Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

56 https://internationalpubls.com 

On Arithmetical Traits of Doubt Fuzzy T-Ideals beneath the 

Normalization is a T-Algebra 

 

CT. Nagaraj(a) , M. Premkumar , Y. Immanuel(b), Abdul Salam(c) ,  M. S Franklin Thamil 

Selvi(d) , M. I. Mary Metilda(e) and J. Juliet Jeyapackiam(f) 

(a)Department of Mathematics, Sree Sevugan Annamalai College, Devakottai-630303, India*(1a)[0000-0002-8637-063X] 

(*1a, b, d & e) Department of Mathematics, Sathyabama Institute of Science and Technology  

(Deemed To Be University) Chennai-600119, Tamilnadu, India. (B)[0000-0003-0719-375X] 

(c)Gulf Asian English School, Sharjah, United Arab Emirates. 

(f)Department of Mathematics, Jayaraj Annapackiam CSI College of Engineering 

Nazareth, Tuticorin-628617, India. 

(a)mathsnagaraj.ct@gmail.com, (*1a) mprem.maths3033@gmail.com, (b)y_immanuel@yahoo.com, 
(c)abdulsalam.maths@gmail.com, (d)thamizanand@gmail.com, (e)metilda81@gmail.com 

(f) jeyasjjjeyas@gmail.com, Corresponding Author Email Id: (*1a) mprem.maths3033@gmail.com[0000-0003-4656-3370] 

 

Article History: 

Received: 06-07-2024 

Revised: 21-08-2024 

Accepted: 03-09-2024 

Abstract:  

The normal doubt fuzzy T-ideal and poset under the set of inclusion principle in T-algebra 

are defined in this article, along with several algebraic properties and instances that are 

covered in detail. 

Keywords: Doubt Fuzzy Set (DFS), Doubt Fuzzy Subset (DFSb),T-Algebra, T-Ideal, Doubt 

Fuzzy T-ideal (DFTI), Normal Doubt Fuzzy T-ideal (NDFTI). 

Classification of Subject : MSC2020-zbMATH-03B52 

 

I. Introduction 

Abu Ayub Ansari[1] introduced the novel idea of T-FβSA of β-algebras in 2014. Prasanna, A., et al. 

[2&3]. outlined the new FBI Normalization notation in B-Algebra and presented the idea of FBGI 

Normalization in BG-Algebra in 2018. Priya's FPSIs and FPSSAs for PS-algebras were standardized 

in 2015[4]. In 2015, Rajam[5] presented the idea of L-FTI in β-algebras. In 2016, Sithar Selvam[6] 

learned about the FPMSA Normalization study. Tamil created FSA and FTI in TM-Algebras in 

2011[7].  Zadeh[8] introduced fuzzy sets for the first time in 1965. 

This work describes the normal fuzzy T-ideal and poset under the set of inclusion principle over T-

algebra and explores some algebraic characteristics. 

II. Preliminaries 

Basic Reference: 2.1 [8] 

 Let 𝑋 be a non-empty set .  A 𝐹𝑆𝑏 of the set X is a mapping 𝜇 : 𝑋→ [0, 1]. 

Basic Reference: 2.2[7] 

 A FS 𝜇 in a BP-algebra X is called a 𝐹𝑇𝐼  of X if it satisfies the following conditions: 

(i)        𝜇(0) ≥ 𝜇(𝑥) 

mailto:mathsnagaraj.ct@gmail.com
mailto:*1a)%20mprem.maths3033@gmail.com
mailto:y_immanuel@yahoo.com
mailto:abdulsalam.maths@gmail.com
mailto:thamizanand@gmail.com
mailto:metilda81@gmail.com
mailto:jeyasjjjeyas@gmail.com
mailto:mprem.maths3033@gmail.com


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(ii) 𝜇(𝑥 ∗ 𝑧) ≥ 𝑚𝑖𝑛{𝜇((𝑥 ∗ 𝑦) ∗ 𝑧), 𝜇(𝑦)}, ∀ 𝑥, 𝑦 ∈ 𝑋. 

