Eigen Neutrosophic Z- Set and Neutrosophic Z- Relation

Neu osophic Se s and Sys ems, Vol. 97, 2026
Uni e si y o New Mexico
P. Sheeba Maybell, M.M. Shanmugap iya, Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
P. Sheeba Maybell1*, M.M. Shanmugap iya2
1 Depa men o Ma hema ics, Ka pagam Academy o Highe Educa ion, Coimba o e, Tamil Nadu, India;
[email p o ec ed]
2 Dep . o Ma hema ics, Ka pagam Academy o Highe Educa ion, Coimba o e, Tamil Nadu, India;
[email p o ec ed]
* Co espondence: Seeba Maybell, Email: [email p o ec ed]
Abs ac : This pape in oduces an inno a i e amewo k o compu ing he G ea es Eigen
Neu osophic Z-se and he Leas Eigen Neu osophic Z-se using he composi ion ope a o s, namely
max-min-min and min-max-max. The p oposed Eigen Neu osophic Z-se , along wi h he
Neu osophic Z- ela ion, emains cons an ac oss di e en compu a ional pe spec i es. This s udy
add esses he limi a ion o exis ing neu osophic and uzzy models ha ail o e ec i ely cap u e
eigen-based ela ionships unde unce ain y by in oducing he Eigen Neu osophic Z-se amewo k
o mo e consis en and in e p e able decision analysis. Fu he mo e, Neu osophic Z-ma ices a e
de eloped, and hei p ope ies a e examined in ela ion o Neu osophic Z- ela ions. In his pape
se e al simila i y ela ions among Neu osophic Z-ma ices a e p esen ed, along wi h discussions on
hei pe mu a ions and he in e ibili y cha ac e is ics. Two dis inc algo i hms a e o mula ed o
es ablish he G ea es Eigen Neu osophic Z-se and he Leas Eigen Neu osophic Z-se ,
accompanied by a nume ical example. Addi ionally, a p ac ical applica ion is p o ided o
demons a e he enhancemen o sco e alue while add essing bo h e ec i eness and unce ain y o
u u e ad ancemen s o ho el managemen decision-making sys ems.
Keywo ds: Neu osophic Z-se , Neu osophic Z- ela ion, Neu osophic Z-Ma ices, Eigen
Neu osophic Z-se , Composi ion ope a o s, Decision-making unce ain y modelling.
1. In oduc ion
Zadeh [ 1] p oposed a no ion namely Z-numbe , which is an o de ed pai o uzzy numbe s 𝑍=
(𝑉,𝑅) in 2011.The eliabili y and es ic ion o uzzy is mainly ocused in Z-numbe [2]. Sma andache
[3] in oduced ano he concep o imp ecise da a called Neu osophic da a which deals wi h
complica ing aspec s o p ocess imp ecision, agueness, and unce ain y in da a. Sanjib Mondal e .al.,
[4] de eloped simila i y ela ions o In ui ionis ic uzzy ma ices. Neu osophic se was la e
de eloped o Quad i pa i ioned neu osophic so se , uzzy neu osophic so ma ices and uzzy
Quad i pa i ioned neu osophic so ma ix [5, 6, 7] which was mo e use ul in decision making.
Neu osophic quali ies and neu osophic me ics o assess us wo hiness a e uni ed in he
neu osophic z-numbe se echnique p oposed by Shigui Du e al. [8] as a gene aliza ion o he z-
numbe s and he neu osophic se . The h ee o de ed pai s o neu osophic numbe s along wi h hei
eliabili y measu es in inde e mina e and inconsis en si ua ions can be esol ed by he sugges ed
neu osophic z-numbe se [9]. In z-numbe s and hei se , he mul i c i e ia decision-making
echnique (MCDM) is eadily emb aced [10, 11, 12] la e on MCDM de eloped o neu osophic z-
numbe s. A uzzy ela ions eigen uzzy se was p esen ed by Sanchez [13]. He p o ided h ee main
algo i hms o ind he G ea es Eigen Fuzzy Se (GEFS) linked wi h uzzy ela ions using max-min
Neu osophic Se s and Sys ems, Vol. 97, 2026 688
P. Sheeba Maybell, M.M. Shanmugap iya, Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
composi ion so ha 𝑅∘A=A. Eigen uzzy se s ha e been success ully used in a numbe o eal-wo ld
applica ions in decision-making, gene ic algo i hms, image analysis, and medicine. Gule ia and Bajaj
[14] la e p oposed eigen sphe ical uzzy se s and applica ions. Using sphe ical uzzy se , hey ha e
p oduced an as ounding achie emen by iden i ying wo dis inc echniques o inding eigen
sphe ical uzzy se s. Fu he T-sphe ical uzzy se o simila i y measu e also ound o decision-
making [15].
