Uni e si y o New Mexico
Fil e s in Hoops based on Lukasiewicz Neu osophic se
N. Abi ami1and M. Ma y Jansi ani2,∗
1Resea ch schola , School o Sciences, Di ision o Ma hema ics, SRM- Ins i u e o Science and Technology,
Ti uchi appalli campus, SRM Naga , T ichy-Chennai highway, nea Samayapu am, Ti uchi appalli-621105,
Tamil Nadu, India; [email p o ec ed],[email p o ec ed]
2School o Sciences, Di ision o Ma hema ics, SRM- Ins i u e o Science and Technology, Ti uchi appalli
campus, SRM Naga , T ichy-Chennai highway, nea Samayapu am, Ti uchi appalli-621105, Tamil Nadu,
India; an hu [email p o ec ed]
∗Co espondence: an hu [email p o ec ed]
Abs ac .The Lukasiewicz neu osophic se (LN S) is de eloped using he ideas o Lukasiewicz -no m and
s-no m. This se is hen applied o he hoop s uc u e and in oduces he Lukasiewicz neu osophic il e
(LN F). The p ope ies o his il e , as well as i s in e connec ion wi h he Lukasiewicz uzzy il e (LFF),
a e subsequen ly in es iga ed.
Keywo ds: Lukasiewicz neu osophic se ; Lukasiewicz neu osophic il e ; Lukasiewicz uzzy il e ; Neu o-
sophic poin ; Hoop
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1. In oduc ion
P o esso Lo i A. Zadeh, he a he o uzzy sys ems heo y, p oposed he idea o uzzy logic
in 1965. Ex ending his, A anasso in oduced in ui ionis ic uzzy se s by adding he deg ee
o belongingness and non-belongingness. As a gene aliza ion o he Zadeh and A anasso sys-
em, Flo en in Sma andache in oduced he neu osophic se , which deals wi h belongingness,
in de e minacy, and non-belongingness, in 1998 [15]. The concep o neu osophic poin and
i s p ope ies was in oduced by Gau am Chand a Ray and Sudeep Dey [13]. Neu osophic
algeb aic s uc u es we e s udied by Kandasamy and Sma andache [10,17]. Many esea che s
ha e since explo ed neu osophic il e s in di e en algeb aic amewo ks [5–7,12,14,16,18,19].
Hoops is an algeb aic s uc u e in oduced by Bosbach [3]. Lukasiewicz logic is he many-
alued logic, which is he ex ension o classical bina y logic. Y.B. Jun in oduced he
Lukasiewicz uzzy se using Lukasiewicz -no m and he Lukasiewicz in ui ionis ic uzzy se
N. Abi ami, M. Ma y Jansi ani, Fil e s in Hoops based on Lukasiewicz Neu osophic se
Neu osophic Se s and Sys ems, Vol. 97, 2026
using Lukasiewicz -no m and he dual o Lukasiewicz -no m (Lukasiewicz -cono m) and
applied hem in BCK algeb as [9, 11], while Mohseni Takallo e al. s udied he Lukasiewicz
uzzy il e s in hoops [11] and Jun e al. ex ended hese ideas o BE-algeb as [8]. Howe e ,
hese il e s canno adequa ely cap u e inde e minacy, which plays a c ucial ole in unce ain y
modeling.
In his pape , we add ess his gap by in oducing he Lukasiewicz neu osophic se , which
ex ends Lukasiewicz uzzy and in ui ionis ic uzzy se s by cap u ing he inde e minacy com-
ponen , which has emained unexplo ed in hoop s uc u es. We in oduced he Lukasiewicz
neu osophic il e and s udied i s cha ac e is ics using he neu osophic poin . We p o ide a
ela ionship be ween he Lukasiewicz uzzy il e and he Lukasiewicz neu osophic il e .
1.1. Compa a i e Analysis
Lukasiewicz in ui ionis ic uzzy il e handles unce ain y be e han he Lukasiewicz uzzy
il e . Howe e , bo h o hese il e s ail o cap u e inde e minancy. Ou p oposed Lukasiewicz
neu osophic il e plays a signi ican ole whe e inde e minancy occu s, and is an ad ancemen
o exis ing heo ies.
