Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 421 https://internationalpubls.com New Operators using Β-Open Sets in a Quadripartitioned Neutrosophic Topological Spaces 1Mohanarao Navuluri, 2V Sathishkumar 1Department Of Mathematics, Annamalai University, Annamalainagar, Tamilnadu, India. (Deputed To Government College Of Engineering, Theni, Tamilnadu, India.) 2Department Of Mathematics, Rajalakshmi Institute Of Technology (Autonomous), Chennai; Department Of Mathematics, Annamalai University, Annamalainagar, Tamilnadu, India. Mohanaraonavuluri@Gmail.Com1, Vsathishkumar2020@Gmail.Com2 Article History: Received: 27-09-2024 Revised: 29-11-2024 Accepted: 09-12-2024 Abstract: In this paper, we introduce some new operators called Quadri partitioned neuromorphic β frontier, quadri- partitioned neutrosophic β border and Quadri partitioned neuromorphic β exterior with the help of quadripar- titioned neutrosophic β-open sets in quadripartitioned neutrosophic topological space. Also, we discuss the important properties of them and the relations between them. Keywords: Quadri partitioned neutrosophic β-open, Quadri partitioned neutrosophic β frontier, quadriparti- tioned neuromorphic β border, quadripartitioned neutrosophic β exterior. 1. Introduction In mathematics, Zadeh26 was first presented a idea of fuzzy set between the intervals in order of logic and set hypothesis. The fuzzy set was attempted in general topology by Chang2 as fuzzy topological space. The intu- itionistic fuzzy set which contains a membership and non-membership values was introduced by Atanassov1 in 1983. Coker4 made intuitionistic fuzzy set in a topology entitled as intuitionistic fuzzy topological spaces. The ideas of neutrosophy and neutrosophic set was presented by Smarandache16,17 toward the start of 20th century. Salama and Alblowi14,15 in 2012, originated neutrosophic set and neutrosophic crisp set in a neutrosophic topological space. In the year 2016, Chatterjee et al.3 grounded the idea of quadripartitioned neutrosophic set and defined several similarity measures between two quadripartitioned neutrosophic sets. Iswaraya and Bageerathi9 studied the concept of neutrosophic semi-open sets and neutrosophic semi-closed sets. Push- palatha and Nandhini12grounded the idea of neutrosophic generalized closed sets in NTS’s. The notion of neutrosophic b-open sets in NTS’s was presented by Ebenanjar et al.8 Rao and Srinivasa13 grounded the concept of pre-open set and pre closed set via neutrosophic topological spaces. Thereafter, Maheswari et al.10 studied the neutrosophic generalized b-closed sets in NTS’s. In the year 2019, Mohammed Ali Jaffer and Ramesh11 studied the concept of neutrosophic generalized pre-regular closed sets. The generalized neutro- sophic b-open sets in NTS’s was