Ha ik ishnan e al. [16] in hei wo k min-max composi ions o neu osophic uzzy ma ices
was demons a ed hei applica ion in diagnosing diseases. Thei wo k shows ha changing
composi ion ope a o s al e s diagnos ic ou comes, bu hey do no add ess eigen-based s abili y o
Z- ela ions. Bu his shows he impo ance o composi ion ope a o s bu lacks in o ma ion abou
eigen Z-se heo y. Kam an e al. [17] examined he use o neu osophic Z-numbe s in AHP-based
p io i iza ion and Z- ough s uc u es o anking al e na i es unde unce ain y. Al hough Z-
numbe s p o ide iche ep esen a ion o unce ain y, his wo k does no de ine eigen Z-se s o
algo i hms o s able ela ion e alua ion. This wo k has a s ong Z-numbe backg ound bu doesn’
ocus on simila i y o eigen p ope ies.
Mish a & Kuma [18] in es iga ed algeb aic p ope ies o neu osophic ma ices, including
in e ibili y and de e minan s o decision-o ien ed sys ems. Thei heo e ical wo k add esses
classical neu osophic ma ices bu does no ex end hese esul s o neu osophic Z-ma ices o eigen
compu a ions. This wo k doesn’ in ol e he Z-ma ix amewo k; only he ounda ion o ma ix
algeb a is used o analysis.
Al-Fai i e al. [19] applied neu osophic and pli hogenic models o mul i-c i e ia decision-
making o unce ain p e e ence s uc u es. While hey imp o e decision accu acy, hey ely on
dis ance and sco e measu es and do no conside eigen-based consis ency.
Saha & Abdel-Basse e al. [20] explo ed spec al measu es such as he “ene gy” o neu osophic
ma ices o ne wo k analysis and clus e ing. Thei esul s demons a e he use ulness o spec al
neu osophic p ope ies, bu hey do no p opose algo i hms o g ea es /leas eigen Z-se s o Z-
ela ions. The Z-se de ini ion and he composi ion s abili y a e missing.
2. Resea ch Gap
The Eigen uzzy se was in oduced by Sanchez [13], along wi h he concep o uzzy ela ions.
This me hod es ablished he G ea es Eigen Fuzzy Se using he max-min composi ion me hod.
Nume ous esea che s ha e applied his max-min composi ion o image e ie al, gene ic
algo i hms, and in he medicinal ield. Subsequen ly, he Eigen Sphe ical Fuzzy Se was in oduced
by Gule ia and Bajaj [14]. They o e ed wo echniques o iden i ying he G ea es Eigen Sphe ical
Fuzzy Se and he Leas Eigen Sphe ical Fuzzy Se . The Neu osophic Z-se is a no el me hod used
o assess unce ain y in eal-li e scena ios. We ha e p oposed a new composi ion ope a o o he
Neu osophic Z-se along wi h i s Neu osophic Z- ela ion. Many esea che s ha e ex ensi ely
explo ed Neu osophic Fuzzy ma ices, hei ela ions, and simila i y measu es. Ou s udy is
signi ican because we ex ended simila i y ela ions o Neu osophic uzzy ma ices o Neu osophic
Z-ma ices.
3. Con ibu ion o his p oposed wo k
• In oduc ion o new composi ion ope a o o neu osophic z-se : Two dis inc composi ion
ope a o s o neu osophic z-se s,speci ically max-min-min and min-max-max, ha e been
de eloped alongside he concep o neu osophic z- ela ion. These composi ion ope a o s
iden i y he G ea es Eigen Neu osophic Z-se s (GENZS) and he Leas Eigen Neu osophic Z-
se s (LENZS), which a e ailo ed o yield sui able alues in si ua ions o unce ain y.
• Neu osophic Z- ela ion o Neu osophic Z-ma ices: A Neu osophic Z-ma ix has been
in oduced oge he wi h he Neu osophic Z- ela ion. Va ious p ope ies o simila i y ela ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 689
P. Sheeba Maybell, M.M. Shanmugap iya, Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
we e examined, o e ing a ounda ional amewo k o Neu osophic Z-ma ices, wi h po en ial
o u he ad ancemen s h ough hese ma ices.
• Algo i hm o s a egy inding: This p oposed wo k in oduced wo algo i hms o each
composi ion ope a o wi hin he amewo k o Neu osophic Z-se s. The p ima y objec i e o
hese algo i hms is o de e mine he eigen neu osophic z-se .
• Nume ical Example and Applica ion: The e icacy o he p oposed me hod is illus a ed using
a nume ical example. E ec i eness and unce ain y a e assessed in a eal-wo ld se ing. The
decision-making scena io p o ides an in-dep h comp ehension ega ding he
me hods easibili y and adap abili y.
Table 1, depic s he compa ison o he exis ing wo ks in neu osophic Z numbe s and uzzy ma ices
wi h he p oposed wo k Eigen Neu osophic Z se .
