Neutrosophic Hyper KU-Ideals

University of New Mexico Neutrosophic Hyper KU-Ideals Ramesh Kumar D 1∗and Vasu M 2 1Department of Mathematics, Government Arts and Science College, Komarapalayam - 638 183, Tamil Nadu, India; [email protected]. 2Department of Mathematics, Government Arts College for Women, Sivagangai - 630 562, Tamil Nadu, India; mv[email protected]. ∗Correspondence: Ramesh Kumar D ([email protected]); Tel.: (+91 9865769145). Abstract.We define neutrosophic hyper KU-ideal (strong, weak, s-weak) and refelexive neutrosophic hyper KU-ideal. A few key properties and their relationships are highlighted. The study of the neutrosophic (weak) hyper KU-ideal is considered. The conditions for a neutrosophic set to be a NSHKUI and a (reflexive) neutrosophic hyper KU-ideal are discussed. There are also circumstances for a NWHKUI to be a NsWHKUI, as well as conditions for a NSHKUI to be a RNHKUI. Keywords: Hyper KU-algebra; hyper KU-ideals; NHKUI;NSHKUI;NWHKUI;NsWHKUI;RNHKUI. —————————————————————————————————————————- 1. Introduction Prabpayak and Leerawat created a novel algebraic structure known as KU-algebras [12, 13]. In KU-algebras, they worked at ideals and congruences. They also established the concept of KU-algebra homomorphism and looked into various related features. They also deduced some simple consequences of the quotient KU-algebras and isomorphism relationships. Marty [9] presented the hyper structure theory (also known as multialgebras) at the 8th Congress of Scandinvian Mathematiciens in 1934. Several authors, primarily in France and the United States, but also in Italy, Russia, and Japan, worked on hyper groups in the 1940’s. Hyperstructures offer a wide range of applications in both pure and applied sciences. Jun et al. extended hyper structures to BCK-algebras, proposed the idea of a hyper BCK-algebra, which is a generalization of a BCK-algebra, and looked into several associated characteristics in [8]. They also defined a hyper BCK-ideal and a weak hyper BCK-ideal, as well as relationships between hyper BCK-ideals and weak hyper BCK-ideals. Jun et al. [7] proposed the notions of a strong hyper BCK-ideal, a weak hyper BCK-ideal, and a reflexive hyper BCK-ideal, as well as a requirement for a hyper BCK-algebra to be a BCK-algebra. Every strong hyper Ramesh Kumar D and Vasu M, Neutrosophic Hyper KU-Ideals Neutrosophic Sets and Systems, Vol. 98, 2026 BCK-ideal is a hyper sub-algebra, a weak hyper BCK-ideal, and a hyper BCK-ideal, and every reflexive hyper BCK-ideal is a strong hyper BCK-ideal, they established. Smarandache [14–16] developed the neutrosophic set, which is a more general platform that extends the notions of classic set, (intuitionistic) fuzzy set, and interval valued (intuitionistic) fuzzy