introduced by Das and Pramanik.6 Das and Pramanik7 also defined the neutrosophic Φ-open sets and neutrosophic Φ- continuous mappings via NTS’s. Vadivel and Sundar defined γ open sets,18 γ continuous maps,21,22 β- open sets19 and β continuous maps23–25 in N -neutrosophic crisp topological spaces and defined some operators20 in NTS’s. mailto:mohanaraonavuluri@gmail.com1 mailto:mohanaraonavuluri@gmail.com1 mailto:vsathishkumar2020@gmail.com2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 422 https://internationalpubls.com ∪ ∈ ∩ In this paper we introduce quadripartitioned neutrosophic β frontier, quadripartitioned neutrosophic β bor- der and quadripartitioned neutrosophic β exterior and discuss their properties in quadripartitioned neutrosophic topological spaces. 2. Preliminaries The needful basic definitions & properties are discussed in this section. Definition 2.1. 3 Let Z be a fixed set. Then, a quadripartitioned neutrosophic set (in-short, Q-Nss) U over Z is defined by U = {(u, TU (u), CU (u), IU (u), FU (u)): u ∈ Z} where TU, CU, IU and FU (∈ [0, 1]) are the truth, contradiction, ignorance, and falsity membership values of u ∈ Z. So, 0 ≤ TU (u) + CU (u) + IU (u) + FU (u) ≤ 4. Definition 2.2. 3 Let Z be a non-empty set & the Q-Nss’s U & Uo in the form U = {(u, TU (u), CU (u),IU (u), FU (u)) : u ∈ Z}, Uo = {(u, TUo (u), CUo (u), IUo (u), FUo ) : u ∈ Z}, then (i) 0QNs = (u, 0, 0, 1, 1) and 1QNs = (u, 1, 1, 0, 0), (ii) U ⊆ Uo iff TU (u) ≤ TUo (u), CU (u) ≤ CUo (u), IU (u) ≥ IUo (u) & FU (u) ≥ FUo (u) : u ∈ Z, (iii) 1QNs − U = {(u, FU (u), IU (u), CU (u), TU (u)) : u ∈ Z} = Uc, (iv) U ∪ Uo = {(u, max (TU (u), TUo (u)), max (CU (u), CUo (u)), min (IU (u), IUo (u)), min(FU (u), FUo (u))) : u ∈ Z}, (v) U ∩ Uo = {(u, min (TU (u), TUo (u)), min (CU (u), CUo (u)), max (IU (u), IUo (u)), max(FU (u), FUo (u) )) : u ∈ Z}. Definition 2.3. 5 Let Z be a fixed set. A collection ΓQ of some Q-Nss’s over Z is called a quadripartitioned neutrosophic topology (in-short, Q-Nst) on Z, if the following conditions holds: (i) 0N , 1N ∈ ΓQ. (ii) Gϕ ∩ Gφ ∈ ΓQ for any Gϕ, Gφ ∈ ΓQ. (iii) Gϕ ∈ ΓQ, ∀ {Gϕ : ϕ ∈ Z} ⊆ ΓQ. Then (Z, ΓQ) is called a quadripartitioned neutrosophic topological space (in-short, Q-Nsts) in Z. Every element of ΓQ are called a quadripartitioned neutrosophic open sets (in-short, Q-Nso set). If C ΓQ, then Cc is called a quadripartitioned neutrosophic closed sets (in-short, Q-Nsc set). Definition 2.4. 5 Let (Z, ΓQ) be Q-Nsts on Z and U be an Q-Nss on Z, then a quadripartitioned neutrosophic interior (resp. closure) of U (in-short, Q-Nsint(U ) (resp. Q-Nscl(U ))) are defined as Q-Nsint(U ) = ∪{Uo : Uo ⊆ U & Uo is a Q-Nso in Z}, Q-Nscl(U ) =∩{Uo : U ⊆ Uo & Uo is a Q-Nsc in Z}, Definition 2.5. 