III On Arithmetical Traits of Doubt Fuzzy T-Ideals beneath the Normalization of T-Algebra 

Definition: 3.1 

 Let 𝐷𝐹𝑇𝐼 Ώ of ᾎ is called to be 𝑁𝐷𝐹𝑇𝐼 if ∃ 𝜃 ∈ ᾎ s.t Ώ(0) = 1. 

Example: 3.1.1 

 Let ᾎ = {0, 𝑎, 𝑏, 𝑐, 𝑑} be a T-Algebra 

* 𝟎 𝒂 𝒃 𝑪 𝒅 

𝟎 0 𝑎 𝑏 𝑐 𝑑 

𝒂 0 0 0 0 𝑎 

𝒃 0 𝑐 0 𝑐 𝑑 

𝒄 0 𝑎 𝑏 0 𝑎 

𝒅 0 0 0 0 0 

Then (ᾎ,∗ ,0) is a T-Algebra. Define 𝐷𝐹𝑆 Ώ in ᾎ by  Ώ(0) = 0.9,  

 Ώ(𝑎) = 0.7, Ώ(𝑏) = 0.8, Ώ(𝑐) = 0.6 𝑎𝑛𝑑 Ώ(𝑑) = 0.5. 

⇒Then Ώ is a 𝑁𝐷𝐹𝑇𝐼 of ᾎ. 

Remark: 3.2 

 Let 𝑁𝐷𝐹𝑇𝐼 Ώ of ᾎ if and only if Ώ(0) = 1. 

Theorem: 3.3 

 Let any 𝐷𝐹𝑇𝐼 Ώ of ᾎ, we can generate the 𝑁𝐷𝐹𝑇𝐼 of ᾎ ⊂ Ώ. 

Proof: 

 Let Ώ be a 𝐷𝐹𝑇𝐼 of ᾎ. 

Define a 𝐷𝐹𝑆 Ώn of ᾎ as Ώ𝑛(ã) = Ώ(ã) + Ώ𝑐(0), ∀ã ∈ ᾎ.   

 Let ã, ɓ ∈ ᾎ 

(i) Ώ𝑛(0) = Ώ(0) + Ώ𝑐(0) ≤ Ώ(ã) + Ώ𝑐(0)   = Ώ𝑛(ã) 

                                                      ⇒ Ώ𝑛(0) ≤ Ώ𝑛(ã) 

(ii) Ώ𝑛(ã ∗ ĉ) = Ώ((ã ∗ ɓ) ∗ ĉ) + Ώ𝑐(0)    ≤ 𝑚𝑎𝑥{Ώ((ã ∗ ɓ) ∗ ĉ), Ώ(ɓ)} + Ώ𝑐(0) 

= 𝑚𝑎𝑥{[Ώ((ã ∗ ɓ) ∗ ĉ) + Ώ𝑐(0)], [Ώ(ɓ) + Ώ𝑐(0)]} 

= 𝑚𝑎𝑥{Ώ𝑛((ã ∗ ɓ) ∗ ĉ), Ώ𝑛 (ɓ)} 

      ⇒Ώ𝑛(ã ∗ ĉ) ≤ 𝑚𝑎𝑥{Ώ𝑛((ã ∗ ɓ) ∗ ĉ), Ώ𝑛 (ɓ)} 

Also Ώ𝑛(0) = Ώ(0) + Ώ𝑐(0) = Ώ(0) + 1 − Ώ(0) = 1. 

∴ Ώ𝑛 is a 𝑁𝐷𝐹𝑇𝐼 of ᾎ. 



Communications on Applied Nonlinear Analysis 

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Lemma: 3.4 

 Let Ώ𝑛 be an 𝐷𝐹𝑆 in ᾎ defined by Ώ𝑛(ã) = Ώ(𝜃) + Ώc(0), ∀ ã ∈ ᾎ. If ∃  element ã ∈ ᾎ in s.t 

Ώ𝑛(ã) = 0,  then Ώ(ã) = 0. 

