Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 562 https://internationalpubls.com On Neutrosophic Transitivity and Absorbent Filters of Basic Logic Algebras A. Ibrahim1, S. Karunya Helen Gunaseeli2 1Assistant Professor, P.G. and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India. Email: dribrahimaadhil@gmail.com 2Research Scholar, P.G. and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India. Email: karjes821@gmail.com Article History: Received: 16-05-2024 Revised: 20-06-2024 Accepted: 12-07-2024 Abstract: The vital objective of this article is to explore the neutrosophic nature of transitive and absorbent filters in Basic Logic (BL) algebras. We establish the notion of neutrosophic transitive and absorbent filters in BL-algebras with suitable illustrations and examine a few of their properties. Also, we prove that every neutrosophic transitive filter in BL- algebras is a neutrosophic filter. In addition, we confer some necessary and sufficient conditions for a neutrosophic filter to be a transitive filter and an extension property. Further, we obtain (i) Every neutrosophic associative filter is an absorbent filter. (ii) 𝐢 is a neutrosophic positive implicative filter if and only if it is a neutrosophic absorbent filter. (iii) If 𝐢 is a neutrosophic absorbent filter, then it is a neutrosophic fantastic filter. In the future, the above research can be extended to deductive filters. Moreover, these filters can be applied in fields such as information technology and systems. Keywords: BL-algebra; Filter; Neutrosophic filter; Neutrosophic transitive filter; Neutrosophic absorbent filter. 1. Introduction The Greek term for knowledge of neutral thought is neutrosophy. It is predicated on an examination of both opposing arguments and the neutralities that exist between them. Since there is uncertainty in everything in the world, the neutrosophic has emerged and found a home in science. In order to investigate, from a semantic perspective, the logical system whose propositional value is given in a lattice. Lattice implication algebras were introduced and some of their features were addressed by Y. Xu [1]. The idea of filters was first presented by Y. Xu and K.Y. Qin [2]. The concepts of transitive and absorbent filters were established and their features were examined by M. Sambasiva Rao [3] in lattice implication algebras. The authors were inspired to investigate this idea in Basic Logic (BL)algebras by this. Recently, the authors studied the neutrosophication of filters of BL-algebras [4]. They then developed the concept to fantastic, positive implicative and associative filters [5, 6]. Our major contributions: ➒ The ideas of neutrosophic transitive and absorbent filters are applied in BL-algebras. We obtain a few equivalent requirements for a neutrosophic filter to be neutrosophic transitive and absorbent. Finally, we establish the relationship among the neutrosophic transitive, absorbent and associative filters in BL-algebras. mailto:karjes821@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 563 https://internationalpubls.com 2. Preliminaries In this part, few of the definitions and findings from the literature are referred to progress the major conclusions. Definition 2.1[7,8] A BL-algebra (𝒒, ∨, ∧, ∘, β†’, 0,1) of type (2, 2, 2, 2, 0, 0) such that the subsequent requirements are persuaded for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒, (i) (𝒒, ∨, ∧, 0,1) is a bounded lattice, (ii) (𝒒, ∘, 1) is a commutative monoid, (iii) β€² ∘ β€² , β€² β†’ β€² is an adjoint pair, that is, 𝛾5 ≀ 𝛼5β†’ 𝛽5if and only if 𝛼5 ∘ 𝛾5 ≀ 𝛽5for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒, (iv) 𝛼5βˆ§π›½5= 𝛼5 ∘ (𝛼5→𝛽5), (v) (𝛼5→𝛽5) ∨ (𝛽5→𝛼5) = 1. Proposition 2.2[9,10]The succeeding requirements are persuaded in a BL- algebra 𝒒 for all 𝛼5, 𝛽5, 𝛾5∈ 𝒒, (i) 𝛽5β†’ (𝛼5→𝛾5) = 𝛼5β†’ (𝛽5→𝛾5) = (𝛼5 ∘ 𝛽5) →𝛾5, (ii) 1 β†’ 𝛼5 =𝛼5, (iii) 𝛼5≀ 𝛽5 if and only if 𝛼5 β†’ 𝛽5 = 1, (iv) 𝛼5βˆ¨π›½5 = ((𝛼5→𝛽5) →𝛽5) ∧ ((𝛽5→𝛼5) →𝛼5), (v) 𝛼5≀ 𝛽5 implies 𝛽5 β†’ 𝛾5 ≀ 𝛼5 →𝛾5, (vi) 𝛼5≀ 𝛽5 implies 𝛾5 β†’ 𝛼5≀ 𝛾5 →𝛽5, (vii) 𝛼5β†’ 𝛽5 ≀ (𝛾5→𝛼5) β†’ (𝛾5→𝛽5), (viii) 𝛼5β†’ 𝛽5≀ (𝛽5→𝛾5) β†’ (𝛼5→𝛾5), (ix) 𝛼5≀ (𝛼5→𝛽5) →𝛽5, (x) 𝛼5 ∘ (𝛼5→𝛽5) = 𝛼5βˆ§π›½5, (xi) 𝛼5 ∘ 𝛽5 ≀ 𝛼5βˆ§π›½5 (xii) 𝛼5β†’ 𝛽5 ≀ (𝛼5 ∘ 𝛾5) β†’ (𝛽5 ∘ 𝛾5), (xiii) 𝛼5 ∘ (𝛽5→𝛾5) ≀ 𝛽5β†’ (𝛼5 ∘ 𝛾5), (xiv) (𝛼5→𝛽5) ∘ (𝛽5→𝛾5) ≀ 𝛼5 →𝛾5, (xv) (𝛼5 ∘ 𝛼5 βˆ—) = 0. Definition 2.3[11,12] A neutrosophic subset 𝐢 of the universe π‘ˆ is a triple (𝑇𝐢, 𝐼𝐢 , 𝐹𝐢 ) where 𝑇𝐢: π‘ˆβ†’[0,1] ,𝐼𝐢 : π‘ˆβ†’[0,1]and 𝐹𝐢 : π‘ˆ β†’ [0,1] represents truth membership, indeterminacy and false membership functions respectively where 0 ≀ 𝑇𝐢(𝛼5) + 𝐼𝐢(𝛼5) + 𝐹𝐢(𝛼5) ≀ 3 for all 𝛼5 ∈ π‘ˆ. Definition 2.4[4] A neutrosophic set 𝐢 of an algebra𝒒 is called a neutrosophic filter, if it persuades the requirements: (i) 𝑇𝐢(𝛼5) ≀ 𝑇𝐢(1), 𝐼𝐢 (𝛼5) β‰₯ 𝐼𝐢 (1) and 𝐹𝐢(𝛼5) β‰₯ 𝐹𝐢(1), (ii) min {𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛼5)}≀ 𝑇𝐢(𝛽5),min{𝐼𝐢 (𝛼5 β†’ 𝛽5), 𝐼𝐢 (𝛼5)} β‰₯ 𝐼𝐢 (𝛽5) and min {𝐹𝐢( 𝛼5 β†’ 𝛽5), 𝐹𝐢(𝛼5)}β‰₯ 𝐹𝐢(𝛽5)} for all𝛼5, 𝛽5 ∈ 𝒒. Proposition 2.5[4] Let 𝐢 be a neutrosophic filter of 𝒒 if