Table 1 Compa ison o exis ing and p oposed wo k
Dimension
Exis ing Wo ks
(Neu osophic, Z-numbe s,
Fuzzy ma ices)
P oposed Wo k
Handling o
unce ain y
Uses u h, inde e minacy,
alsi y alues; Z-numbe s add
eliabili y bu no eigen
cha ac e iza ion
In oduces Eigen Neu osophic Z-se o
measu e s able ela ion alues unde
unce ain y
Composi ion
ope a o s
S udies max–min / min–max
amilies sepa a ely
Demons a es bo h max–min–min and
min–max–max ope a o s and p o es
eigen-se consis ency ac oss composi ions
Ma ix amewo k
Classical neu osophic ma ices
used o simila i y o sco ing
I de ines Neu osophic Z-ma ices,
explo es in e ibili y, pe mu a ions,
simila i y ela ions
Eigen-based analysis
Mos ly absen ; spec al
analysis exis s bu no o Z-
ela ions
P o ides algo i hms o G ea es and
Leas Eigen Neu osophic Z-se s wi h
nume ical examples
P ac ical decision-
making
Sco e o dis ance-based
anking
Uses eigen Z-se s o enhance sco e and
in e p e abili y in ho el managemen
decision-making
Rep oducibili y
O en concep ual o quali a i e
Deli e s wo algo i hms, ope a o
consis ency p oo , and implemen a ion
s eps
The pape is o ganized as ollows: Key de ini ions and concep s a e examined in Sec ion 3.
Neu osophic Z-ma ices, Neu osophic Z- ela ions, in e ibili y equi emen s, and simila i y
ela ions a e in oduced in Sec ion 4. NZM idempo en is also aken o conside a ion in his pa .
Nume ous cha ac e is ics and indings pe aining o neu osophic Z-ma ices a e examined. The
composi ion ope a o and he Neu osophic Z- ela ion a e de ined in Sec ion 5. The no ions o he
Neu osophic Se s and Sys ems, Vol. 97, 2026 690
P. Sheeba Maybell, M.M. Shanmugap iya, Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
Leas Eigen Neu osophic Z-se s (LENZS) and he G ea es Eigen Neu osophic Z-se s (GENZS) a e
p esen ed in his s udy along wi h an algo i hm. The p inciples o GENZS and LENZS a e explained
wi h he use o a nume ical example. We show how he sugges ed echnique can be used in p ac ical
si ua ions in Sec ion 6. In Sec ion 7 he wo k is concluded and u u e esea ch di ec ions a e
discussed.
3 P elimina ies
3.1 De ini ion
Le X be a uni e se se hen a Neu osophic Z-numbe se (NZNs) in a uni e se se X is de ined in
he ollowing as
𝑆𝑍=(<x,T(V,R)(x),I(V,R)(x),F(V,R)(x)>𝑥∈X)
(1)
he e
T(V,R)(x)=(𝑇𝑉(x),𝑇𝑅(x)) ,I(V,R)(x)=(𝐼𝑉(x),𝐼𝑅(x)),F (V,R)(x)=(𝐹𝑉(x),𝐹𝑅(x)):X→[0,1]2
(2)
a e he o de pai s o neu osophic alues o u h ulness, inde e minacy, and alsehood; he i s
componen consis s o he neu osophic alues in a uni e se se X, and he second componen consis s
o neu osophic eliabili y measu es, wi h he ule o
0≤𝑇𝑉(x)+ 𝐼𝑉(x)+𝐹𝑉(x)≤3 and 0≤𝑇𝑅(x)+ 𝐼𝑅(x)+𝐹𝑅(x)≤3
(3)
3.2 De ini ion
Le X be a uni e se se and F be a se o pa ame e s. Conside a nonemp y se 𝑆𝑍, 𝑆𝑍 ∈𝐹. Le P(X)
be he collec ion o all neu osophic z- numbe se s o X. The se (E, 𝑆𝑍 ) be e med as neu osophic
z- numbe se s (NZNs) o e X, whe e 𝐸∶ 𝑆𝑍→𝑃(𝑋). Conside S as neu osophic z-ma ices
(NZMs) o e X ins ead o (E, 𝑆𝑍).
