Filters in Hoops based on Lukasiewicz Neutrosophic set

University of New Mexico Filters in Hoops based on Lukasiewicz Neutrosophic set N. Abirami1and M. Mary Jansirani2,∗ 1Research scholar, School of Sciences, Division of Mathematics, SRMInstitute of Science and Technology, Tiruchirappalli campus, SRM Nagar, Trichy-Chennai highway, near Samayapuram, Tiruchirappalli-621105, Tamil Nadu, India; [email protected],[email protected] 2School of Sciences, Division of Mathematics, SRMInstitute of Science and Technology, Tiruchirappalli campus, SRM Nagar, Trichy-Chennai highway, near Samayapuram, Tiruchirappalli-621105, Tamil Nadu, India; anthuv[email protected] ∗Correspondence: anthuv[email protected] Abstract.The Lukasiewicz neutrosophic set (LN S) is developed using the ideas of Lukasiewicz t-norm and s-norm. This set is then applied to the hoop structure and introduces the Lukasiewicz neutrosophic filter (LN F). The properties of this filter, as well as its interconnection with the Lukasiewicz fuzzy filter (LFF), are subsequently investigated. Keywords: Lukasiewicz neutrosophic set; Lukasiewicz neutrosophic filter; Lukasiewicz fuzzy filter; Neutrosophic point ; Hoop —————————————————————————————————————————- 1. Introduction Professor Lofti A. Zadeh, the father of fuzzy systems theory, proposed the idea of fuzzy logic in 1965. Extending this, Atanassov introduced intuitionistic fuzzy sets by adding the degree of belongingness and non-belongingness. As a generalization of the Zadeh and Atanassov system, Florentin Smarandache introduced the neutrosophic set, which deals with belongingness, in determinacy, and non-belongingness, in 1998 [15]. The concept of neutrosophic point and its properties was introduced by Gautam Chandra Ray and Sudeep Dey [13]. Neutrosophic algebraic structures were studied by Kandasamy and Smarandache [10,17]. Many researchers have since explored neutrosophic filters in different algebraic frameworks [5–7,12,14,16,18,19]. Hoops is an algebraic structure introduced by Bosbach [3]. Lukasiewicz logic is the manyvalued logic, which is the extension of classical binary logic. Y.B. Jun introduced the Lukasiewicz fuzzy set using Lukasiewicz t-norm and the Lukasiewicz intuitionistic fuzzy set N. Abirami, M. Mary Jansirani, Filters in Hoops based on Lukasiewicz Neutrosophic set Neutrosophic Sets and Systems, Vol. 97, 2026 using Lukasiewicz t-norm and the dual of Lukasiewicz t-norm (Lukasiewicz t-conorm) and applied them in BCK algebras [9, 11], while Mohseni Takallo et al. studied the Lukasiewicz fuzzy filters in hoops [11] and Jun et al. extended these ideas to BE-algebras [8]. However, these filters cannot adequately capture indeterminacy, which plays a crucial role in uncertainty modeling. In this paper, we address this