and only if (i) If 𝛼5 ≀ 𝛽5then 𝑇𝐢(𝛼5) ≀ 𝑇𝐢(𝛽5), 𝐼𝐢(𝛼5) β‰₯ 𝐼𝐢(𝛽5) and 𝐹𝐢(𝛼5) β‰₯ 𝐹𝐢(𝛽5) , Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 564 https://internationalpubls.com (ii) 𝑇𝐢(𝛼5 ∘ 𝛽5) β‰₯ min {𝑇𝐢(𝛼5),𝑇𝐢(𝛽5) }, 𝐼𝐢 (𝛼5 ∘ 𝛽5)≀ min{ 𝐼𝐢 (𝛼5), 𝐼𝐢 (𝛽5)} and 𝐹𝐢(𝛼5 ∘ 𝛽5) ≀ min { 𝐹𝐢(𝛼5), 𝐹𝐢(𝛽5)} for all 𝛼5, 𝛽5 ∈ 𝒒. Proposition 2.6[4, 5]Let 𝐢 be a neutrosophicfilter of 𝒒 for all𝛼5, 𝛽5, 𝛾5 ∈ 𝒒 then the following hold. (i) 𝑇𝐢(𝛼5 β†’ 𝛽5) = 𝑇𝐢(1), then 𝑇𝐢(𝛼5) ≀ 𝑇𝐢(𝛽5), 𝐼𝐢(𝛼5 β†’ 𝛽5) = 𝐼𝐢(1), then 𝐼𝐢(𝛼5) β‰₯ 𝐼𝐢(𝛽5), 𝐹𝐢(𝛼5 β†’ 𝛽5) = 𝐹𝐢(1), then 𝐹𝐢(𝛼5) β‰₯ 𝐹𝐢(𝛽5) (ii) 𝑇𝐢(𝛼5 ∧ 𝛽5) = min{𝑇𝐢(𝛼5), 𝑇𝐢(𝛽5)}, 𝐼𝐢(𝛼5 ∧ 𝛽5) = min{𝐼𝐢(𝛼5), 𝐼𝐢(𝛽5)}, 𝐹𝐢(𝛼5 ∧ 𝛽5) =min{𝐹𝐢(𝛼5), 𝐹𝐢(𝛽5)} (iii) 𝑇𝐢(𝛼5 ∘ 𝛽5) = min{𝑇𝐢(𝛼5), 𝑇𝐢(𝛽5)}, 𝐼𝐢(𝛼5 ∘ 𝛽5) = min{𝐼𝐢(𝛼5), 𝐼𝐢(𝛽5)}, 𝐹𝐢(𝛼5 ∘ 𝛽5) =min{𝐹𝐢(𝛼5), 𝐹𝐢(𝛽5)} (iv) 𝑇𝐢(0) = min{𝑇𝐢(𝛼5), 𝑇𝐢(𝛼5 βˆ—)}, 𝐼𝐢(0) = min{𝐼𝐢(𝛼5), 𝐼𝐢(𝛼5 βˆ—)}, 𝐹𝐢(0) = min{𝐹𝐢(𝛼5), 𝐹𝐢(𝛼5 βˆ—)} Definition 2.7[5] Let 𝐢 be called a neutrosophic fantastic filter of 𝒒, if it persuades the subsequent requirements for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒, (i) 𝑇𝐢(1) β‰₯ 𝑇𝐢(𝛼5), 𝐼𝐢(1) ≀ 𝐼𝐢(𝛼5), 𝐹𝐢(1) ≀ 𝐹𝐢(𝛼5). (ii) min{𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛼5)}≀ 𝑇𝐢(𝛽5),min{𝐼𝐢 (𝛼5 β†’ 𝛽5), 𝐼𝐢 (𝛼5)} β‰₯ 𝐼𝐢 (𝛽5) and min {𝐹𝐢( 𝛼5 β†’ 𝛽5), 𝐹𝐢(𝛼5)}β‰₯ 𝐹𝐢(𝛽5)} . (iii) 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛽5) β†’ 𝛼5) β‰₯ min{𝑇𝐢(𝛾5 β†’ (𝛽5 β†’ 𝛼5)), 𝑇𝐢(𝛾5)}, 𝐼𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛽5) β†’ 𝛼5) ≀ min {𝐼𝐢(𝛾5 β†’ (𝛽5 β†’ 𝛼5)), 𝐼𝐢(𝛾5)}, 𝐹𝐢 ((𝛼5 β†’ 𝛽5) β†’ 𝛽5) β†’ 𝛼5) ≀ min {𝐹𝐢(𝛾5 β†’ (𝛽5 β†’ 𝛼5)), 𝐹𝐢(𝛾5)}. Proposition 2.8[5] Let 𝐢 be a neutrosophic fantastic filter of 𝒒 if and only if 𝑇𝐢 (((𝛼5 β†’ 𝛽5) β†’ 𝛽5) β†’ 𝛼5) β‰₯ 𝑇𝐢(𝛽5 β†’ 𝛼5), 𝐼𝐢(((𝛼5 β†’ 𝛽5) β†’ 𝛽5) β†’ 𝛼5) ≀ 𝐼𝐢(𝛽5 β†’ 𝛼5)and𝐹𝐢 (((𝛼5 β†’ 𝛽5) β†’ 𝛽5) β†’ 𝛼5) ≀ 𝐹𝐢(𝛽5 β†’ 𝛼5)for all 𝛼5, 𝛽5 ∈ 𝒒. Definition 2.9[6] Let 𝐢 be a neutrosophic filter of a BL-algebra 𝒒. 𝐢 is called a neutrosophic positive implicative filter if it persuades the following, (i) 𝑇𝐢(𝛼5) ≀ 𝑇𝐢(1), 𝐼𝐢 (𝛼5) β‰₯ 𝐼𝐢 (1) and 𝐹𝐢(𝛼5) β‰₯ 𝐹𝐢(1), (ii) min{ 𝑇𝐢(𝛼5 β†’ ((𝛽5 β†’ 𝛾5) β†’ 𝛽5), 𝑇𝐢(𝛼5)} ≀ 𝑇𝐢(𝛽5), min{ 𝐼𝐢(𝛼5 β†’ ((𝛽5 β†’ 𝛾5) β†’ 𝛽5), 𝐼𝐢(𝛼5)} β‰₯ 𝐼𝐢(𝛽5),min{ 𝐹𝐢(𝛼5 β†’ ((𝛽5 β†’ 𝛾5) β†’ 𝛽5), 𝐹𝐢(𝛼5)} β‰₯ 𝐹𝐢(𝛽5) for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒. Definition 2.10[6] Let 𝐢 be a neutrosophic filter of a BL-algebra 𝒒. 