅∩𝑆(𝑘1 ) ≤ min{ 𝐹𝑅∩𝑆(𝑙1) ,𝐹𝑅∩𝑆(𝑗1) }. Thus 𝑅 ∩ 𝑆 is a neutrosophic filter of ℬ. ∎ Corollary 3.12 Let 𝑃𝑖 be a family of neutrosophic filters of ℬ, where 𝑖 ∈ 𝐼, 𝐼 is a index set, then ∩𝑖∈𝐼 𝑃𝑖 is a neutrosophic filter of ℬ. ∎ 4. Discussion and Conclusions In this paper, we have introduced the notion of a neutrosophic filter in BL- algebras with illustrations. Moreover, several features of the neutrosophic filters are conferred. Also, we have derived a few equivalent conditions for a neutrosophic set of a BL-algebra to be a neutrosophic filter. In BL-algebras, fuzzy, vague, and many additional filters have already been defined. The main focus of this paper is to establish the neutrosophical nature of BL-algebras. Further, research on BL-algebras structure and the creation of the associated many-valued logical system will benefit from the aforementioned study. 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