gap by introducing the Lukasiewicz neutrosophic set, which extends Lukasiewicz fuzzy and intuitionistic fuzzy sets by capturing the indeterminacy component, which has remained unexplored in hoop structures. We introduced the Lukasiewicz neutrosophic filter and studied its characteristics using the neutrosophic point. We provide a relationship between the Lukasiewicz fuzzy filter and the Lukasiewicz neutrosophic filter. 1.1. Comparative Analysis Lukasiewicz intuitionistic fuzzy filter handles uncertainty better than the Lukasiewicz fuzzy filter. However, both of these filters fail to capture indeterminancy. Our proposed Lukasiewicz neutrosophic filter plays a significant role where indeterminancy occurs, and is an advancement of existing theories. Term Notation Lukasiewicz fuzzy set LFS Lukasiewicz fuzzy filter LFF Lukasiewicz neutrosophic set LNS Lukasiewicz neutrosophic filter LNF 2. Preliminaries Definition 2.1. [11] If an algebra (H,⊛,⇝,1) satisfies the axioms (H1,H2,H3,H4), then it is said to be a hoop. H1 : n⇝n= 1 for all n ∈ H H2 : n⊛(n⇝s) = s⊛(s⇝n)for all n, s ∈ H H3 : n⇝(s⇝p) = (n⊛s)⇝p for all n, s, p ∈ H H4:(H,⊛,1) is a commutative monoid. In a hoop H, we say that an element nis less than or equal to s, that is n≤sif and only if n⇝s= 1. Proposition 2.2. [3] All of the following claims are met by every hoop H. N. Abirami, M. Mary Jansirani, Filters in Hoops based on Lukasiewicz Neutrosophic set Neutrosophic Sets and Systems, Vol. 97, 2026 710 n⊛s≤p⇐⇒ n≤s⇝p(1) n⊛s≤n, s (2) n≤s⇝n(3) n⇝1=1,1⇝n=n(4) (n⇝s)⊛(s⇝p)≤n⇝p(5) n≤s=⇒n⊛p≤s⊛p(6) n⊛(n⇝s)≤s(7) ∀n, s, p ∈ H. Definition 2.3. [13] Let X be the universe of discourse. A neutrosophic set Nover X is defined by N={(x, TN(x), IN(x), F N(x))|x∈X} where TN, IN, F N:X→[0,1] and 0 ≤TN(x) + IN(x) + FN(x)≤3. Definition 2.4. [13] A neutrosophic set P={(x, T P(x), IP(x), F P(x))|x∈X}is called a neutrosophic point if it is of the form P(y) =    (α, β, γ),if y=x (0,1,1),for y=x where 0 < α ≤1,0≤β < 1,0≤γ < 1. For the neutrsophic point P={(x, T P(x), IP(x), F P(x))|x∈X}with support x will be denoted by Px α,β,γ or p < x, α, β, γ > or xα,β,γ. Definition 2.5. [13] Let Nbe a neutrosophic set over X and xα,β,γ be the neutrosophic point in X. Then xα,β,γ is said to belong to N, denoted by xα,β,γ ∈ N if and only if α≤TN(x), β ≥IN(x), γ ≥FN(x). Definition 2.6. [11] Consider a fuzzy set Fin Hand let ε∈(0,1). A function from Hto [0,1] defined by Lε F(x) = max{0,F(x) + ε−1} is called the Lukasiewicz fuzzy set (LFS) of Fin H. Definition 2.7. [11] ALFS Lε Fis called a Lukasiewicz fuzzy filter (LFF) of Hif it satisfies: Lε F(1) is an upper bound of {Lε F(x)|x∈ H} (8) N. Abirami, M. Mary Jansirani, Filters in Hoops based on Lukasiewicz Neutrosophic set Neutrosophic Sets and Systems, Vol. 97, 2026 711 Lε