𝐢 is called a neutrosophic associative filter if it satisfies the following, (i) 𝑇𝐢(𝛼5) ≀ 𝑇𝐢(1), 𝐼𝐢 (𝛼5) β‰₯ 𝐼𝐢 (1) and 𝐹𝐢(𝛼5) β‰₯ 𝐹𝐢(1), (ii) min{ 𝑇𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5)), 𝑇𝐢(𝛼5 β†’ 𝛽5)} ≀ 𝑇𝐢(𝛾5), min{ 𝐼𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5)), 𝐼𝐢(𝛼5 β†’ 𝛽5)} β‰₯ 𝐼𝐢(𝛾5), min{ 𝐹𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5)), 𝐹𝐢(𝛼5 β†’ 𝛽5)} β‰₯ 𝐹𝐢(𝛾5) for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒. Proposition 2.11[6] Let 𝐢 be a neutrosophic filter of 𝒒. Then, 𝐢 is a neutrosophic associative filter if and only if it satisfies 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛾5) β‰₯ 𝑇𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5)), 𝐼𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛾5) ≀ 𝐼𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5)),𝐹𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛾5) ≀ 𝐹𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5)) for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 565 https://internationalpubls.com Proposition 2.12[6]Everyneutrosophic positive implicative filter of 𝒒 is a neutrosophic fantastic filter. 3. Neutrosophic transitive filter Here, we put forward the conception of a neutrosophic transitive filter and confer its features with illustrations. Definition 3.1 Let 𝐢 be called a neutrosophic transitive filter of 𝒒, if it persuades the subsequent requirements for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒, (i) 𝑇𝐢(1) β‰₯ 𝑇𝐢(𝛼5), 𝐼𝐢(1) ≀ 𝐼𝐢(𝛼5), 𝐹𝐢(1) ≀ 𝐹𝐢(𝛼5). (ii) min{𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛼5)}≀ 𝑇𝐢(𝛽5),min{𝐼𝐢 (𝛼5 β†’ 𝛽5), 𝐼𝐢 (𝛼5)} β‰₯ 𝐼𝐢 (𝛽5) and min {𝐹𝐢( 𝛼5 β†’ 𝛽5), 𝐹𝐢(𝛼5)}β‰₯ 𝐹𝐢(𝛽5)} . (iii) 𝑇𝐢(𝛼5 β†’ 𝛾5) β‰₯ min{𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛽5 β†’ 𝛾5)}, 𝐼𝐢(𝛼5 β†’ 𝛾5) ≀ min {𝐼𝐢(𝛼5 β†’ 𝛽5), 𝐼𝐢(𝛽5 β†’ 𝛾5)}, 𝐹𝐢(𝛼5 β†’ 𝛾5) ≀ min{𝐹𝐢(𝛼5 β†’ 𝛽5), 𝐹𝐢(𝛽5 β†’ 𝛾5)}. Example 3.2 Let 𝐢 ={0, πœ–1, πœ‡1, 𝜌1, 1}.The binary operations Β° and β†’ are given by the subsequent tables (3.1) and (3.2). Table 3.1: β€² Β° β€² Operation Table 3.2: β€² β†’ β€²Operation Then, (𝒒, ∨, ∧, ∘, β†’, 0, 1) is a BL- algebra. Define a neutrosophic set 𝐢 of 𝒒 as follows: 𝐢 = {(0, [0.5,0.7,0.7]), (πœ–1, [0.5,0.7,0.7]), (πœ‡1, [0.5,0.7,0.7]), (𝜌1, [0.5,0.7,0.7]), (1, [0.6,0.7,0.7])}. It is evident that 𝐢 assures the conditions (i) and (ii) of the definition 3.1 and hence is a neutrosophic transitive filter. Proposition 3.3 Every neutrosophic transitive filter of a BL-algebra 𝒒 is a neutrosophic filter with respect to 1. Proof: Let 𝐢 be a neutrosophic transitive filter of a BL-algebra𝒒. Taking 𝛼5 = 1 in (iii) of the Definition 3.1, we get 𝑇𝐢(1 β†’ 𝛾5) β‰₯ min{𝑇𝐢(1 β†’ 𝛽5), 𝑇𝐢(𝛽5 β†’ 𝛾5)} Β° 0 πœ–1 πœ‡1 𝜌1 1 0 0 0 0 0 0 