3.3 De ini ion
Le 𝑆𝐴 be a 𝑁𝑍𝑀𝑚×𝑛 and 𝑆𝐵 be a 𝑁𝑍𝑀𝑛×𝑝 hen he composi ion o 𝑆𝐴 and 𝑆𝐵 is de ined as
𝑆𝐴∘𝑆𝐵=(<(∑(𝑇𝑉𝑖𝑘
𝐴∧ 𝑇𝑉𝑘𝑗
𝐵)
𝑛
𝑖=1 ,(∑(𝑇𝑅𝑖𝑘
𝐴∧𝑇𝑅𝑘𝑗
𝐵)),(∏(𝐼𝑉𝑖𝑘
𝐴∨ 𝐼𝑉𝑘𝑗
𝐵))
𝑛
𝑖=1 ,
𝑛
𝑖=1
( ∏(𝐼𝑅𝑖𝑘
𝐴 ∨𝐼𝑅𝑘𝑗
𝐵))
𝑛
𝑖=1 ,( ∏(𝐹𝑉𝑖𝑘
𝐴 ∨𝐹𝑉𝑘𝑗
𝐵)),(
𝑛
𝑖=1 ∏(𝐹𝑅𝑖𝑘
𝐴∨𝐹𝑅𝑘𝑗
𝐵))
𝑛
𝑖=1 >)
(4)
Equi alen ly i can be w i en as
𝑆𝐴∘𝑆𝐵=(<(⋃(𝑇𝑉𝑖𝑘
𝐴∧ 𝑇𝑉𝑘𝑗
𝐵)),
𝑛
𝑖=1 (⋃(𝑇𝑅𝑖𝑘
𝐴∧𝑇𝑅𝑘𝑗
𝐵))
𝑛
𝑖=1 ,(⋀(𝐼𝑉𝑖𝑘
𝐴∨ 𝐼𝑉𝑘𝑗
𝐵)),
𝑛
𝑖=1
(⋀(𝐼𝑅𝑖𝑘
𝐴∨𝐼𝑅𝑘𝑗
𝐵)),
𝑛
𝑖=1 (⋀(𝐹𝑉𝑖𝑘
𝐴 ∨𝐹𝑉𝑘𝑗
𝐵)),
𝑛
𝑖=1 (⋀(𝐹𝑅𝑖𝑘
𝐴∨𝐹𝑅𝑘𝑗
𝐵))>)
𝑛
𝑖=1
(5)
Neu osophic Se s and Sys ems, Vol. 97, 2026 691
P. Sheeba Maybell, M.M. Shanmugap iya, Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
I he numbe o 𝑆𝐴 columns equal he numbe o ows 𝑆𝐵, hen he p oduc is de ined. This
mul iplica ion p ocedu e is called as max-min composi ion ope a o . Consequen ly, 𝑆𝐴∘𝑆𝐵 and a e
conside ed con o mable o mul iplica ion, a he han using 𝑆𝐴∘𝑆𝐵 i is deno ed as 𝑆𝐴𝑆𝐵, whe e
∑(𝑇𝑉𝑖𝑘
𝐴∧ 𝑇𝑉𝑘𝑗
𝐵)
𝑛
𝑖=1 means max- min ope a ion and ∏(𝐼𝑉𝑖𝑘
𝐴∨ 𝐼𝑉𝑘𝑗
𝐵))
𝑛
𝑖=1 means min-max ope a ion.
4 Neu osophic Z- ela ion
4.1 De ini ion
Le 𝐻(𝐴,𝐴) be an Neu osophic Z- ela ion (NZR) on a se 𝑆𝐴. Le 𝑇𝑉,𝑅:𝑆𝐴 ⟶[0,1]2, 𝐼𝑉,𝑅:𝑆𝐴 ⟶
[0,1]2,𝑎𝑛𝑑 𝐹𝑉,𝑅:𝑆𝐴 ⟶[0,1]2a e he h ee membe ship unc ion and 𝑀𝐻 be he co esponding
Neu osophic Z- Ma ices (NZM) in ela ion H.
4.2 De ini ion
The ela ion 𝐻(𝐴,𝐴)is e lexi e i he diagonal en ies o 𝑀𝐻 is [<(1,1),(0,0),(0,0)>]whe e
𝑇(𝑉,𝑅)𝐻(𝑥,𝑥)=(1,1), 𝐼(𝑉,𝑅)𝐻(𝑥,𝑥)=(0,0) and 𝐹(𝑉,𝑅)𝐻(𝑥,𝑥)=(0,0) o all 𝑥 ∈ 𝑆𝐴 .
4.3 De ini ion
The ela ion 𝐻(𝐴,𝐴)is symme ic i 𝑀𝐻=𝑀𝐻
𝑇whe e 𝑀𝐻
𝑇is he anspose o 𝑀𝐻 such ha
𝑇(𝑉,𝑅)𝐻(𝑥,𝑦)=𝑇(𝑉,𝑅)𝐻(𝑦,𝑥), 𝐼(𝑉,𝑅)𝐻(𝑥,𝑦)=𝐼(𝑉,𝑅)𝐻(𝑦,𝑥) and 𝐹(𝑉,𝑅)𝐻(𝑥,𝑦)=𝐹(𝑉,𝑅)𝐻(𝑦,𝑥) o all 𝑥,𝑦 ∈
𝑆𝐴 .