set. On BL-algebras, Borzooei et al. [4] investigated neutrosophic deductive filters. Zhang et al. [19] discussed neutrosophic regular filters and fuzzy regular filters when applying the concept of neutrosophic set to pseudo-BCI algebras. Neutrosophic set theory has been applied to a variety of areas, and many studies have been conducted to develop, improve, and expand the theory ( [1–3,5,6,10,17] and [18]). The goal of this paper is to introduce the concepts of neutrosophic (strong, weak, s-weak) hyper KU-ideal, as well as RNHKUI. We look at their connections and properties. Characterizations of neutrosophic (weak) hyper KU-ideal are discussed. We define exactly for a neutrosophic set to be a NSHKUI and a (reflexive) neutrosophic hyper KU-ideal. We’re looking for some provisions that will allow a NSHKUI to become a RNHKUI. We go over the conditions for a NWHKUI to be a NsWHKUI. 2. Preliminaries The basic definitions of hyper KU-ideals and neutrosophic set are given in this section. Let Hbe a non-empty set and let “◦” be a mapping ◦:H×H→P(H)\{∅} which is said to be hyperoperation. For any two subsets Aand B, denote by A◦B, the set ∪ {l01 ◦l02|l01 ∈A, l02 ∈B}. We shall use l01 ◦l02 instead of {l01} ◦ l02, l01 ◦ {l02}or {l01}◦{l02}. By a hyper KU-algebra H( [11]), we mean a non-empty set Hwith a special element 0 and a hyperoperation ◦, for all l01, l02, l03 ∈H, that satisfies the following axioms: (HKU1) (l02 ◦l03)◦(l01 ◦l03)l01 ◦l02, (HKU2) l01 ◦0 = {0}, (HKU3) 0◦l01 ={l01}, (HKU4) if l01 l02 and l02 l01 imply l01 =l02,for all l01, l02, l03 ∈H, where l01 l02 is defined by 0∈l02 ◦l01 and for any A, B ⊆H, A Bis defined by ∀r∈A, ∃t∈Bsuch that rt. Proposition 2.1. [11] Let Hbe a hyper KU-algebra. Then for all l01, l02, l03 ∈H, the following statements hold: (i) A⊆Bimplies AB, for all nonempty subsets A, B of H. (ii) 0◦0 = {0}. Ramesh Kumar D and Vasu M, Neutrosophic Hyper KU-Ideals Neutrosophic Sets and Systems, Vol. 98, 2026 42 (iii) 0l01. (iv) l03 l03. (v) l01 ◦l03 l03. (vi) A◦0 = {0}. (vii) 0◦A=A. (viii) (0 ◦0) ◦l01 ={l01}and (l01 ◦(0 ◦l01)) = {0}. (ix) l01 ◦l01 ={l01} ⇔ l01 = 0. (x) l03 ◦(l02 ◦l01) = l02 ◦(l03 ◦l01)for all l01, l02, l03 ∈H. Definition 2.2. [11] Let (H, ◦)be a hyper KU-algebra. A subset Aof His called: •A hyper KU-ideal (briefly, HKUI) of Hif (1) 0∈A, (2) l02 ◦l01 Aand l02 ∈Aimply l01 ∈A, forall l01, l02 ∈H. •A weak hyper KU-ideal (briefly, WHKUI) of Hif it satisfies (1) and (3) l02 ◦l01 ⊆A, l02 ∈A⇒l01 ∈A, ∀l01, l02 ∈H, •A strong hyper KU-ideal (briefly, SHKUI) of Hif it satisfies (1) and (4) (l02 ◦l01)∩A6=∅, l02 ∈A⇒l01 ∈A, ∀l01, l02 ∈H, A subset Iof a hyper KU-algebra His said to be reflexive if (l◦l)⊆Ifor all l∈H. Let Hbe a non-empty set. A neutrosophic set (NS) in H(See [16] ) is a structure of the form: L:= {hl;LT(l), LI(l), LF(l)i | l∈H} where LT:H→[0,1] is a truth membership function, LI:H→[0,1] is an indeterminate membership function, and LF:H→[0,1] is a false membership function. For abbreviation, we continue to write L= (LT, LI, LF)for the NS L:= {hl;LT(l), LI(l), LF(l)i | l∈H}. Given a NS L = (LT, LI, LF)in a hyper KU-algebra Hand a subset Vof H, by ∗LT,∗LT,∗LI,∗LI,∗LFand ∗LFwe mean ∗LT(V) = infv∈VLT(v)and ∗LT(V) = supv∈VLT(v), ∗LI(V) = infv∈VLI(v)and ∗LI(V) = supv∈VLI(v), ∗LF(V) = infv∈VLF(v)and ∗LF(V) = supv∈VLF(v), respectively. Notation. From now on, in this paper, we assume that His a hyper KU-algebra. 3. Neutrosophic hyper KU-ideals We discussed the features of neutrosophic (strong, weak, s-weak) hyper KU-ideal and reflexive neutrosophic hyper KU-ideal in this part. Ramesh Kumar D and Vasu M, Neutrosophic Hyper KU-Ideals Neutrosophic Sets and Systems, Vol. 98, 2026 43 Definition 3.1. Let L= (LT, LI, LF)be a NS in H. Then Lis said to be a neutrosophic hyper KU-ideal (briefly, NHKUI) of Hif it satisfies the following assertions for all l01, l02 ∈H,      l01 l02 ⇒          LT(l01)≥LT(l02) LI(l01)≥LI(l02) LF(l01)≤LF(l02)      ,(1)    LT(l01)≥min{∗LT(l02 ◦l01), LT(l02)} LI(l01)≥min{∗LI(l02 ◦l01), LI(l02)} LF(l01)≤max{∗LF(l02 ◦l01), LF(l02)}   (2) Example 3.2. Let H={l0, la, lb}be a hyper KU-algebra. The hyper operation “◦′′ on H described by Table 1. Table 1 : Cayley table for the binary operation “◦′′ ◦l0lalb l0{l0} {la} {lb} la{l0} {l0, la} {la, lb} lb{l0} {l0, la} {l0, la, lb} We define a NS L = (LT, LI, LF)on Hby Table 2. Table 2 : Tabular representation of L= (LT, LI, LF) H LT(l)LI(l)LF(l) l00.77 0.65 0.08 la0.55 0.47 0.57 lb0.11 0.27 0.69 It is easy to check that L= (LT, LI, LF)is a NHKUI of H. Proposition 3.3. For any NHKUI L = (LT, LI, LF)of H, the following assertions are valid. (i) L= (LT, LI, LF)satisfies (∀l01 ∈H)   LT(0) ≥LT(l01) LI(0) ≥LI(l01) LF(0) ≤LF(l01)   .(3) (ii) If L= (LT, LI, LF)satisfies (∀V⊆H)(∃u, v, w ∈V)   LT(u) = ∗LT(V) LI(v) = ∗LI(V) LF(w) = ∗LF(V)   ,(4) then the following assertion is valid. (∀l01, l02 ∈H)(∃u, v, w ∈l02 ◦l01)   LT(l01)≥min{LT(u), LT(l02)} LI(l01)≥min{LI(v), LI(l02)} LF(l01)≤max{LF(w), LF(l02)}   .(5) Ramesh Kumar D and Vasu M, Neutrosophic Hyper KU-Ideals Neutrosophic Sets and Systems, Vol. 98, 2026 44 Proof. By Proposition 2.1 (ii) and (1) we have LT(0) ≥LT(l01), LI(0) ≥LI(l01)and LF(0) ≤LF(l01). Assume that L= (LT, LI, LF)satisfies the condition (4). For all l01, l02 ∈H, there exists u0, v0, w0∈l02 ◦l01 such that LT(u0) = ∗LT(l02 ◦l01), LI(v0) = ∗LI(l02 ◦l01)and LF(w0) = ∗LF(l02 ◦l01). Now condition (2) implies that LT(l01)≥min{∗LT(l02 ◦l01), LT(l02)}= min{LT(u0), LT(l02)} LI(l01)≥min{∗LI(l02 ◦l01), LI(l02)}= min{LI(v0), LI(l02)}. LF(l01)≤max{∗LF(l02 ◦l01), LF(l02)}= max{LF(w0), LF(l02)} This completes the proof. Ξ We define the following sets: U(LT, ξT) := {l01 ∈H|LT(l01)≥ξT}, U(LI, ξI) := {l01 ∈H|LI(l01)≥ξI}, L(LF, ξF) := {l01 ∈H|LF(l01)≤ξF}, where L= (LT, LI, LF)is a NS in Hand ξT, ξI, ξF∈[0,1]. Lemma 3.4. Let Lbe a subset of H. If Iis a HKUI of Hsuch that LI, then Lis contained in I. Proof. Assume that LHand let l∈L. Then 0◦l={l}  Hand so l∈Hby using Definition 2.2 (2). Therefore L⊆H.Ξ Theorem 3.5. ANS L = (LT, LI, LF)is a NHKUI of Hiff the nonempty sets U(LT, ξT), U(LI, ξI)and L(LF, ξF)are HKUI’s of Hfor all ξT, ξI, ξF∈[0,1]. Proof. Assume that L= (LT, LI, LF)is a NHKUI of Hand suppose that U(LT, ξT), U(LI, ξI)and L(LF, ξF)are nonempty for all ξT, ξI, ξF∈[0,1]. It is easy to see that 0∈U(LT, ξT),0∈U(LI, ξI)and 0∈L(LF, ξF). Let l01, l02 ∈Hbe such that l02 ◦l01 U(LT, ξT)and l02 ∈U(LT, ξT). Then LT(l02)≥ξTand for any l∈l02 ◦l01 there exists l0∈U(LT, ξT)such that ll0.We conclude from (1) that LT(l)≥LT(l0)≥ξTfor all l∈l02 ◦l01.Hence ∗LT(l02 ◦l01)≥ξT, and so LT(l01)≥min{∗LT(l02 ◦l01), LT(l02)} ≥ ξT, that is, l01 ∈U(LT, ξT). Similarly, we show that if l02◦l01 U(LI, ξI)and l02 ∈U(LI, ξI), then l01 ∈U(LI, ξI). Hence U(LT, ξT)and U(LI, ξI)are HKUI’s of H. Let l01, l02 ∈Hbe such that Ramesh Kumar D and Vasu M, Neutrosophic Hyper KU-Ideals Neutrosophic Sets and Systems, Vol. 98, 2026 45 l02 ◦l01 L(LF, ξF)and l02 ∈L(LF, ξF). Then LF(l02)≤ξF. Let m∈l02 ◦l01. Then there exists m0∈L(LF, ξF)such that mm0, which implies from (1) that LF(m)≤LF(m0)≤ξF. Thus ∗LF(l02 ◦l01)≤ξF, and so LF(l01)≤max{∗LF(l02 ◦l01), LF(l02)} ≤ ξF. Hence l01 ∈L(LF, ξF)and therefore L(LF, ξF)is a HKUI of H. Conversely, suppose that the nonempty sets U(LT, ξT), U(LI, ξI)and L(LF, ξF)are HKUI’s of Hfor all ξT, ξI, ξF∈[0,1].Let l01, l02 ∈Hbe such that l01 l02. Then l02 ∈U(LT, LT(l02)) ∩U(LI, LI(l02)) ∩L(LF, LF(l02)), and thus l01 U(LT, LT(l02)), l01 U(LI, LI(l02)) and l01 L(LF, LF(l02)). According to Lemma 3.4, we have l01 ∈U(LT, LT(l02)), l01 ∈U(LI, LI(l02)) and l01 ∈L(LF, LF(l02)) which imply that LT(l01)≥LT(l02),LI(l01)≥LI(l02)and LF(l01)≤LF(l02). For any l01, l02 ∈H, let ξT:= min{∗LT(l02 ◦l01), LT(l02)}, ξI:= min{∗LI(l02 ◦l01), LI(l02)}and ξF:= max{∗LF(l02 ◦l01), LF(l02)}.Then l02 ∈U(LT, ξT)∩U(LI, ξI)∩L(LF, ξF), and for each uT, vI, wF∈l02 ◦l01 we have LT(uT)≥∗LT(l02 ◦l01)≥min{∗LT(l02 ◦l01), LT(l02)}=ξT, LI(vI)≥∗LI(l02 ◦l01)≥min{∗LI(l02 ◦l01), LI(l02)}=ξI and LF(wF)≤∗LF(l02 ◦l01)≤max{∗LF(l02 ◦l01), LF(l02)}=ξF. Hence uT∈U(LT, ξT), vI∈U(LI, ξI)and wF∈L(LF, ξF)and so l02 ◦l01 ⊆U(LT, ξT), l02 ◦ l01 ⊆U(LI, ξI)and l02 ◦l01 ⊆L(LF, ξF). By Proposition 2.1, we have l02 ◦l01 U(LT, ξT), l02 ◦ l01 U(LI, ξI)and l02 ◦l01 L(LF, ξF). It follows from Definition 2.2 (2) that l01 ∈U(LT, ξT)∩U(LI, ξI)∩L(LF, ξF). Hence LT(l01)≥ξT= min{∗LT(l02 ◦l01), LT(l02)}, LI(l01)≥ξI= min{∗LI(l02 ◦l01), LI(l02)} and LF(l01)≤ξF= max{∗LF(l02 ◦l01), LF(l02)}. Therefore L= (LT, LI, LF)is a NHKUI of H.