Te m No a ion
Lukasiewicz uzzy se LFS
Lukasiewicz uzzy il e LFF
Lukasiewicz neu osophic se LNS
Lukasiewicz neu osophic il e LNF
2. P elimina ies
De ini ion 2.1. [11]
I an algeb a (H,⊛,⇝,1) sa is ies he axioms (H1,H2,H3,H4), hen i is said o be a hoop.
H1 : n⇝n= 1 o all n ∈ H
H2 : n⊛(n⇝s) = s⊛(s⇝n) o all n, s ∈ H
H3 : n⇝(s⇝p) = (n⊛s)⇝p o all n, s, p ∈ H
H4:(H,⊛,1) is a commu a i e monoid.
In a hoop H, we say ha an elemen nis less han o equal o s, ha is n≤si and only i
n⇝s= 1.
P oposi ion 2.2. [3]
All o he ollowing claims a e me by e e y hoop H.
N. Abi ami, M. Ma y Jansi ani, Fil e s in Hoops based on Lukasiewicz Neu osophic se
Neu osophic Se s and Sys ems, 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.
De ini ion 2.3. [13]
Le X be he uni e se o discou se. A neu osophic se No e X is de ined by
N={(x, TN(x), IN(x), F N(x))|x∈X}
whe e TN, IN, F N:X→[0,1] and 0 ≤TN(x) + IN(x) + FN(x)≤3.
De ini ion 2.4. [13]
A neu osophic se P={(x, T P(x), IP(x), F P(x))|x∈X}is called a neu osophic poin i i
is o he o m
P(y) =
(α, β, γ),i y=x
(0,1,1), o y=x
whe e 0 < α ≤1,0≤β < 1,0≤γ < 1.
Fo he neu sophic poin P={(x, T P(x), IP(x), F P(x))|x∈X}wi h suppo x will be
deno ed by Px
α,β,γ o p < x, α, β, γ > o xα,β,γ.
De ini ion 2.5. [13]
Le Nbe a neu osophic se o e X and xα,β,γ be he neu osophic poin in X.
Then xα,β,γ is said o belong o N, deno ed by xα,β,γ ∈ N i and only i
α≤TN(x), β ≥IN(x), γ ≥FN(x).
De ini ion 2.6. [11]
Conside a uzzy se Fin Hand le ε∈(0,1). A unc ion om H o [0,1] de ined by
Lε
F(x) = max{0,F(x) + ε−1}
is called he Lukasiewicz uzzy se (LFS) o Fin H.
De ini ion 2.7. [11]
ALFS Lε
Fis called a Lukasiewicz uzzy il e (LFF) o Hi i sa is ies:
Lε
F(1) is an uppe bound o {Lε
F(x)|x∈ H} (8)
N. Abi ami, M. Ma y Jansi ani, Fil e s in Hoops based on Lukasiewicz Neu osophic se
Neu osophic Se s and Sys ems, Vol. 97, 2026 711
Lε
F(y)≥ {Lε
F(x), Lε
F(x⇝y)}(9)
3. Lukasiewicz Neu osophic Fil e s
The concep s o Lukasiewicz neu osophic se (LN S) and Lukasiewicz neu osophic
il e (LNF) in he hoop s uc u e we e p esen ed in his sec ion, and hei p ope ies we e
examined.
De ini ion 3.1.
Le Nbe a neu osophic se in a hoop Hand le ε, δ, θ ∈[0,1] be such ha 0 ≤ε+δ+θ≤3.
A mapping de ined as an objec in he o m below
LN={(x, Lε
TN, Lδ
IN, Lθ
FN)|x∈ H}
whe e
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 ha 0 ≤Lε
TN+Lδ
IN+Lθ
FN≤3∀x∈ H
is called he LNS o Nin Hdeno ed by LN= (Lε
TN, Lδ
IN, Lθ
FN).