4.4 De ini ion
The ela ion 𝐻(𝐴,𝐴) is ansi i e i 𝑀𝐻≥𝑀𝐻
2 i.e.,
𝑇(𝑉,𝑅)𝐻(𝑥,𝑧)≥max (min ((𝑇(𝑉,𝑅)𝐻(𝑦,𝑥),𝑇(𝑉,𝑅)𝐻(𝑦,𝑧))),
𝐼(𝑉,𝑅)𝐻(𝑥,𝑧)≤min (max ((𝐼(𝑉,𝑅)𝐻(𝑦,𝑥),𝐼(𝑉,𝑅)𝐻(𝑦,𝑧))) and
𝐹(𝑉,𝑅)𝐻(𝑥,𝑧)≤min (max((𝐹(𝑉,𝑅)𝐻(𝑦,𝑥),𝐹(𝑉,𝑅)𝐻(𝑦,𝑧))) o all pai (𝑥,𝑧)∈𝑆𝐴×𝑆𝐴.
4.5 De ini ion
Le 𝐻(𝐴,𝐴) ela ion is e lexi e, symme ic and ansi i e hen 𝐻(𝐴,𝐴) ela ion is called as
simila i y ela ion.
4.6 P oposi ion
Fo any 𝑆𝐴∈𝑁𝑍𝑀𝑛×𝑛, 𝑆𝐴 is e lexi e i 𝑆𝐴≥𝐼𝑛 .
p oo
Sinc 𝑆𝐴≥𝐼𝑛, hen ma ix en ies which is diagonal o 𝑆𝐴 a e [<(1,1),(0,0),(0,0)>].
 𝑆𝐴 is a e lexi e ma ix.
Hence he p oo .
4.7 De ini ion
Fo an 𝑆𝐴∈𝑁𝑍𝑀𝑛×𝑛 , we de ine

Neu osophic Se s and Sys ems, Vol. 97, 2026 692
P. Sheeba Maybell, M.M. Shanmugap iya, Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
• 𝑆𝐴 is Re lexi e i 𝑆𝐴≥𝐼𝑛
• 𝑆𝐴 is Weekly e lexi e i 𝑆𝐴≥𝑆𝐴
• 𝑆𝐴 is Symme ic 𝑆𝐴=𝑆𝐴𝑇
• 𝑆𝐴 is Idempo en 𝑆𝐴=𝑆𝐴2
• 𝑆𝐴 is T ansi i e 𝑆𝐴2≤𝑆𝐴
4.8 P oposi ion
Le 𝑆𝐴∈𝑁𝑍𝑀𝑛×𝑛 be a e lexi e o NZM. Then
I. 𝑆𝐴𝑇 is e lexi e NZM, whe e𝑆𝐴𝑇 is anspose o 𝑆𝐴.
II. 𝑆𝐴𝐾 is e lexi e NZM o posi i e in ege k.
III. 𝑆𝐴𝑆𝐵 ≥ 𝑆𝐵 o 𝑆𝐵∈𝑁𝑍𝑀𝑛×𝑛
IV. 𝑆𝐵𝑆𝐴 ≥ 𝑆𝐵 o 𝑆𝐵∈𝑁𝑍𝑀𝑛×𝑛
V. 𝑆𝐴𝑆𝐵 and 𝑆𝐵𝑆𝐴 a e e lexi e NZMs i 𝑆𝐵 is e lexi e
VI. 𝑆𝐴𝑆𝐴𝑇 and 𝑆𝐴𝑇𝑆𝐴 a e e lexi e NZMs.
P oo :
I. Since 𝑆𝐴 has e lexi e p ope ies only when i s diagonal en ies a e [<(1,1),(0,0),(0,0)>].
Hence 𝑆𝐴𝑇 is e lexi e.
II. Since 𝑆𝐴 is e lexi e, 𝑆𝐴≥𝐼𝑛 hen 𝑆𝐴2≥𝑆𝐴≥𝐼𝑛 (mul iplying on bo h sides). P oceeding o
(k-1) imes we ge 𝑆𝐴𝑘≥ 𝑆𝐴𝑘−1≥⋯…..≥𝑆𝐴2≥𝑆𝐴 ≥𝐼𝑛. The esul holds o any scala k. hen
𝑆𝐴𝑘 is e lexi e.
III. 𝑆𝐴≥𝐼𝑛 hen, 𝑆𝐴𝑆𝐵≥𝐼𝑛𝑆𝐵 ⇒𝑆𝐴𝑆𝐵≥𝑆𝐵
IV. 𝑆𝐴≥𝐼𝑛 hen, 𝑆𝐵𝑆𝐴≥𝐼𝑛𝑆𝐵 ⇒𝑆𝐵𝑆𝐴≥𝑆𝐵
V. Since 𝑆𝐵 is e lexi e 𝑆𝐵≥𝐼𝑛 hen 𝑆𝐴𝑆𝐵≥𝑆𝐵≥ 𝐼𝑛 and 𝑆𝐵𝑆𝐴≥𝑆𝐵≥ 𝐼𝑛 om (III) and
(IV). Hence 𝑆𝐴𝑆𝐵 and 𝑆𝐵𝑆𝐴 a e e lexi e.