5 Let (Z, ΓQ) be Q-Nsts on Z and U be an Q-Nss on Z. Then U is said to be a quadripartitioned neutrosophic pre (resp. semi, α & b) open set (in-short, Q-Ns ƿo set (resp. Q-Ns Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 423 https://internationalpubls.com o set, Q-Nsαo set & Q-Nsbo set)) if U ⊆Q-Nsint(Q-Nscl(U )) (resp. U Q-Nscl(Q-Nsint(U )), U⊆Q-Nsint(Q-Nscl(Q-Nsint(U ))) & U ⊆ Q-Nscl(Q-Nsint(U )) Q-Nsint(Q-Nscl(U ))). The complement of an Q-Ns ƿo set (resp. Q-Ns o set, Q-Nsαo set & Q-Nsbo set) is called a quadripar titioned neutrosophic pre (resp. semi, α & b) closed set (in-short, Q-Ns Ѕc set (resp. Q-Ns c set, Q-Nsαc set & Q-Nsbc set)) in Z. The family of all Q-NsPo set (resp. Q-NsPc set, Q-NsSo set, Q-NsSc set, Q-Nsαo set, Q-Nsαc set, Q-Nsbo set & Q-Nsbc set) of Z is denoted by Q-NsPOS(Z) (resp. Q-NsPCS(Z), Q- NsSOS(Z), Q- NsSCS(Z), Q-NsαOS(Z), Q-NsαCS(Z), Q-NsbOS(Z) & Q-NsbCS(Z)). Definition 2.6. Let (Z, ΓQ) be Q-Nsts on Z and U be a Q-Nss on Z. Then U is said to be a quadriparti- tioned neutrosophic β open set (in-short, Q-Nsβo) set if U Q-Nscl(Q-Nsint(Q- Nscl(U ))). The complement of an Q-Nsβo set is called a quadripartitioned neutrosophic β closed set (in-short, Q- Nsβc set in Z. The family of all Q-Nsβo set (resp. Q-Nsβc set) of Z is denoted by Q-NsβOS(Z) (resp. Q- NsβCS(Z)). Definition 2.7. The Q-Nsβ interior of U (briefly, Q-Nsβint(U )) and Q-Nsβ closure of U (briefly, Q-Nsβcl(U )) are defined as (i) Q-Nsβint(U ) = ∪{Uo : Uo ⊆ U & Uo is a Q-Nsβo set in Z}. (ii) Q-Nsβcl(U ) = ∩{Uo : U ⊆ Uo & Uo is a Q-Nsβc set in Z}. Theorem 2.8. Let (Z, ΓQ) be Q-Nsts on Z and G be a Q-Nss on Z. Then (i) Q-Nsβcl(1 − G) = 1 − Q-Nsβint(G). (ii) Q-Nsβint(1 − G) = 1 − Q-Nsβcl(G). Theorem 2.9. Let (Z, ΓQ) be Q-Nsts on Z and G be an Q-Nss on Z. Then (i) Q-Nsβint(G) ⊆ G. (ii) G is Q-Nsβo iff Q-Nsβint(G) = G. (iii) Q-Nsβint(Q-Nsβint(G)) = Q-Nsβint(G). Theorem 2.10. Let (Z, ΓQ) be Q-Nsts on Z. Let G and T be quadripartitioned neutrosophic subsets of Z, then the following statements hold. (i) G ⊆ Q-Nsβcl(G). (ii) G is Q-Nsβc iff Q-Nsβcl(G) = G. (iii) Q-Nsβcl(Q-Nsβcl(G)) = Q-Nsβcl(G). (iv) G ⊆ T ⇒ Q-Nsβcl(G) ⊆ Q-Nsβcl(T ). (v) Q-Nsβcl(G ∩ T ) ⊆ Q-Nsβcl(G) ∩ Q-Nsβcl(T ). (vi) Q-Nsβcl(G ∪ T ) = Q-Nsβcl(G) ∪ Q-Nsβcl(T ). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 424 https://internationalpubls.com 3 Quadripartitioned neutrosophic β frontier In this section, we introduce quadripartitioned neutrosophic β frontier and discuss their properties in quadri- partitioned neutrosophic topological spaces. Definition 3.1. Let (Z, ΓQ) be a Q-Nsts with respect to F where F is a quadripartitioned neutrosophic subset of Z. Let A be a neutrosophic subset of Z. Then the quadripartitioned neutrosophic β frontier of a quadripartitioned neutrosophic subset A is denoted by Q-NsβFr(A) and is defined by Q- NsβFr(A) = Q- Nsβcl(A) ∩ Q-Nsβcl(Ac). Remark 