πœ–1 0 πœ–1 𝜌1 𝜌1 πœ–1 πœ‡1 0 𝜌1 πœ‡1 𝜌1 πœ‡1 𝜌1 0 𝜌1 𝜌1 𝜌1 𝜌1 1 0 πœ–1 πœ‡1 𝜌1 1 β†’ 0 πœ€1 πœ‡1 𝜌1 1 0 1 1 1 1 1 πœ€1 0 1 πœ‡1 πœ‡1 1 πœ‡1 0 πœ–1 1 πœ–1 1 𝜌1 0 1 1 1 1 1 0 πœ–1 πœ‡1 𝜌1 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 566 https://internationalpubls.com 𝑇𝐢(𝛾5) β‰₯ min{𝑇𝐢(𝛽5), 𝑇𝐢(𝛽5 β†’ 𝛾5)}for all 𝛼5, 𝛽5 ∈ 𝒒. Similarly, we can prove for 𝐼𝐢 , 𝐹𝐢 . The converse part may not be true. This can be proved by an example. Example 3.4 Let 𝐷 = {0, πœ–1, πœ‡1, 𝜌1, 1}. The bi-fold operations are specified by the tables (3.1) and (3.2). Let 𝐷 = {(0, [0.3,0.7,0.7]), (πœ–1, [0.5,0.7,0.7]), (πœ‡1, [0.3,0.7,0.7]), (𝜌1, [0.3,0.7,0.7]), (1, [0.4,0.6,0.6])}. Here, 𝐷 is not a neutrosophic transitive filter. Since,𝑇𝐷(1) = 0.4 ≱ 0.5 = 𝑇𝐷(πœ€1). Proposition 3.5 Let 𝐢 be a neutrosophic filter of a BL-algebra 𝒒. Then, 𝐢 is a neutrosophic transitive filter of 𝒒 if and only if it satisfies the following conditions for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒, 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛾5)) β‰₯ min{𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛽5 β†’ 𝛾5), 𝑇𝐢((𝛼5 β†’ 𝛽5)}, 𝐼𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛾5)) ≀ min{𝐼𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛽5 β†’ 𝛾5), 𝐼𝐢((𝛼5 β†’ 𝛽5)} 𝐹𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛾5)) ≀ min{𝐹𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛽5 β†’ 𝛾5), 𝐹𝐢((𝛼5 β†’ 𝛽5)} for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒. Proof: Let 𝐢 be a neutrosophic filter of a BL-algebra 𝒒. Assume that 𝐢 is a neutrosophic transitive filter of 𝒒. 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛾5)) β‰₯ min{𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛽5 β†’ 𝛾5)), 𝑇𝐢((𝛽5 β†’ 𝛾5) β†’ (𝛼5 β†’ 𝛾5))}, β‰₯ min{𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛽5 β†’ 𝛾5)), 𝑇𝐢(𝛼5 β†’ 𝛽5)}, Similarly, we can prove for 𝐼𝐢 , 𝐹𝐢 . Conversely, consider 𝑇𝐢(𝛼5 β†’ 𝛾5) β‰₯ 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛾5)) β‰₯ min{ 𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛽5 β†’ 𝛾5))} β‰₯ min{ 𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛽5 β†’ 𝛾5)}for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒. Similarly, we can prove for 𝐼𝐢 , 𝐹𝐢 . Hence, 𝐢 is a neutrosophic transitive filter of𝒒. Proposition 3.6 Let 𝐢 be a neutrosophic filterof𝒒 . Then, 𝐢 is a neutrosophic transitive filter of 𝒒 if it satisfies 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛾5) β‰₯ 𝑇𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5)), 