VI. Using (I) in (V) eplace 𝑆𝐴𝑇 in he place o 𝑆𝐵 we de i e he desi e esul .
Hence he p oo
4.9 P oposi ion
I 𝑆𝐴∈𝑁𝑍𝑀𝑛×𝑛 be ansi i e and also i is e lexi e hen 𝑆𝐴 is idempo en .
P oo :
I is known ha 𝑆𝐴 is e lexi e, 𝑆𝐴≥𝐼𝑛
𝑆𝐴2≥𝑆𝐴≥𝐼𝑛
(6)
Also, 𝑆𝐴 is ansi i e
𝑆𝐴2≤𝑆𝐴
(7)
Combining (6) & (7) 𝑆𝐴2= 𝑆𝐴
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Hence 𝑆𝐴 is idempo en
No e: Con e se is no ue.
Example
Le 𝑆𝐴 = [<(0.8,0.3),(0.3,0.4),(0.4,0.4)> <(0.8,0.3),(0.3,0.4),(0.4,0.4)>
<(0.5,0.3),(0.3,0.4),(0.4,0.4)> <(0.8,0.3),(0.3,0.4),(0.4,0.4)>] I2
Hence 𝑆𝐴 is no e lexi e, Bu
𝑆𝐴2=𝑆𝐴𝑆𝐴 (max-min)
= [<(0.8,0.3),(0.3,0.4),(0.4,0.4)> <(0.8,0.3),(0.3,0.4),(0.4,0.4)>
<(0.5,0.3),(0.3,0.4),(0.4,0.4)> <(0.8,0.3),(0.3,0.4),(0.4,0.4)>] = 𝑆𝐴
𝑆𝐴 is idempo en bu no e lexi e
4.10 P oposi ion
I 𝑆𝐴 and 𝑆𝐵 a e wo symme ic NZMs o o de n x n such ha 𝑆𝐴𝑆𝐵= 𝑆𝐵𝑆𝐴, hen 𝑆𝐴𝑆𝐵 is
symme ic NZM.
I can be p o ed easily
No e:
I 𝑆𝐴 is symme ic in 𝑁𝑍𝑀𝑛×𝑛 hen 𝑆𝐴𝐾 is also symme ic o any scala k.
4.11 P oposi ion
Le 𝑆𝐴 , 𝑆𝐵 ∈𝑁𝑍𝑀𝑛×𝑛 is ansi i e, such ha 𝑆𝐴𝑆𝐵= 𝑆𝐵𝑆𝐴 , hen 𝑆𝐴 𝑆𝐵 will be ansi i e.
P oo
We know 𝑆𝐴 and 𝑆𝐵 bo h ansi i e 𝑆𝐴2≤𝑆𝐴 and 𝑆𝐵2≤𝑆𝐵. Now
(𝑆𝐴𝑆𝐵)2=(𝑆𝐴𝑆𝐵)(𝑆𝐴𝑆𝐵)
=𝑆𝐴(𝑆𝐵𝑆𝐴)𝑆𝐵
=𝑆𝐴(𝑆𝐴𝑆𝐵)𝑆𝐵
=(𝑆𝐴𝑆𝐴)(𝑆𝐵𝑆𝐵)
=𝑆𝐴2𝑆𝐵2
⇒(𝑆𝐴𝑆𝐵)2≤𝑆𝐴𝑆𝐵
hence 𝑆𝐴𝑆𝐵 is ansi i e.
No e:
I 𝑆𝐴 is ansi i e in 𝑁𝑍𝑀𝑛×𝑛 hen 𝑆𝐴𝑘 is also ansi i e o any scala k.
4.12 P oposi ion:
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P. Sheeba Maybell, M.M. Shanmugap iya, Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
I 𝑆𝐴=[𝑆𝐴𝑝𝑞]=[<(𝑇𝑉𝑝𝑞
𝐴,𝑇𝑅𝑝𝑞
𝐴),(𝐼𝑉𝑝𝑞
𝐴,𝐼𝑅𝑝𝑞
𝐴),(𝐹𝑉𝑝𝑞
𝐴,𝐹𝑅𝑝𝑞
𝐴)>] ∈𝑁𝑍𝑀𝑛×𝑛 is symme ic and ansi i e
hen 𝑆𝐴𝑝𝑞 ≤ 𝑆𝐴𝑝𝑝 o p,q ∈ { 1,2, 3………n}.