3.2. For a quadripartitioned neutrosophic subset A of Z, Q-NsβFr(A) is a Q-Nsβc. Theorem 3.3. For a quadripartitioned neutrosophic subset A in Q-Nsts (Z, ΓQ), Q-NsβFr(A) = Q- NsβFr(Ac). Proof. Let A be a quadripartitioned neutrosophic subset in Q-Nsts (Z, ΓQ). Then by Definition 3.1, Q- NsβFr(A) = Q-Nsβcl(A) ⋂ Q-Nsβcl(Ac) = Q-Nsβcl(Ac)⋂Q-Nsβcl(A) = Q-Nsβcl(Ac)⋂(Q- Nsβcl(Ac)c). Again by Definition 3.1, this is equal to Q-NsβFr(Ac). Hence Q-NsβFr(A) = Q- NsβFr(Ac). Theorem 3.4. Let A be a quadripartitioned neutrosophic subset in Q-Nsts (Z, ΓQ). Then Q- NsβFr(A) = Q-Nsβcl(A) − Q-Nsβint(A). Proof. Let A be a quadripartitioned neutrosophic subset in Q-Nsts (Z, ΓQ). By Theorem 2.8 (ii), (Q- Nsβcl(Ac))c = Q-Nsβint(A) and by Definition 3.1, Q-NsβFr(A) = Q-Nsβcl(A) ∩ (Q- Nsβcl(Ac)) = Q- Nsβcl(A) ∩ (Q-Nsβ int(Ac))c. By using A − B = A ∩ Bc, Q-NsβFr(A) = Q- Nsβcl(A) − Q-Nsβint(A). Hence Q-NsβFr(A) = Q-Nsβcl(A) − Q-Nsβint(A). Theorem 3.5. A quadripartitioned neutrosophic subset A is Q-Nsβc set in Z if and only if Q- NsβFr(A) ⊆A. Proof. Let A be a Q-Nsβc set in the Q-Nsts (Z, ΓQ). Then by Definition 3.1, Q-NsβFr(A) = Q- Nsβcl(A) ∩ Q-Nsβcl(Ac) ⊆ Q-Nsβcl(A). By using Theorem 2.10 (ii), Q-Nsβcl(A) = A. Hence Q- NsβFr(A) ⊆ A, if A is Q-Nsβc in Z. Conversely, Assume that, Q-NsβFr(A) ⊆ A. Then Q-Nsβcl(A) − Q-Nsβint(A) ⊆ A. Since Q- Nsβint(A) ⊆ A, we conclude that Q-Nsβcl(A) = A and hence A is Q-Nsβc. Theorem 3.6. If A is a Q-Nsβo set in Z, then Q-NsβFr(A) ⊆ Ac. Proof. Let A be a Q-Nsβo set in the Q-Nsts (Z, ΓQ). By Definition 2.6, Ac is Q-Nsβc set in Z. By Theorem 3.5, Q-NsβFr(Ac) ⊆ Ac and by Theorem 3.5, we get Q-NsβFr(A) ⊆ Ac. Theorem 3.7. Let A ⊆ B and B be any Q-Nsβc set in Z. Then Q-NsβFr(A) ⊆ B. Proof. By Theorem 2.10 (iv), A ⊆ B, Q-Nsβcl(A) ⊆ Q-Nsβcl(B). By Definition 3.1, Q- NsβFr(A) = Q-Nsβcl(A) ∩ Q-Nsβcl(Ac) ⊆ Q-Nsβcl(B) ∩ Q-Nsβcl(Ac) ⊆ Q-Nsβcl(B). Then by Remark ??, this is equal to B. Hence Q-NsβFr(A) ⊆ B. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 425 https://internationalpubls.com c ∪ ∪ Theorem 3.8. Let A be a quadripartitioned neutrosophic subset in the Q-Nsts (Z, ΓQ). Then (Q- NsβFr(A))c =Q-Nsβint(A) ∪ Q-Nsβint(Ac). Proof. Let A be a quadripartitioned neutrosophic subset in the Q-Nsts (Z, ΓQ). Then by Definition 3.1, (Q-NsβFr(A))c = (Q-Nsβcl(A)∩Q-Nsβcl(Ac))c = ((Q-Nsβcl(A))c∪(Q-Nsβcl(Ac))c. By Theorem(ii), which is equal to Q-Nsβint(Ac) Q-Nsβint(A). Hence (Q-NsβFr(A))c = Q- Nsβint(A)∪Q- Nsβint(Ac). Theorem 3.9. For a quadripartitioned neutrosophic subset A in the Q-Nsts (Z, ΓQ), then Q- NsβFr(A)⊆Q- NsFr(A). Proof. Let A be a quadripartitioned neutrosophic subset in the Q-Nsts (Z, ΓQ). Then by Definition 2.7, Q-Nsβcl(A)⊇Q-Nsβcl(A) and Q-Nsβcl(Ac)⊆Q-Nscl(Ac). By Definition 3.1, Q-NsβFr(A) = Q- Nsβcl(A)∩Q-Nsβcl(Ac)⊆Q-Nscl(A)∩Q-Nscl(Ac), this