Le LN= (Lε
TN, Lδ
IN, Lθ
FN) be he LN S o Nin H. I (ε, δ, θ) = (1,0,0), hen
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 ha i (ε, δ, θ) = (1,0,0), hen he LN S,LN= (Lε
TN, Lδ
IN, Lθ
FN) o Nis he
classical neu osophic se Nin H.
I (ε, δ, θ) = (0,1,1), hen max{0, TN(x) + ε−1}= 0 and min{1, IN(x) + δ}= 1,
min{1, FN(x) + θ}= 1.Thus i (ε, δ, θ) = (0,1,1), hen he LNS,LN= (Lε
TN, Lδ
IN, Lθ
FN)
o Nis he cons an unc ion wi h he alue(0,1,1).The e o e, in handling he LNS, he alue
o (ε, δ, θ) can always be conside ed o be in (0,1) x (0,1) x (0,1).
De ini ion 3.2.
ALNS LNin His called a LNF o Hi i sa is ies:
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)
o all n, s ∈ H,0< α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1.
Example 3.3.
Conside a Hoop s uc u e Hwi h he bina y ope a ions ′′ ⊛′′ and ′′ ⇝′′ as below.
N. Abi ami, M. Ma y Jansi ani, Fil e s in Hoops based on Lukasiewicz Neu osophic se
Neu osophic Se s and Sys ems, 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
De ine he neu osophic se Nin Has ollows,
N=
(0.63,0.23,0.35),i x=n
(0.63,0.05,0.30),i x=s
(0.70,0.01,0.27),i x=p
(0.91,0,0.2),i x= 1
Fo (ε,δ,θ)=(1,0.5,0.6), he LNS is gi en as ollows
LN=
(0.63,0.73,0.95),i x=n
(0.63,0.55,0.90),i x=s
(0.70,0.51,0.87),i x=p
(0.91,0.5,0.8),i x= 1
Now we can check ha i is a LN F o H.
Theo em 3.4.
ALNS LNin His a LNF o Hi and only i he ollowing condi ions a e alid.
LTε
N(1) is an uppe bound o {LTε
N(n)|n∈ H}
Lδ
IN(1) is a lowe bound o {Lδ
IN(n)|n∈ H}
Lθ
FN(1) is a lowe bound o {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)
o all n, s ∈ H,∀0< α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1.
P oo .
Suppose ha LNis a LN F o H. I LTε
N(1) is no an uppe bound o {LTε
N(n)|n∈ H}, hen
LTε
N(1) < LTε
N(m) o some m∈ H. since m≤1 and mLTε
N(m),Lδ
IN(m),Lθ
FN(m)∈LN, i ollows
N. Abi ami, M. Ma y Jansi ani, Fil e s in Hoops based on Lukasiewicz Neu osophic se
Neu osophic Se s and Sys ems, Vol. 97, 2026 713
om (11) ha 1LTε
N(m),Lδ
IN(m),Lθ
FN(m)∈LN. Tha 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 uppe bound o {LTε
N(n)|n∈ H},
Lδ
IN(1) is a lowe bound o {Lδ
IN(n)|n∈ H}, and
Lθ
FN(1) is a lowe bound o {Lθ
FN(n)|n∈ H}.
Le 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 ha e (n⊛(n⇝s))α1∧α2,β1∨β2,γ1∨γ2∈LN. Since
(n⊛(n⇝s)) ≤s, by (11) we ha e ha , sα1∧α2,β1∨β2,γ1∨γ2∈LN.
Assume ha LNsa is ies (12) and (13).
Le n, s ∈ H,0< α ≤1,0≤β < 1,0≤γ < 1,be such ha n≤sand nα,β,γ ∈LN. Then i
ollows om (12) ha
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 condi ion (13) leads o sα,β,γ ∈Lε
Nwhich p o es (11).
Le n, s ∈ H, and 0< α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<be such ha 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 ha e
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 ha e, (s⇝n⊛s)α1∧α2,β1∨β2,γ1∨γ2∈LN.