P oo
Le 𝑆𝐴 is symme ic, 𝑆𝐴𝑝𝑞 = 𝑆𝐴𝑞𝑝 o all p, q ∈ {1,2, 3………n}
Also, since 𝑆𝐴 is ansi i e
𝑆𝐴2≤𝑆𝐴⟹𝑆𝐴≥ 𝑆𝐴2
Thus
𝑆𝐴𝑝𝑞 ≥ 𝑚𝑎𝑥⏟
𝑟 [min(𝑆𝐴𝑝𝑟,𝑆𝐴𝑟𝑗)] o p= q and ∈ {1,2, 3………n}
𝑆𝐴𝑝𝑞 ≥ 𝑚𝑎𝑥⏟
𝑟 [min(𝑆𝐴𝑝𝑟,𝑆𝐴𝑟𝑞)] o p= q and ∈ {1,2, 3………n}
≥ min(𝑆𝐴𝑝𝑟,𝑆𝐴𝑟𝑞) o =q and each p
𝑆𝐴𝑝𝑝 ≥ 𝑆𝐴𝑝𝑞 (since 𝑆𝐴𝑝𝑞 = 𝑆𝐴𝑞𝑝)
Hence p o ed.
4.13 De ini ion
Le 𝑆𝐴 ∈𝑁𝑍𝑀𝑛 and 𝑆𝐵 is said o be in e ible i and only i he e exis 𝑆𝐵 ∈𝑁𝑍𝑀𝑛 such ha
𝑆𝐴𝑆𝐵= 𝑆𝐵𝑆𝐴=𝐼𝑛 .
4.14 De ini ion
An 𝑆𝐴 ∈𝑁𝑍𝑀𝑛 is called Neu osophic Z-Pe mu a ion ma ix (NZPM) i bo h ow and column
con ains exac ly one en y I and all o he en ies a e  .
4.15 P oposi ion
I 𝑆𝐴 be a 𝑁𝑍𝑀𝑛 o an NZPM hen 𝑆𝐴𝑆𝐴𝑇= 𝑆𝐴𝑇𝑆𝐴=𝐼𝑛
P oo :
𝑆𝐴=(<(𝑇𝑉𝑖𝑗
𝐴,𝑇𝑅𝑖𝑗
𝐴),(𝐼𝑉𝑖𝑗
𝐴,𝐼𝑅𝑖𝑗
𝐴),(𝐹𝑉𝑖𝑗
𝐴,𝐹𝑅𝑖𝑗
𝐴)>)
Then 𝑆𝐴𝑇=(<(𝑇𝑉𝑗𝑖
𝐴,𝑇𝑅𝑗𝑖
𝐴),(𝐼𝑉𝑗𝑖
𝐴,𝐼𝑅𝑗𝑖
𝐴),(𝐹𝑉𝑗𝑖
𝐴,𝐹𝑅𝑗𝑖
𝐴)>)
now, i, j h en ies o 𝑆𝐴𝑆𝐴𝑇 is
∑𝑆𝐴𝑖𝑘𝑆𝐵𝑘𝑗
𝑛
𝑘=1 = ∑𝑆𝐴𝑖𝑘𝑆𝐴𝑘𝑗
𝑛
𝑘=1 ={ 𝑖𝑓 𝑖≠𝑗
𝐼 𝑖𝑓 𝑖=𝑗
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P. Sheeba Maybell, M.M. Shanmugap iya, Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
(since 𝑆𝐴𝑆𝐵 is NZPM, ∑𝑆𝐴𝑖𝑘𝑆𝐴𝑖𝑘
𝑛
𝑘=1 =𝐼 )
Hence 𝑆𝐴𝑆𝐴𝑇 is an 𝐼𝑛.
con e se can be p o ed easily.
Hence he p oo .
4.19 P oposi ion
Le 𝑆𝐴 be a NZMn, 𝑆𝐴 is in e ible i and only i 𝑆𝐴 is an NZPM.