is equal to Q-NsFr(A). Hence Q- NsβFr(A)⊆Q-NsFr(A). Theorem 3.10. For a quadripartitioned neutrosophic subset A in the Q-Nsts (Z, ΓQ), Q-Nsβcl(Q- NsβFr(A))⊆ Q-NsβFr(A). Proof. Let A be the quadripartitioned neutrosophic subset in the Q-Nsts (Z, ΓQ). Then by Definition 3.1,Q-Nsβcl(Q-NsβFr(A)) = Q-Nsβcl(Q-Nsβcl(A) (Q-Nsβcl(Ac)))⊆(Q-Nsβcl(Q- Nsβcl(A)))∩(Q-Nsβcl(Q-Nsβcl(Ac))). By Theorem 2.10 (iii), Q-Nsβcl(Q-Nsδ Fr(A)) = Q- Nsβcl(A)∩(Q-Nsβcl(Ac)). By Definition 3.1, this is equal to Q-NsβFr(A). Theorem 3.11. For a quadripartitioned neutrosophic subset A in the Q-Nsts (Z, ΓQ), Q-NsβFr(Q- Nsβint(A)) Q-NsβFr(A). Proof. Let A be the quadripartitioned neutrosophic subset in the Q-Nsts (Z, ΓQ). Then Q−NsβFr(Q−Nsβint(A)=Q−Nsβcl(Q−Nsint(A))∩(Q−Nsβcl(Q−Nsβint(A))c)[by Definition 3.1] =Q−Nsβcl(Q−Nsβint(A))∩(Q−Nsβcl(Q−Nsβcl(Ac)))[by Theorem 2.8(i)] =Q−Nsβcl(Q−Nsβint(A)) ∩ (Q−Nsβcl(Ac))[ by Theorem 2.10 (iii)] ⊆Q−Nsβcl(A) ∩ Q−Nsβcl(Ac)[ by Theorem 2.9 (i)] =Q−NsβFr(A)[ by Definition 3.1]. Hence Q-NsβFr(Q-Nsβint(A)) ⊆ (Q-NsβFr(A)). Theorem 3.12. For a quadripartitioned neutrosophic subset A in the Q-Nsts (Z, ΓQ), then Q- NsβFr(Q- Nsβcl(A)) ⊆ Q-NsβFr(A). Proof. Let A be a quadripartitioned neutrosophic subset in the Q-Nsts (Z, ΓQ). Then Q−NsβFr(Q−Nsβcl(A))=Q−Nsβcl(Q−Nsβcl(A))∩(Q−Nsβcl(Q−Nsβcl(A))c)[byDefinition 3.1] =Q−Nsβcl(A)∩(Q−Nsβcl(Q−Nsβint(Ac)))[byTheorem2.8(ii)and2.10(iii)&(iv)] ⊆Q−Nsβcl(A) ∩ Q−Nsβcl(A )[by Theorem 2.9 (i)] =Q−NsβFr(A)[by Definition 3.1] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 426 https://internationalpubls.com Hence Q-NsβFr(Q-Nsβcl(A))⊆ Q-NsβFr(A). Theorem 3.13. Let A be a quadripartitioned neutrosophic subset in the Q-Nsts (Z, ΓQ). Then Q- Nsβint(A) ⊆A − Q-NsβFr(A). Proof. Let A be a quadripartitioned neutrosophic subset in the Q-Nsts(Z,ΓQ).Now by Definition 3.1, A − Q−NsβFr(A) =A ∩ (Q−NsβFr(A))c =A ∩ [Q−Nsβcl(A) ∩ Q−Nsβcl(Ac)]c =A ∩ [Q−Nsβint(Ac) ∪ Q−Nsβint(A)] =[A ∩ Q−Nsβint(Ac)] ∪ [A ∩ Q−Nsβint(A)] =[A ∩ Q−Nsβint(Ac)] ∪ Q−Nsβint(A) ⊇ Q−Nsβint(A) Hence Q-Nsβint(A) ⊆ A − Q-NsβFr(A). Theorem 3.14. Let A and B be quadripartitioned neutrosophic subsets in the Q-Nsts (Z, ΓQ). Then Q- NsβFr(A ∪ B) ⊆ Q-NsβFr(A) ∪ Q-NsβFr(B). Proof. Let A and B be quadripartitioned neutrosophic subsets in the Q-Nsts (Z, ΓQ). Then Q−NsβFr(A ∪ B) =Q−Nsβcl(A ∪ B) ∩ Q−Nsβcl(A ∪ B)c[by Definition 3.1] =Q−Nsβcl(A ∪ B) ∩ Q−Nsβcl(Ac ∩ Bc) ⊆(Q−Nsβcl(A) ∪ Q−Nsβcl(B) ∩ ((Q−Nsβcl(A ))) ∩ (Q−Nsβcl(B )) [by Theorem 2.10 (v) & (vi)] =[(Q−Nsβcl(A) ∪ (Q−Nsβcl(B)) ∩ (Q−Nsβcl(Ac)))] ∩ [(Q−Nsβcl(A)∪ (Q−Nsβcl(B)) ∩ (Q−Nsβcl(Bc)))] =[(Q−Nsβcl(A) ∩ Q−Nsβcl(Ac)) ∪ ((Q−Nsβcl(B) ∩ (Q−Nsβcl(Ac))))]∩ [(Q−Nsβcl(A) ∩ (Q−Nsβcl(Bc))) ∪ ((Q−Nsβcl(B) ∩ (Q−Nsβcl(Bc))))] =[Q−NsβFr(A) ∪ (Q−Nsβcl(B)) ∩ (Q−Nsβcl(Ac))] ∩ [(Q−Nsβcl(A)∩ (Q−Nsβcl(Bc))) ∪ (Q−NsβFr(B))][by Definition 