Ξ Ramesh Kumar D and Vasu M, Neutrosophic Hyper KU-Ideals Neutrosophic Sets and Systems, Vol. 98, 2026 46 Theorem 3.6. If L= (LT, LI, LF)is a NHKUI of H, then the set J:= {l01 ∈H|LT(l01) = LT(0), LI(l01) = LI(0), LF(l01) = LF(0)}(6) is a HKUI of H. Proof. It is easy to check that 0∈J. Let l01, l02 ∈Hbe such that l02 ◦l01 Jand l02 ∈J. Then LT(l02) = LT(0), LI(l02) = LI(0) and LF(l02) = LF(0). Let l∈l02 ◦l01. Then there exists l0∈Jsuch that ll0,and thus by (1), LT(l)≥LT(l0) = LT(0), LI(l)≥LI(l0) = LI(0) and LF(l)≤LF(l0) = LF(0). It follows from (2) that LT(l01)≥min{∗LT(l02 ◦l01), LT(l02)} ≥ LT(0), LI(l01)≥min{∗LI(l02 ◦l01), LI(l02)} ≥ LI(0) and LF(l01)≤max{∗LF(l02 ◦l01), LF(l02)} ≤ LF(0). Hence LT(l01) = LT(0), LI(l01) = LI(0) and LF(l01) = LF(0), that is, l01 ∈J. Therefore Jis aHKUI of H.Ξ We define the situation under which a NS L = (LT, LI, LF)is a NHKUI of H. Theorem 3.7. Let Hsatisfy |l02 ◦l01|<∞for all l01, l02 ∈H, and let {Jβ|β∈Λ⊆[0,0.5]} be a collection of HKUI’s of Hsuch that H=∪ β∈Λ Jβ,(7) (∀α, β ∈Λ)(α > β ⇔Jα⊂Jβ).(8) Then a NS L = (LT, LI, LF)in Hdefined by LT:H→[0,1], l01 7→ sup{β∈Λ|l01 ∈Jβ}, LI:H→[0,1], l01 7→ sup{β∈Λ|l01 ∈Jβ}, LF:H→[0,1], l01 7→ inf{β∈Λ|l01 ∈Jβ} is a NHKUI of H. Proof. We first shows that ρ∈[0,1] ⇒∪ δ∈Λ,δ≥ρ Jδisa HKUI of H. (9) It is clear that 0∈∪ δ∈Λ,δ≥ρ Jδfor all ρ∈[0,1]. Let l01, l02 ∈Hbe such that l02 ◦l01 = {l1, l2,· · · , ln}, l02 ◦l01 ∪ δ∈Λ,δ≥ρ Jδand l02 ∈∪ δ∈Λ,δ≥ρ Jδ. Then l02 ∈Jγfor some γ∈Λwith ρ≤γ, and for any li∈l02 ◦l01 there exists mi∈∪ δ∈Λ,δ≥ρ Jδ, and so mi∈Jβifor some βi∈Λ with ρ≤βi, such that limi. If we let β:= min{βi|i∈ {1,2,· · · , n}} then Jβi⊂Jβfor Ramesh Kumar D and Vasu M, Neutrosophic Hyper KU-Ideals Neutrosophic Sets and Systems, Vol. 98, 2026 47 all i∈ {1,2,· · · , n}and so l02 ◦l01 Jβwith ρ≤β. We may assume that γ > β without loss of generality, and so Jγ⊂Jβ. By Definition 2.2 (2), we have l01 ∈Jβ⊂∪ δ∈Λ,δ≥ρ Jδ.Hence ∪ δ∈Λ,δ≥ρ Jδis a HKUI of H. Next, we consider the following two cases: (i) β= sup{ρ∈Λ|ρ < β},(ii) β6= sup{ρ∈Λ|ρ < β}.(10) If the first case is valid, then l01 ∈U(LT, β)⇔l01 ∈Jρfor all ρ < β ⇔l01 ∈∩ ρ<β Jρ, and so U(LT, β) = ∩ ρ<β Jρwhich is a HKUI of H. Similarly, we know that U(LI, β)is a HKUI of H. For the second case, we will show that U(LT, β) = ∪ ρ≥β Jρ. If l01 ∈∪ ρ≥β Jρ, then l01 ∈Jρfor some ρ≥β. Thus LT(l01)≥ρ≥β, and so l01 ∈U(LT, β)which shows that ∪ ρ≥β Jρ⊆U(LT, β). Assume that l01 6∈ ∪ ρ≥β Jρ. Then l01 6∈ Jρfor all ρ≥β, and so there exist δ > 0such that (β−δ, β)∩Λ = ∅. Thus l01 6∈ Jρfor all ρ>β−δ, that is, if l01 ∈Jρthen ρ≤β−δ < β. Hence l01 6∈ U(LT, β). This shows that U(LT, β) = ∪ ρ≥β Jρwhich is a HKUI of Hby (9). Similarly we can prove that U(LI, β)is a HKUI of H. Now we consider the following two cases: α= inf{γ∈Λ|α < γ}and α6= inf{γ∈Λ|α < γ}.