F(y)≥ {Lε F(x), Lε F(x⇝y)}(9) 3. Lukasiewicz Neutrosophic Filters The concepts of Lukasiewicz neutrosophic set(LN S) and Lukasiewicz neutrosophic filter(LNF) in the hoop structure were presented in this section, and their properties were examined. Definition 3.1. Let Nbe a neutrosophic set in a hoop Hand let ε, δ, θ ∈[0,1] be such that 0 ≤ε+δ+θ≤3. A mapping defined as an object in the form below LN={(x, Lε TN, Lδ IN, Lθ FN)|x∈ H} where Lε TN:H → [0,1], Lε TN(x) = max{0, TN(x) + ε−1} Lδ IN:H → [0,1], Lδ IN(x) = min{1, IN(x) + δ} Lθ FN:H → [0,1], Lθ FN(x) = min{1, FN(x) + θ} such that 0 ≤Lε TN+Lδ IN+Lθ FN≤3∀x∈ H is called the LNS of Nin Hdenoted by LN= (Lε TN, Lδ IN, Lθ FN). Let LN= (Lε TN, Lδ IN, Lθ FN) be the LN S of Nin H. If (ε, δ, θ) = (1,0,0), then max{0, TN(x)+ε−1}=TN(x) and min{1, IN(x)+δ}=IN(x), min{1, FN(x)+θ}=FN(x). This shows that if (ε, δ, θ) = (1,0,0), then the LN S,LN= (Lε TN, Lδ IN, Lθ FN) of Nis the classical neutrosophic set Nin H. If (ε, δ, θ) = (0,1,1), then max{0, TN(x) + ε−1}= 0 and min{1, IN(x) + δ}= 1, min{1, FN(x) + θ}= 1.Thus if (ε, δ, θ) = (0,1,1), then the LNS,LN= (Lε TN, Lδ IN, Lθ FN) of Nis the constant function with the value(0,1,1).Therefore, in handling the LNS, the value of (ε, δ, θ) can always be considered to be in (0,1) x (0,1) x (0,1). Definition 3.2. ALNS LNin His called a LNF of Hif it satisfies: nα1,β1,γ1∈LN, sα2,β2,γ2∈LN⇒(n⊛s)α1∧α2,β1∨β2,γ1∨γ2∈LN(10) n≤s, nα1,β1,γ1∈LN⇒sα1,β1,γ1∈LN(11) for all n, s ∈ H,0< α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1. Example 3.3. Consider a Hoop structure Hwith the binary operations ′′ ⊛′′ and ′′ ⇝′′ as below. N. Abirami, M. Mary Jansirani, Filters in Hoops based on Lukasiewicz Neutrosophic set Neutrosophic Sets and Systems, Vol. 97, 2026 712 Table 3.1 ⊛n s p 1 nn n n n sn n s s pn s p p 1 n s p 1 Table 3.2 ⇝n s p 1 n1 1 1 1 ss 1 1 1 pn s 1 1 1 n s p 1 Define the neutrosophic set Nin Has follows, N=                (0.63,0.23,0.35),if x=n (0.63,0.05,0.30),if x=s (0.70,0.01,0.27),if x=p (0.91,0,0.2),if x= 1 For (ε,δ,θ)=(1,0.5,0.6), the LNS is given as follows LN=                (0.63,0.73,0.95),if x=n (0.63,0.55,0.90),if x=s (0.70,0.51,0.87),if x=p (0.91,0.5,0.8),if x= 1 Now we can check that it is a LN F of H. Theorem 3.4. ALNS LNin His a LNF of Hif and only if the following conditions are valid. LTε N(1) is an upper bound of {LTε N(n)|n∈ H} Lδ IN(1) is a lower bound of {Lδ IN(n)|n∈ H} Lθ FN(1) is a lower bound of {Lθ FN(n)|n∈ H} (12) n(α1,β1,γ1)∈LN,(n⇝s)α2,β2,γ2∈LN⇒sα1∧α2,β1∨β2,γ1∨γ2∈LN(13) for all n, s ∈ H,∀0< α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1. Proof. Suppose that LNis a LN F of H. If LTε N(1) is not an upper bound of {LTε N(n)|n∈ H},then LTε N(1) < LTε N(m) for some m∈ H. since m≤1 and mLTε N(m),Lδ IN(m),Lθ FN(m)∈LN, it