𝐼𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛾5) ≀ 𝐼𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5)),𝐹𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛾5) ≀ 𝐹𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5)) for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒. Every neutrosophic transitive filter of a BL-algebra𝒒 is an associative filter. Proof: Let𝐢 be a neutrosophicfilter of 𝒒 satisfying the given condition. Then, min{𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛽5 β†’ 𝛾5)} ≀ min {𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5))} ≀ min {𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛾5)} ≀ 𝑇𝐢(𝛾5) ≀ 𝑇𝐢(𝛼5 β†’ 𝛾5) 𝑇𝐢(𝛼5 β†’ 𝛾5) β‰₯ min{𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛽5 β†’ 𝛾5)} Similarly, we can prove for 𝐼𝐢 , 𝐹𝐢 . Hence, 𝐢 is a neutrosophic transitive filter of𝒒. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 567 https://internationalpubls.com Corollary 3.7 Let 𝐢 be a neutrosophic filter of a BL-algebra 𝒒. If 𝐢 is a neutrosophic associative filter, then it is a neutrosophic transitive filter. Proof: By the proposition 3.6 and 2.11, the proof is obvious. Proposition 3.8 Each neutrosophic filter 𝐢 of a BL-algebra 𝒒 is a neutrosophic transitive filter if it satisfies𝑇𝐢(𝛾5) β‰₯ min{𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5))} 𝐼𝐢(𝛾5) ≀ min{𝐼𝐢(𝛼5 β†’ 𝛽5), 𝐼𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5))} 𝐹𝐢(𝛾5) ≀ min{𝐹𝐢(𝛼5 β†’ 𝛽5), 𝐹𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5))}for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒. Proof: Let𝐢 be a neutrosophic filter of a BL-algebra 𝒒. Assume that 𝑇𝐢(𝛾5) β‰₯ min{𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5))}. Consider 𝑇𝐢(𝛼5 β†’ 𝛾5) β‰₯ 𝑇𝐢(𝛾5) β‰₯ min {𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛾5)} β‰₯ min {𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛼5 β†’ (𝛽5 β†’ 𝛾5))} β‰₯ min{𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛽5 β†’ 𝛾5)} Similarly, we can prove for 𝐼𝐢 , 𝐹𝐢 . Hence, 𝐢 is a neutrosophic transitive filter of𝒒. Proposition 3.9 Let 𝐢 and 𝐷 be two neutrosophic filters of 𝒒 and 𝐢 be a neutrosophic transitive filter of 𝒒 . If 𝐢 βŠ† 𝐷, then 𝑇𝐢(1) = 𝑇𝐷(1), 𝐼𝐢(1) = 𝐼𝐷(1), 𝐹𝐢(1) = 𝐹𝐷(1) and 𝐷 is also a neutrosophic transitive filter. Proof: Let 𝐢 be a neutrosophic transitive filter of 𝒒. 