(11) The first case implies that l01 ∈L(LF, α)⇔l01 ∈Jγfor all α < γ ⇔l01 ∈∩ α<γ Jγ, and so L(LF, α) = ∩ α<γ Jγwhich is a HKUI of H. For the second case, there exists δ > 0such that (α, α +δ)∩Λ = ∅. If l01 ∈∪ α≥γ Jγ, then l01 ∈Jγfor some α≥γ. Thus LF(l01)≤γ≤α, that is, l01 ∈L(LF, α). Hence ∪ α≥γ Jγ⊆L(LF, α). If l01 6∈ ∪ α≥γ Jγ, then l01 6∈ Jγfor all γ≤α and thus l01 6∈ Jγfor all aγ < α +δ. This shows that if l01 ∈Jγthen γ≥α+δ. Hence LF(l01)≥α+δ > α, i.e., l01 6∈ L(LF, α). Therefore L(LF, α)⊆∪ α≥γ Jγ. Consequently, L(LF, α) = ∪ α≥γ Jγwhich is a HKUI of Hby (9). It follows from Theorem 3.5 that L= (LT, LI, LF)is a NHKUI of H.Ξ Definition 3.8. ANS L = (LT, LI, LF)in His called a neutrosophic strong hyper KU-ideal (briefly, NSHKUI) of Hif it satisfies the following assertions. ∗LT(l01 ◦l01)≥LT(l01)≥min{sup u0∈l02◦l01 LT(u0), LT(l02)}, ∗LI(l01 ◦l01)≥LI(l01)≥min{sup v0∈l02◦l01 LI(v0), LI(l02)} ∗LF(l01 ◦l01)≤LF(l01)≤max{inf w0∈l02◦l01 LF(w0), LF(l02)} (12) Ramesh Kumar D and Vasu M, Neutrosophic Hyper KU-Ideals Neutrosophic Sets and Systems, Vol. 98, 2026 48 for all l01, l02 ∈H. Example 3.9. Consider a hyper KU-algebra H={l0, la, lb}with the hyper operation “◦′′ which is given by Table 3. Table 3 : Cayley table for the binary operation “◦′′ ◦l0lalb l0{l0} {la} {lb} la{l0} {l0, la} {lb} lb{l0} {lb} {l0, lb} Let L= (LT, LI, LF)be a NS in Hwhich is described in Table 4. Table 4 : Tabular representation of L= (LT, LI, LF) H LT(l)LI(l)LF(l) l00.77 0.65 0.08 la0.55 0.47 0.57 lb0.11 0.27 0.69 It is routine to verify that L= (LT, LI, LF)is a NSHKUI of H. Theorem 3.10. For any NSHKUI L = (LT, LI, LF)of H, the following assertions are valid. (i) L= (LT, LI, LF)satisfies the conditions (1) and (3). (ii) L= (LT, LI, LF)satisfies (∀l01, l02 ∈H)(∀u, v, w ∈l02 ◦l01)   LT(l01)≥min{LT(u), LT(l02)} LI(l01)≥min{LI(v), LI(l02)} LF(l01)≤max{LF(w), LF(l02)}   .(13) Proof. (i) Since l01 l01, i.e., 0∈l01 ◦l01 for all l01 ∈H, we get LT(0) ≥∗LT(l01 ◦l01)≥LT(l01), LI(0) ≥∗LI(l01 ◦l01)≥LI(l01), LF(0) ≤∗LF(l01 ◦l01)≤LF(l01), which shows that (3) is valid. Let l01, l02 ∈Hbe such that l01 l02. Then 0∈l02 ◦l01, and so ∗LT(l02 ◦l01)≥LT(0),∗LI(l02 ◦l01)≥LI(0) and ∗LF(l02 ◦l01)≤LF(0). It follows from (3) that LT(l01)≥min{∗LT(l02 ◦l01), LT(l02)} ≥ min{LT(0), LT(l02)}=LT(l02), LI(l01)≥min{∗LI(l02 ◦l01), LI(l02)} ≥ min{LI(0), LI(l02)}=LI(l02), LF(l01)≤max{∗LF(l02 ◦l01), LF(l02)} ≤ max{LF(0), LF(l02)}=LF(l02). Hence L= (LT, LI, LF)satisfies the condition (1). Ramesh Kumar D and Vasu M, Neutrosophic Hyper KU-Ideals Neutrosophic Sets and Systems, Vol. 98, 2026 49 4. R. A. 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