follows N. Abirami, M. Mary Jansirani, Filters in Hoops based on Lukasiewicz Neutrosophic set Neutrosophic Sets and Systems, Vol. 97, 2026 713 from (11) that 1LTε N(m),Lδ IN(m),Lθ FN(m)∈LN. That is, LTε N(1) ≥LTε N(m), Lδ IN(1) ≤Lδ IN(m),Lθ FN(1) ≤Lθ FN(m). Thus LTε N(1) is an upper bound of {LTε N(n)|n∈ H}, Lδ IN(1) is a lower bound of {Lδ IN(n)|n∈ H}, and Lθ FN(1) is a lower bound of {Lθ FN(n)|n∈ H}. Let n, s ∈ H and 0 < α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1, n(α1,β1,γ1)∈LNand (n⇝s)α2,β2,γ2∈LN. Then by (10) we have (n⊛(n⇝s))α1∧α2,β1∨β2,γ1∨γ2∈LN. Since (n⊛(n⇝s)) ≤s, by (11) we have that, sα1∧α2,β1∨β2,γ1∨γ2∈LN. Assume that LNsatisfies (12) and (13). Let n, s ∈ H,0< α ≤1,0≤β < 1,0≤γ < 1,be such that n≤sand nα,β,γ ∈LN. Then it follows from (12) that LTε N(n⇝s) = LTε N(1) ≥LTε N(n)≥α Lδ IN(n⇝s) = Lδ IN(1) ≤Lδ IN(n)≤β Lθ FN(n⇝s) = Lθ FN(1) ≤Lθ FN(n)≤γ ⇒(n⇝s)α,β,γ ∈Lε N. The condition (13) leads to sα,β,γ ∈Lε Nwhich proves (11). Let n, s ∈ H, and 0< α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<be such that n(α1,β1,γ1)∈LN and sα2,β2,γ2∈LN. Since, n⇝(s⇝n⊛s) = (n⊛s)⇝(n⊛s) (by H3) = 1 (by H1). we have LTε N(n⇝(s⇝n⊛s)) = LTε N(1) ≥LTε N(s)≥α2 Lδ IN(n⇝(s⇝n⊛s)) = Lδ IN(1) ≤Lδ IN(s)≤β2 Lθ FN(n⇝(s⇝n⊛s)) = Lθ FN(1) ≤Lθ FN(s)≤γ2 i.e., (n⇝(s⇝n⊛s))α2,β2,γ2∈LN Hence by (13) we have, (s⇝n⊛s)α1∧α2,β1∨β2,γ1∨γ2∈LN. Again by (13) we have, (n⊛s)α1∧α2,β1∨β2,γ1∨γ2∈LNwhich shows (10). ∴LNin His a LNF of H. Theorem 3.5. ALN S LNin His a LN F of Hif and only if it satisfies: nα,β,γ ∈LN⇒1α,β,γ ∈LN(14) N. Abirami, M. Mary Jansirani, Filters in Hoops based on Lukasiewicz Neutrosophic set Neutrosophic Sets and Systems, Vol. 97, 2026 714 ∀n∈ H,0< α ≤1,0≤β < 1,0≤γ < 1. LTε N(s)≥min{LTε N(n), LTε N(n⇝s)} Lδ IN(s)≤max{Lδ IN(n), Lδ IN(n⇝s)} Lθ FN(s)≤max{Lθ FN(n), Lθ FN(n⇝s)} (15) forall n, s ∈ H. Proof. Assume that LNis a LNF of H. Let n∈ H and 0 < α ≤1,0≤β < 1,0≤γ < 1 be such that nα,β,γ ∈LN. The condition (12) leads to LTε N(1) ≥LTε N(n)≥α Lδ IN(1) ≤Lδ IN(n)≤β Lθ FN(1) ≤Lθ FN(n)≤γ which implies that 1α,β,γ ∈LN. we know that nLTε N(n),Lδ IN(n),Lθ FN(n)∈LNand (n⇝s)LTε N(n⇝s),Lδ IN(n⇝s),Lθ FN(n⇝s)∈LN∀n, s ∈ H. It follows from (13) that sLTε N(n)∧LTε N(n⇝s),Lδ IN(n)∨Lδ IN(n⇝s),Lθ FN(n)∨Lθ FN(n⇝s)∈LNand hence LTε N(s)≥min{LTε N(n), LTε N(n⇝s)} Lδ IN(s)≤max{Lδ IN(n), Lδ IN(n⇝s)} Lθ FN(s)≤max{Lθ FN(n), Lθ FN(n⇝s)} for all n,s ∈ H. Conversely suppose that LNsatisfies (14) and (15). Since nLTε N(n),Lδ IN(n),Lθ FN(n)∈LNfor all n∈ H, we have by (14) that 1LTε N(n),Lδ IN(n),Lθ FN(n)∈LNand so LTε N(1) ≥LTε N(n), Lδ IN(1) ≤Lδ IN(n), Lθ FN(1) ≤Lθ FN(n) for all n∈ H. Hence LTε N(1) is an upper bound of {LTε N(n)|n∈ H} Lδ IN(1) is a lower bound of {Lδ IN(n)|n∈ H} Lθ FN(1) is a lower bound of {Lθ FN(n)|n∈ H} Let n, s ∈ H and 0 < α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1 