3.1] =(Q−NsβFr(A) ∪ Q−NsβFr(B)) ∩ [(Q−Nsβcl(B) ∩ (Q−Nsβcl(Ac)))∪ ((Q−Nsβcl(A) ∩ Q−Nsβcl(Bc)))] ⊆Q−NsβFr(A) ∪ Q−NsβFr(B). Hence, Q-NsβFr(A ∪ B) ⊆ Q-NsβFr(A) ∪ Q-NsβFr(B). Theorem 3.15. For any quadripartitioned neutrosophic subsets A and B in the Q-Nsts (Z, ΓQ), Q- NsβFr(A∩ B) ⊆ (Q-NsβFr(A) ∩ (Q-Nsβcl(B))) ∪ (Q-NsβFr(B) ∩ Q-Nsβcl(A)). Proof. Let A and B be quadripartitioned neutrosophic subsets in the Q-Nsts (Z, ΓQ). Then Q−NsβFr(A ∩ B) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 427 https://internationalpubls.com =Q−Nsβcl(A ∩ B) ∩ (Q−Nsβcl(A ∩ B)c)[by Definition 3.1] =Q−Nsβcl(A ∩ B) ∩ (Q−Nsβcl(Ac ∪ Bc)) ⊆(Q−Nsβcl(A) ∩ Q−Nsβcl(B))∩ (Q−Nsβcl(A ) ∪ Q−Nsβcl(B ))[by Theorem 2.10 (v) & (vi)] =[(Q−Nsβcl(A)∩Q−Nsβcl(B))∩Q−Nsβcl(Ac)]∪[(Q−Nsβcl(A)∩Q−Nsβcl(B))∩ Q−Nsβcl(Bc)] =(Q−NsβFr(A) ∩ Q−Nsβcl(B)) ∪ (Q−NsβFr(B) ∩ Q−Nsβcl(A))[by Definition 3.1]. Hence Q-NsβFr(A ∩ B)⊆((Q-NsβFr(A)∩(Q-Nsβcl(B)))∪(Q-NsβFr(B)∩(Q-Nsβcl(A)))). Corollary 3.16. For any quadripartitioned neutrosophic subsets A and B in the Q-Nsts (Z, ΓQ), Q-NsβFr(A∩ B) ⊆ Q-NsβFr(A) ∪ Q-NsβFr(B). Proof. Let A and B be quadripartitioned neutrosophic subsets in the Q-Nsts (Z, ΓQ). Then Q−NsβFr(A ∩ B) =Q−Nsβcl(A ∩ B) ∩ (Q−Nsβcl(A ∩ B)c)[by Definition 3.1] =Q−Nsβcl(A ∩ B) ∩ (Q−Nsβcl(Ac∪Bc) ⊆(Q−Nsβcl(A) ∩ Q−Nsβcl(B)) ∩ (Q−Nsβcl(A) ∪ Q−Nsβcl(B ))[by Theorem 2.10 (v) & (vi)] =(Q−Nsβcl(A)∩Q−Nsβcl(B))∩(Q−Nsβcl(Ac)∪(Q−Nsβcl(A)∩Q−Nsβcl(B))∩ (Q−Nsβcl(Bc))) =(Q−NsβFr(A) ∩ Q−Nsβcl(B)) ∪ (Q−Nsβcl(A) ∩ Q−NsβFr(B))[by Definition 3.1] ⊆Q−NsβFr(A) ∪ (Q−NsβFr(B). Hence Q-NsβFr(A ∩ B) ⊆ Q-NsβFr(A) ∪ Q-NsβFr(B). Theorem 3.17. For any quadripartitioned neutrosophic subset A in the Q-Nsts (Z, ΓQ), (i) Q-NsβFr(Q-NsβFr(A)) ⊆ Q-NsβFr(A), (ii) Q-NsβFr(Q-NsβFr(Q-NsβFr(A))) ⊆ Q-NsβFr(Q-NsβFr(A)). Proof. (i) Let A be a quadripartitioned neutrosophic subset in the Q-Nsts (Z, ΓQ). Then Q−NsβFr(Q−NsβFr(A)) =Q−Nsβcl(Q−NsβFr(A)) ∩ Q−Nsβcl(Q−NsβFr(A)c) by [Definition 3.1] =Q−Nsβcl(Q−Nsβcl(A) ∩ (Q−Nsβcl (Ac)) ∩ (Q−Nsβcl(Q−Nsβcl(A)) ∩ (Q−Nsβcl (Ac)) c)) by [ Definition 3.1] ⊆(N N δcl(Q−Nsβcl(A)) ∩ (Q−Nsβcl(Q−Nsβcl(A)))∩(Q−Nsβcl(Q−Nsβint(A)))∪ (Q−Nsβint(A)))[by Theorem 2.10 (iii) & (v)] =(Q−Nsβcl(A) ∩ (Q−Nsβcl(Ac)) ∩ (Q−Nsβcl(Q−Nsβint(A) ∪ Q−Nsβint(A)))) [by Theorem 2.10 (iii)] ⊆Q−Nsβcl(A) ∩ Q−Nsβcl(Ac) =Q−NsβFr(A) [by Definition 3.1]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 428 https://internationalpubls.com Therefore Q-NsβFr(Q-NsβFr(A)) ⊆ Q-NsβFr(A). (ii) Again, Q-NsβFr(Q-NsβFr(Q-NsβFr(A))) ⊆ Q-NsβFr(Q-NsβFr(A)). 