Lemma: 3.5 

 Let Ώ be 𝐷𝐹𝑇𝐼 of ᾎ. Then, 

 (i) if Ώ itself is 𝐷𝐹𝑇𝐼 then Ώ(ã) = Ώ𝑛(ã). 

(ii) if Ώ is a 𝐷𝐹𝑇𝐼  of 𝑋 then (Ώ𝑛(ã))
𝑛

= Ώ𝑛(ã). 

Proposition: 3.6 

 Let Ώ  𝐷𝐹𝑇𝐼 ᾎ. If Ώ contains the 𝑁𝐷𝐹𝑇𝐼 of ᾎ, generated by any other 𝐷𝐹𝑇𝐼 of ᾎ then Ώ is normal . 

Proof: 

 Let 𝛿 be a 𝐷𝐹𝑇𝐼 of ᾎ. 

by the. 3.3, let 𝛿𝑛 is a 𝐷𝐹𝑇𝐼 of ᾎ 

∴ 𝛿𝑛(0) = 1( lem. 3.4) 

Let Ώ be a  𝐷𝐹𝑇𝐼 of ᾎ s.t 𝛿𝑛 ⊂  Ώ. 

⇒ Ώ(ã) ≤ 𝛿𝑛(ã), ∀ã ∈ ᾎ 

Put ã = 0 ⇒ Ώ(0) ≤ 𝛿𝑛(0) = 1 

⇒ Ώ(0) ≤ 1 ∴ Ώ is normal 

Theorem: 3.7 

 A set 𝑁Ώ = {ã ∈ 𝑋/Ώ(ã) = Ώ(0)}. Let Ώ and 𝛿 be 𝑁𝐷𝐹𝑇𝐼 of ᾎ. If Ώ ⊂ 𝛿 then 𝑁Ώ ⊂ 𝑁𝛿 . 

Proof: 

 Let ã ∈ 𝑁Ώ 

Since Ώ ⊂ 𝛿, 𝛿(ã) ≤ Ώ(ã) = Ώ(0) = 1 = 𝛿(0) 

⇒ã ∈ 𝑁𝛿  

∴ 𝑁Ώ ⊂ 𝑁𝛿 

Theorem: 3.8 

 Let Ώ be the  𝐷𝐹𝑇𝐼 of ᾎ. Let 𝑓: [0, Ώ(0)] → [0,1] be an increasing function.  Let’s define a 𝐷𝐹𝑆  

Ώ𝑓: ᾎ → [0,1] by Ώ𝑓(ã) = 𝑓(Ώ(ã)), ∀ ã ∈ ᾎ. Therefore 

(i) If Ώ𝑓 is a 𝐷𝐹𝑇𝐼 of ᾎ 

(ii) If 𝑓(Ώ(0)) = 1, then Ώ𝑓 is normal 

(iii) If 𝑓(𝑡) ≤ 𝑡, ∀ 𝑡 ∈ [0, Ώ(0)]  then Ώ ⊂ Ώ𝑓. 

Proof: 



Communications on Applied Nonlinear Analysis 

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Let Ώ be 𝐷𝐹𝑇𝐼  of  ᾎ.  

Let 𝑓: [0, Ώ(0)] → [0,1] be an increasing function.   

Define a 𝐷𝐹𝑆 Ώ𝑓: ᾎ → [0,1] by Ώ𝑓(ã) = 𝑓(Ώ(ã)), ∀ ã ∈ ᾎ. 

(i) (a)    Ώ𝑓(0) = 𝑓(Ώ (0)) ≤ 𝑓(Ώ(ã))   = Ώ𝑓(ã) 

        ⇒ Ώ𝑓(0) ≤ Ώ𝑓(ã) 

     (b)     Ώ𝑓(ã ∗ ĉ) = 𝑓(Ώ (ã ∗ ĉ)) ≤ 𝑓 {𝑚𝑎𝑥{Ώ((ã ∗ ɓ) ∗ ĉ), Ώ(ɓ)}} 

= 𝑚𝑎𝑥{𝑓(Ώ(ã ∗ ɓ) ∗ ĉ), 𝑓(Ώ(ɓ))} 

     = 𝑚𝑎𝑥{Ώ𝑓((ã ∗ ɓ) ∗ ĉ), Ώ𝑓(ɓ)} 

⇒Ώ𝑓 is a 𝐹𝑇𝐼. 