be such that n(α1,β1,γ1)∈LN and (n⇝s)α2,β2,γ2∈LN. Then LTε N(n)≥α1,LTε N(n⇝s)≥α2 Lδ IN(n)≤β1,Lδ IN(n⇝s)≤β2 Lθ FN(n)≤γ1,Lθ FN(n⇝s)≤γ2 which imply from (15) that, N. Abirami, M. Mary Jansirani, Filters in Hoops based on Lukasiewicz Neutrosophic set Neutrosophic Sets and Systems, Vol. 97, 2026 715 LTε N(s)≥min{LTε N(n), LTε N(n⇝s)} ≥ min{α1, α2} Lδ IN(s)≤max{Lδ IN(n), Lδ IN(n⇝s)} ≤ max{β1, β2} Lθ FN(s)≤max{Lθ FN(n), Lθ FN(n⇝s)} ≤ max{γ1, γ2} Thus sα1∧α2,β1∨β2,γ1∨γ2∈LN. Therefore LNis a LNF of Hby (3.4). Proposition 3.6. Every LNF LNof Hsatisfies the following condition. p≤n⇝s, n(α1,β1,γ1)∈LN, pα2,β2,γ2∈LN⇒sα1∧α2,β1∨β2,γ1∨γ2∈LN(16) forall n, s, p ∈ H,0< α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1. Proof. Let n, s, p ∈ H and 0 < α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1 be such that p≤n⇝s, n(α1,β1,γ1)∈LNand pα2,β2,γ2∈LN. Then p⇝(n⇝s) = 1. LTε N(n)≥α1,Lδ IN(n)≤β1,Lθ FN(n)≤γ1 LTε N(p)≥α2,Lδ IN(p)≤β2,Lθ FN(p)≤γ2 Hence LTε N(s)≥min{LTε N(n), LTε N(n⇝s)} ≥min{LTε N(n), min{LTε N(p⇝(n⇝s)), LTε N(p)}} =min{LTε N(n), min{LTε N(1), LTε N(p)}} =min{LTε N(n), LTε N(p)} ≥min{α1, α2} and Lδ IN(s)≤max{Lδ IN(n), Lδ IN(n⇝s)} ≤max{Lδ IN(n), max{Lδ IN(p⇝(n⇝s)), Lδ IN(p)}} =max{Lδ IN(n), max{Lδ IN(1), Lδ IN(p)}} =max{Lδ IN(n), Lδ IN(p)} ≤max{β1, β2} N. Abirami, M. Mary Jansirani, Filters in Hoops based on Lukasiewicz Neutrosophic set Neutrosophic Sets and Systems, Vol. 97, 2026 716 and Lθ FN(s)≤max{Lθ FN(n), Lθ FN(n⇝s)} ≤max{Lθ FN(n), max{Lθ FN(p⇝(n⇝s)), Lθ FN(p)}} =max{Lθ FN(n), max{Lθ FN(1), Lθ FN(p)}} =max{Lθ FN(n), Lθ FN(p)} ≤max{γ1, γ2} Thus we have that sα1∧α2,β1∨β2,γ1∨γ2∈LN. 4. Relation between Lukasiewicz Neutrosophic Filter and Lukasiewicz Fuzzy Filter In this section, we discussed the relationship between LNF and LFF. Theorem 4.1. ALNS LNin a hoop is a LN F if and only if LNsatisfies the following three conditions. (1) Lε TNis a LFF of H (2) 1 −Lδ INis a LFF of H (3) 1 −Lθ FNis a LFF of H, where (1 −Lδ IN)(n) = 1 −Lδ IN(n),(1 −Lθ FN)(n)=1−Lθ FN(n) Proof. Suppose that LNis a LN F in H. From (14) we have that Lε TN(1) is an upper bound of {Lε TN(n)|n∈ H} and by (15) we have Lε TN(s)≥min{Lε TN(n), Lε TN(n⇝s)}. Hence Lε TNis a LFF of Hby (2.7). Similarly from (14) we have that Lδ IN(1) is a lower bound of {Lδ IN(n)|n∈ H}. That is, Lδ IN(1) ≤Lδ IN(n). 1−Lδ IN(1) ≥1−Lδ IN(n) (1 −Lδ IN)(1) ≥(1 −Lδ IN)(n) ∴(1 −Lδ IN)(1) is an upper bound of {(1 −Lδ IN)(n)|n∈ H}. By (15), we have Lδ IN(s)≤max{Lδ IN(n), Lδ IN(n⇝s)} 1−Lδ IN(s)≥1−max{Lδ IN(n), Lδ IN(n⇝s)} =min{1−Lδ IN(n),1−Lδ IN(n⇝s)}. Thus (1 −Lδ IN)(s)≥min{(1 −Lδ IN)(n),(1 −Lδ IN)(n⇝s)} ∴1−Lδ INis a LFF of Hby (2.7). N. Abirami, M. Mary Jansirani, Filters in Hoops based on Lukasiewicz Neutrosophic set Neutrosophic Sets and Systems, Vol. 97, 2026 717