4 Quadripartitioned neutrosophic β border and quadripartitioned neutrosophic β exterior In this section, we introduce the quadripartitioned neutrosophic β border, quadripartitioned neutrosophic β exterior using quadripartitioned neutrosophic β open sets and their properties are discussed in Q-Nsts’s. Definition 4.1. Let A be a quadripartitioned neutrosophic subset of Q-Nsts (Z, ΓQ). Then the set Q- NsBr(A) = A Q-Nsint(A) (resp. Q-NsβBr(A) = A -Q-Nsβint(A)) is called the quadripartitioned neutrosophic (resp. quadripartitioned neutrosophic β) border of A. Theorem 4.2. If a subset A of Z is Q-Nsβc, then Q-NsβBr(A) = Q-NsβFr(A). Proof. Let A be a Q-Nsβc subset of Z. Then by Theorem 2.8 (ii), Q-Nsβcl(A) = A. Now, Q- NsβFr(A) = Q-Nsβcl(A) − Q-Nsβint(A) = A − Q-Nsβint(A) = Q-NsβBr(A). Theorem 4.3. For a quadripartitioned neutrosophic subset A of Z, A = Q-Nsβint(A) ∪ Q- NsβBr(A). Proof. Let x(e1,e2,e3,e4) ∈ A. If x(e1,e2,e3,e4) ∈ Q-Nsβint(A), then the result is obvious. If x(e1,e2,e3,e4) ̸∈ Q-Nsβint(A), then by the definition of Q-NsβBr(A), x(e1,e2,e3,e4) ∈ Q- NsβBr(A). Hence x(e1,e2,e3,e4) ∈ Q-Nsβint(A) ∪ Q-NsβBr(A) and so A ⊆ Q-Nsβint(A) ∪ Q-NsβBr(A). On the other hand, since Q- Nsβint(A) ⊆ A and Q-NsβBr(A) ⊆ A, we have Q- Nsβint(A) ∪ Q-NsβBr(A) ⊆ A. Theorem 4.4.For a quadripartitioned neutrosophic subset A of Z,Q-Nsβint(A)∩Q-NsβBr(A)=0N . Proof. Suppose Q-Nsβint(A)∩Q-NsβBr(A)≠0N .Let x(e1,e2,e3,e4)∈ Q-Nsβint(A)∩Q-NsβBr(A). Then x(e1,e2,e3,e4) ∈ Q-Nsβint(A) and x(e1,e2,e3,e4) ∈ Q-NsβBr(A). Since Q-NsβBr(A)= A − Q- Nsβint(A), then x(e1,e2,e3,e4) ∈ A. But x(e1,e2,e3,e4) ∈ Q-Nsβint(A), x(e1,e2,e3,e4) ∈ A. There is a contra- diction. Hence Q-Nsβint(A) ∩ Q-NsβBr(A) = 0N . Theorem 4.5. For a quadripartitioned neutrosophic subset A of Z, A is a Q-Nsβo set if and only if Q- NsβBr(A) = 0N . Proof. Necessity: Suppose A is Q-Nsβo. Then by Theorem 2.9 (ii), Q-Nsβint(A) = A. Now, Q- NsβBr(A) = A - Q-Nsβint(A) = A - A = 0N . Sufficiency: Suppose Q-NsβBr(A) = 0N . This implies, A - Q-Nsβint(A) = 0N . Therefore A = Q- Nsβint(A) and hence A is Q-Nsβo. Corollary 4.6. For a Q-Nsts, Q-NsβBr(0N ) = 0N and Q-NsβBr(1N ) = 0N . Proof.Since 0N and 1N are Q-Nsβo, by Theorem 4.5,Q-NsβBr(0N)=0N and Q-NsβBr(1N)= 0N Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 429 https://internationalpubls.com Theorem 4.7. For a quadripartitioned neutrosophic subset A of Z, Q-NsβBr(Q-Nsβint(A)) = 0N . Proof. By the definition of Q-Nsβ border, Q-NsβBr(Q-Nsβint(A)) = Q-Nsβint(A) - Q-Nsβint(Q- Nsβint(A)). By Theorem 2.9 (iii), Q-Nsβint(Q-Nsβint(A)) = Q-Nsβint(A) and hence Q- NsβBr(Q- Nsβint(A)) = 0N . Theorem 4.8. For a quadripartitioned neutrosophic subset A of Z, Q-Nsβint(Q-NsβBr(A))=0N . Proof. Let x(e1,e2,e3,e4) ∈ Q-Nsβint(Q-NsβBr(A)). Since Q-NsβBr(A) ⊆ A, by Theorem 2.9 (i), Q-Nsβint(Q-NsβBr(A)) ⊆ Q-Nsβint(A). Hence x(e1,e2,e3,e4) ∈ Q-Nsβint(A). Since Q- Nsβint(Q- NsβBr(A)) ⊆ Q-NsβBr(A), x(e1,e2,e3,e4) ∈ Q-NsβBr(A). Therefore x(e1,e2,e3,e4) ∈ Q-Nsβint(A) ∩ Q- NsβBr(A), x(e1,e2,e3,e4) = 0N . Theorem 4.9. For a quadripartitioned neutrosophic subset A of Z, Q-NsβBr(Q-NsβBr(A)) = Q- NsβBr(A). Proof. By the