P oo :
Fi s pa :
𝑆𝐴𝑆𝐴𝑇= 𝑆𝐴𝑇𝑆𝐴=𝐼𝑛 (by p e ious p oposi ion)
hence 𝑆𝐴 is in e ible and 𝑆𝐴𝑇 is he in e se o 𝑆𝐴 (i.e) 𝑆𝐴−=𝑆𝐴𝑇
Second pa :
Le 𝑆𝐴 be in e ible and 𝑆𝐵 be he in e se o 𝑆𝐴. Thus 𝑆𝐴𝑆𝐵= 𝑆𝐵𝑆𝐴=𝐼𝑛 ollows ha
∑𝑆𝐴𝑝𝑟𝑆𝐵𝑟𝑞
𝑛
𝑘=1 = ∑𝑆𝐵𝑝𝑟𝑆𝐴𝑟𝑞
𝑛
𝑘=1 = o p≠q
∑𝑆𝐴𝑝𝑟𝑆𝐵𝑟𝑞
𝑛
𝑘=1 = ∑𝑆𝐵𝑝𝑟𝑆𝐴𝑟𝑝
𝑛
𝑘=1 =𝐼
𝑆𝐴𝑝𝑞𝑆𝐵𝑞𝑟=𝑆𝐵𝑝𝑟𝑆𝐴𝑟𝑞 = o p≠q and {1,2,…..n}
(8)
𝑆𝐴𝑝𝑟𝑆𝐵𝑟𝑝 =𝑆𝐵𝑝𝑟𝑆𝐴𝑟𝑝 =𝐼 o a leas one {1,2,…..n}
and o each p {1,2,…..n}
(9)
F om (8)
𝑆𝐴𝑝𝑟 = o 𝑆𝐵𝑟𝑞 = o bo h 𝑆𝐴𝑝𝑟 =𝑆𝐵𝑟𝑞 =
o p≠q and {1,2,…..n}
(10)
and
𝑆𝐵𝑝𝑟 = o 𝑆𝐴𝑟𝑞 = o bo h 𝑆𝐵𝑝𝑟 =𝑆𝐴𝑟𝑞 =
o p≠q and {1,2,…..n}
(11)
Also, using (11)
𝑆𝐴𝑝𝑟 =𝑆𝐵𝑟𝑝 =I and 𝑆𝐴𝑟𝑝 =𝑆𝐵𝑝𝑟 =I
o a leas one {1,2,…..n} and o each p {1,2,…..n}
(12)
Le he esul s o (12) equa ion exis s o k=p (say) ha is
𝑆𝐴𝑝𝑘 =𝑆𝐵𝑘𝑝=I = [<(1,1),(0,0),(0,0)>]
Then om (10), we ge 𝑆𝐵𝑝𝑗 = =[<(0,0),(1,1),(1,1)>] o all i≠ j and
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P. Sheeba Maybell, M.M. Shanmugap iya, Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
𝑄3
=[<(0.8,0.7),(0.4,0.3),(0.2,0.1)> <(0.8,0.7),(0.4,0.3),(0.2,0.1)> <(0.8,0.7),(0.4,0.3),(0.2,0.1)>]
Now, Q3=Q2 hen Q2 is he desi ed GENZ se
Nex , algo i hm I o Calcula e LENZS
calcula e Q1
′
𝑄1′
=[<(0.6,0.5),(0.6,0.4),(0.3,0.1)> <(0.5,0.6),(0.5,0.3),(0.4,0.3)> <(0.5,0.6),(0.6 ,0.7),(0.3,0.4)>]
Nex s ep o n=1, 𝑄2′= 𝑄1′ ∙𝐻1
𝑄2′
=[<(0.6,0.6),(0.5,0.3),(0.3,0.1)> <(0.5,0.6),(0.5,0.3),(0.3,0.3)> <(0.5,0.6),(0.6 ,0.4),(0.3,0.1)>]
Now, ind 𝑄3′= 𝑄2′ ∙𝐻1
𝑄3′
=[<(0.6,0.6),(0.5,0.3),(0.3,0.1)> <(0.5,0.6),(0.5,0.3),(0.3,0.3)> <(0.5,0.6),(0.6 ,0.4),(0.3,0.1)>]
Now, 𝑄3′= 𝑄2′ hen 𝑄2′ is he acqui ed LENZS.
Then using (15), (16) and (17) ind 𝑆𝐴𝑚𝑎𝑥 and 𝑆𝐴𝑚in.
The e ec i eness (E) and Unce ain y(U) can be ound using a e age o max, min alue and
di e ence o max, min alue di ided by 2.
Table 2: Resul o E ec i eness and Unce ain y
Table 2 depic s e ec i eness E1 is highe and unce ain y U1 is lowe which makes he ambiance is
good. The highes unce ain y o U3 gi es he eedback o mone a y alue can be conside ed in u u e.
7. Conclusion
In his wo k, neu osophic z- ela ion along wi h neu osophic z- ma ices and hei ce ain
connec ed p ope ies and models a e in oduced. The algo i hms o calcula ing wo ypes o eigen
neu osophic z-se wi h analogous we e shown. A las , a u iliza ion o neu osophic z- se in choice
s a egy p oblem using eigen neu osophic z-se we e ound. As, an ex ension o his wo k in u u e,
Neu osophic Z- ela ions and Z-ma ices may be gene alized o highe -o de s uc u es such as
Neu osophic Z- enso s, enabling he modeling o mul i-dimensional and highly unce ain da a.
Pa ame e s
𝑆𝐴𝑚𝑎𝑥
0.8066
0.8066
0.8066
𝑆𝐴𝑚in
0.7266
0.6866
0.6766
E
0.7666
0.7463
0.7416
U
0.04
0.06
0.065

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P. Sheeba Maybell, M.M. Shanmugap iya, Eigen Neu osophic Z- Se and Neu osophic Z- Rela ion
Eigen neu osophic Z-se in eg a ion wi h machine lea ning, deep lea ning, and hyb id in elligen
sys ems could open up new oppo uni ies o unce ain da a-d i en decision making.
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Recei ed: June 4, 2025. Accep ed: No 15, 2025