𝑇𝐷((𝛽5 β†’ 𝛾5) β†’ ((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛾5))) β‰₯ 𝑇𝐢((𝛽5 β†’ 𝛾5) β†’ ((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛾5))) = 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ ((𝛽5 β†’ 𝛾5) β†’ (𝛼5 β†’ 𝛾5))) = 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ (𝛽5 β†’ 𝛾5) β†’ 𝛾5)) = 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ ((𝛾5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛽5))) = 𝑇𝐢((𝛾5 β†’ 𝛽5) β†’ ((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛽5))) = 𝑇𝐢((𝛾5 β†’ 𝛽5) β†’ 1) = 𝑇𝐢(1) = 𝑇𝐷(1) Since 𝐷 is a neutrosophic filter, 𝑇𝐷((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛾5)) β‰₯ min{𝑇𝐷(𝛽5 β†’ 𝛾5), 𝑇𝐷((𝛽5 β†’ 𝛾5) β†’ ((𝛼5 β†’ 𝛽5) β†’ (𝛼5 β†’ 𝛾5)))} = 𝑇𝐷(𝛽5 β†’ 𝛾5) β‰₯ min{𝑇𝐷(𝛼5 β†’ 𝛽5), 𝑇𝐷((𝛼5 β†’ 𝛽5) β†’ (𝛽5 β†’ 𝛾5))} Similarly, we can prove for 𝐼𝐷 , 𝐹𝐷. Hence, 𝐷 is a neutrosophic transitive filter. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 568 https://internationalpubls.com 4. Neutrosophic absorbent filter Here, we put forward the conception of a neutrosophic absorbent filter and confer its features with illustrations. Definition 4.1 Let 𝐢 be called a neutrosophic absorbent filter of 𝒒, if it persuades the subsequent requirements for all 𝛼5, 𝛽5, 𝛾5 ∈ 𝒒, (i) 𝑇𝐢(1) β‰₯ 𝑇𝐢(𝛼5), 𝐼𝐢(1) ≀ 𝐼𝐢(𝛼5), 𝐹𝐢(1) ≀ 𝐹𝐢(𝛼5). (ii) min{𝑇𝐢(𝛼5 β†’ 𝛽5), 𝑇𝐢(𝛼5)}≀ 𝑇𝐢(𝛽5),min{𝐼𝐢 (𝛼5 β†’ 𝛽5), 𝐼𝐢 (𝛼5)} β‰₯ 𝐼𝐢 (𝛽5) and min {𝐹𝐢( 𝛼5 β†’ 𝛽5), 𝐹𝐢(𝛼5)}β‰₯ 𝐹𝐢(𝛽5)} . (iii) 𝑇𝐢(𝛼5) β‰₯ 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛼5), 𝐼𝐢(𝛼5) ≀ 𝐼𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛼5), 𝐹𝐢(𝛼5) ≀ 𝐹𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛼5) Example 4.2 Let 𝐢 = {0, πœ–1, πœ‡1, 𝜌1, 1}. The bi-fold operations are given by the subsequent tables (4.1) and (4.2). Table 4.1: β€²Β°β€² OperationTable 4.2: β€² β†’ β€² Operation Then, (𝒒, ∨, ∧, ∘, β†’, 0, 1) is a BL- algebra. Consider a neutrosophic set 𝐢: 𝐢 = {(0, [0.5,0.7,0.7]), (πœ–1, [0.5,0.7,0.7]), (πœ‡1, [0.5,0.7,0.7]), (𝜌1, [0.5,0.7,0.7]), (1, [0.6,0.7,0.7])}. It is evident that 𝐢 assures the definition 4.1. Hence, 𝐢 is a neutrosophic absorbent filter. Proposition 4.3 Every neutrosophic associative filter of 𝒒 is a neutrosophic absorbent filter. Proof: Let 𝐢 be a neutrosophic associative filter of 𝒒. 𝑇𝐢(𝛼5) = 𝑇𝐢(1 β†’ 𝛼5) = 𝑇𝐢(((𝛼5 β†’ 𝛽5) β†’ 1) β†’ 𝛼5) β‰₯ 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ (1 β†’ 𝛼5)) = 𝑇𝐢(1 β†’ ((𝛼5 β†’ 𝛽5) β†’ 𝛼5)) = 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛼5) Therefore, 𝑇𝐢(𝛼5) β‰₯ 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛼5) Β° 0 πœ–1 πœ‡1 1 0 0 πœ–1 πœ‡1 πœ‡1 πœ–1 0 0 0 πœ–1 πœ‡1 0 0 0 πœ‡1 1 0 1 1 0 β†’ 0 πœ–1 πœ‡1 1 0 0 0 0 0 πœ–1 πœ–1 0 0 πœ–1 πœ‡1 πœ–1 1 0 πœ‡1 1 1 