definition of Q-Nsβ border, Q-NsβBr(Q-NsβBr(A)) = Q-NsβBr(A) - Q- Nsβint(Q- NsβBr(A)). By Theorem 4.8 Q-Nsβint(Q-NsβBr(A)) = 0N and hence Q-NsβBr(Q- NsβBr(A)) = Q- NsβBr(A). Theorem 4.10. Let A be a quadripartitioned neutrosophic subset of Z. Then, Q-NsβBr(A) = A ∩Q- Nsβcl(Ac). Proof. Since Q-NsβBr(A) = A − Q-Nsβint(A) and by Theorem 2.10, Q-NsβBr(A)=A − (Q- Nsβcl(Ac))c = A ∩ (Q-Nsβcl (Ac)c) = A ∩ Q-Nsβcl(Ac). Theorem 4.11. For a quadripartitioned neutrosophic subset A of Z, Q-NsβBr(A) ⊆ Q-NsβFr(A). Proof. Since A ⊆ Q-Nsβcl(A), A − Q-Nsβint(A) ⊆ Q-Nsβcl(A) − Q-Nsβint(A). That implies,Q-NsβBr(A) ⊆ Q-NsβFr(A). Definition 4.12. Let A be a quadripartitioned neutrosophic subset of a Q-Nsts (Z, ΓQ). The quadriparti- tioned neutrosophic (resp. β) interior of Ac is called the quadripartitioned neutrosophic (resp. quadripartitioned neutrosophic β) exterior of A and it is denoted by Q-NsExt(A) (resp. Q- NsβExt(A)). That is, Q-NsExt(A) = Q-Nsint(Ac) (resp. Q-NsβExt(A) = Q-Nsβint(Ac)). Theorem 4.13. For a quadripartitioned neutrosophic subset A of Z, Q-NsβExt(A)=(Q-Nsβcl(A))c. Proof. We know that, U - Q-Nsβcl(A =Q-Nsβint(Ac), then Q-NsβExt(A)=Q-Nsβint(Ac) = (Q-Nsβcl(A))c. Theorem 4.14. For a quadripartitioned neutrosophic subset A of Z, Q-NsβExt(Q-NsβExt(A)) = Q- Nsβint(Q-Nsβcl(A)) ⊇ Q-Nsβint(A). Proof. Now, Q-NsβExt(Q-NsβExt(A)) = Q-NsβExt(Q-Nsβint(Ac)) = Q-Nsβint((Q- Nsβint(Ac))c) =Q-Nsβint(Q-Nsβcl(A)) ⊇ Q-Nsβint(A). Theorem 4.15. For a quadripartitioned neutrosophic subset A of Z, If A⊆ B, then Q-NsβExt(B) ⊆Q- NsβExt(A). Proof. Suppose A⊆B.Now, Q-NsβExt(B)=Q-Nsβint(Bc) ⊆ Q-Nsβint(Ac)= Q-NsβExt(A). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 430 https://internationalpubls.com Theorem 4.16. For a quadripartitioned neutrosophic subset A of Z, Q-NsβExt(1N ) = 0N and Q- NsβExt(0N ) = 1N . Proof. Now, Q-NsβExt(1N ) = Q-Nsβint((1N )c) = Q-Nsβint(0N ) and Q-NsβExt(0N ) = Q- Nsβint((0N )c) = Q-Nsβint(1N ). Since 0N and 1N are Q-Nsβo sets, then Q-Nsβint(0N ) = 0N and Q-Nsβint(1N ) = 1N . Hence Q-NsβExt(0N ) = 1N and Q-NsβExt(1N ) = 0N . Theorem 4.17. For a quadripartitioned neutrosophic subset A of Z, Q-NsβExt(A) = Q- NsβExt((Q- NsβExt(A))c). Proof. Now, Q-NsβExt((Q-NsβExt(A))c) = Q-NsβExt((Q-Nsβint(Ac))c) = Q- Nsβint((((Q- Nsβint(Ac))c))c) = Q-Nsβint (Q-Nsβint(Ac)) = Q-Nsβint(Ac) = Q- NsβExt(A). Theorem 4.18. For a sub sets A and B of Z, the followings are valid. (i) Q-NsβExt(A ∪ B) ⊆ Q-NsβExt(A) ∩ Q-NsβExt(B). (ii) Q-NsβExt(A ∩ B) ⊇ Q-NsβExt(A) ∪ Q-NsβExt(B). Proof. (i) Q-NsβExt(A ∪ B) = Q-Nsβint((A ∪ B)c) = Q-Nsβint((Ac) ∩ (Bc)) ⊆ Q-Nsβcl(Ac) ∩ Q- Nsβcl(Bc) = Q-NsβExt(A) ∩ Q-NsβExt(B). (ii) Q-NsβExt(A ∩ B) = Q-Nsβint((A ∩ B)c) = Q-Nsβint((Ac) ∪ (Bc)) ⊇ Q-Nsβcl(Ac) ∪ Q- Nsβcl(Bc) = Q-NsβExt(A) ∪ Q-NsβExt(B). 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