Again by (13) we ha e, (n⊛s)α1∧α2,β1∨β2,γ1∨γ2∈LNwhich shows (10).
∴LNin His a LNF o H.
Theo em 3.5. ALN S LNin His a LN F o Hi and only i i sa is ies:
nα,β,γ ∈LN⇒1α,β,γ ∈LN(14)
N. Abi ami, M. Ma y Jansi ani, Fil e s in Hoops based on Lukasiewicz Neu osophic se
Neu osophic Se s and Sys ems, 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)
o all n, s ∈ H.
P oo .
Assume ha LNis a LNF o H.
Le n∈ H and 0 < α ≤1,0≤β < 1,0≤γ < 1 be such ha nα,β,γ ∈LN.
The condi ion (12) leads o
LTε
N(1) ≥LTε
N(n)≥α
Lδ
IN(1) ≤Lδ
IN(n)≤β
Lθ
FN(1) ≤Lθ
FN(n)≤γ
which implies ha 1α,β,γ ∈LN.
we know ha 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. I ollows om (13) ha
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)}
o all n,s ∈ H.
Con e sely suppose ha LNsa is ies (14) and (15).
Since nLTε
N(n),Lδ
IN(n),Lθ
FN(n)∈LN o all n∈ H, we ha e by (14) ha
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) o
all n∈ H. Hence
LTε
N(1) is an uppe bound o {LTε
N(n)|n∈ H}
Lδ
IN(1) is a lowe bound o {Lδ
IN(n)|n∈ H}
Lθ
FN(1) is a lowe bound o {Lθ
FN(n)|n∈ H}
Le n, s ∈ H and 0 < α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1 be such ha 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 om (15) ha ,
N. Abi ami, M. Ma y Jansi ani, Fil e s in Hoops based on Lukasiewicz Neu osophic se
Neu osophic Se s and Sys ems, 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.
The e o e LNis a LNF o Hby (3.4).
P oposi ion 3.6.
E e y LNF LNo Hsa is ies he ollowing condi ion.
p≤n⇝s, n(α1,β1,γ1)∈LN, pα2,β2,γ2∈LN⇒sα1∧α2,β1∨β2,γ1∨γ2∈LN(16)
o all n, s, p ∈ H,0< α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1.
P oo .
Le n, s, p ∈ H and 0 < α1, α2≤1,0≤β1, β2<1,0≤γ1, γ2<1 be such ha 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. Abi ami, M. Ma y Jansi ani, Fil e s in Hoops based on Lukasiewicz Neu osophic se
Neu osophic Se s and Sys ems, 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 ha e ha sα1∧α2,β1∨β2,γ1∨γ2∈LN.
4. Rela ion be ween Lukasiewicz Neu osophic Fil e and Lukasiewicz Fuzzy Fil e
In his sec ion, we discussed he ela ionship be ween LNF and LFF.
Theo em 4.1.
ALNS LNin a hoop is a LN F i and only i LNsa is ies he ollowing h ee condi ions.
(1) Lε
TNis a LFF o H
(2) 1 −Lδ
INis a LFF o H
(3) 1 −Lθ
FNis a LFF o H,
whe e (1 −Lδ
IN)(n) = 1 −Lδ
IN(n),(1 −Lθ
FN)(n)=1−Lθ
FN(n)
P oo .
Suppose ha LNis a LN F in H.
F om (14) we ha e ha Lε
TN(1) is an uppe bound o {Lε
TN(n)|n∈ H} and by (15) we ha e
Lε
TN(s)≥min{Lε
TN(n), Lε
TN(n⇝s)}. Hence Lε
TNis a LFF o Hby (2.7).
Simila ly om (14) we ha e ha Lδ
IN(1) is a lowe bound o {Lδ
IN(n)|n∈ H}.
Tha 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 uppe bound o {(1 −Lδ
IN)(n)|n∈ H}.
By (15), we ha e
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 o Hby (2.7).
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Neu osophic Se s and Sys ems, Vol. 97, 2026 717