1 0 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 569 https://internationalpubls.com Similarly, we can prove for 𝐼𝐢 , 𝐹𝐢 . Hence, 𝐢 is a neutrosophic absorbent filter of𝒒. Proposition 4.4 A neutrosophic filter 𝐢 of a BL-algebra 𝒒 is a neutrosophic positive implicative filter if and only if it is a neutrosophic absorbent filter. Proof: Assume that𝐢 is a neutrosophic positive implicative filter of 𝒒. From the Definition 2.9, we have 𝑇𝐢(𝛼5) β‰₯ min{𝑇𝐢 (1 β†’ ((𝛼5 β†’ 𝛽5) β†’ 𝛼5)) , 𝑇𝐢(1)} = min{𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛼5), 𝑇𝐢(1)} = 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛼5) 𝑇𝐢(𝛼5) β‰₯ 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛼5)for all 𝛼5, 𝛽5 ∈ 𝒒. Similarly, we can prove for 𝐼𝐢 , 𝐹𝐢 . Hence, 𝐢 is a neutrosophic absorbent filter of𝒒. Conversely, 𝐢 is a neutrosophic absorbent filter of 𝒒. 𝑇𝐢(𝛼5) β‰₯ 𝑇𝐢((𝛼5 β†’ 𝛽5) β†’ 𝛼5) β‰₯ min{𝑇𝐢 (𝛾5 β†’ ((𝛼5 β†’ 𝛽5) β†’ 𝛼5)) , 𝑇𝐢(𝛾5)} Therefore, 𝑇𝐢(𝛼5) β‰₯ min {𝑇𝐢(𝛾5 β†’ ((𝛼5 β†’ 𝛽5) β†’ 𝛼5)) , 𝑇𝐢(𝛾5)}. Similarly, we can prove for 𝐼𝐢 , 𝐹𝐢 . Hence, 𝐢 is a neutrosophic positive implicative filter of 𝒒. Corollary 4.5 Let 𝐢 be a neutrosophic filter of a BL-algebra 𝒒. If 𝐢 is a neutrosophic absorbent filter, then it is a neutrosophic fantastic filter of 𝒒. Proof: Let 𝐢 be a neutrosophic absorbent filter of a BL-algebra 𝒒. Then, by the Proposition 4.4, 𝐢 is a neutrosophic positive implicative filter of 𝒒. Then from the Proposition 2.12, 𝐢 is a neutrosophic fantastic filter of 𝒒. 5. Conclusion In the current study, we have put forward the notions of neutrosophic transitive and absorbent filters in Basic Logic algebras and looked into a few associated features. Additionally, we have proved that every neutrosophic transitive filter in BL-algebras is a neutrosophic filter and a neutrosophic associative filter. In addition, we confer some necessary and sufficient condition, extension property for a neutrosophic filter to be a transitive filter. The purpose of this article is two fold, first is to introduce the notions in BL-algebra and then to explore their relationship among various filters. Further, we have obtained (i) Everyneutrosophic associative filter is an absorbent filter. (ii) 𝐢 is a neutrosophic positive implicative filter if and only if it is a neutrosophic absorbent filter. (iii) If 𝐢 is a neutrosophic absorbent filter, then it is a neutrosophic fantastic filter. In the future, the above research can be extended to ultra and deductive filters. 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