(ii) If 𝑓(Ώ (0)) = 1    ⇒Ώ𝑓(0) = 1 

⇒ Ώ𝑓 is normal 

(iii) Let 𝑓(𝑡) ≤ 𝑡, ∀  𝑡 ∈ [0, Ώ(0)] 

                             Then Ώ𝑓(ã) = 𝑓(Ώ(ã) ≤ Ώ(ã)), ∀ ã ∈ ᾎ  ∴ Ώ ⊆ Ώ𝑔. 

Definition: 3.9 

 Let 𝜗 = (
𝜏

𝜏
is the 𝑁𝐷𝐹𝑇𝐼 of ᾎ) then the 𝜗 is called a Poset according to the principle of inclusion. 

Definition: 3.10 

 Let 𝑠 > 0 be a real number. If 𝛽 ∈ [0,1],  𝛽𝑠 be the positive root in case 𝑠 < 1. We define Ώ𝑠: 𝐾 →

[0,1] by Ώ𝑠(ã) = (Ώ(ã))
𝑠
, ∀ã ∈ ᾎ.  

Theorem: 3.11 

 Let, Ώ∈ 𝜗 be a constant s.t it is a maximum element of (𝜗, ⊆). Then, Ώ only accept  the values of  0 

& 1.  

Proof:  Let, Ώ∈ 𝜗. Then Ώ(0) = 1,  

Let, ã ∈ ᾎ s.t Ώ(ã) ≠ 1. 

We claim that Ώ(0) = 0. If not, then ∃ 𝑏 ∈ 𝑋 s.t 0 < Ώ(𝑏) < 1. 

We now define a 𝐷𝐹𝑆, 𝜋: ᾎ → [0,1] by 𝜋(ã) =
1

2
{Ώ(ã) + Ώ(𝑏)}, ∀ ã ∈ ᾎ. 

Then Ώ obviously is well defined 

Now ,(i)                            𝜋(0) =
1

2
{Ώ(0) + Ώ(𝑏)} 

≤
1

2
{Ώ(ã) + Ώ(𝑏)} = 𝜋(ã) 

⇒𝜋(0) ≤ 𝜋(ã) 



Communications on Applied Nonlinear Analysis 

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(ii)       𝜋(ã ∗ ĉ) =
1

2
{Ώ(ã ∗ ĉ) + Ώ(𝑏)} 

≤
1

2
{𝑚𝑎𝑥{Ώ((ã ∗ ĉ) ∗ ĉ), Ώ(ɓ)} + Ώ(𝑏)} 

=
1

2
{𝑚𝑎𝑥({Ώ((ã ∗ ĉ) ∗ ĉ) + Ώ(𝑏)}, {Ώ(ɓ) + Ώ(𝑏)})} 

= 𝑚𝑎𝑥 {
1

2
{Ώ((ã ∗ ĉ) ∗ ĉ) + Ώ(𝑏)},

1

2
{Ώ(ɓ) + Ώ(𝑏)} } 

= 𝑚𝑎𝑥{𝜋((ã ∗ ĉ) ∗ ĉ), 𝜋(ɓ)} 

⇒ 𝜋(ã ∗ ĉ) ≤ 𝑚𝑎𝑥{𝜋((ã ∗ ĉ) ∗ ĉ), 𝜋(ɓ)}. 

⇒𝜋 is a 𝐹𝑇𝐼. 

⇒ 𝜋𝑛   is a 𝑁𝐹𝑇𝐼. 

𝜋𝑛(ã) = 𝜋(ã) + 𝜋𝑐(0) 

= 𝜋(ã) + (1 − 𝜋(0)) 

=
1

2
{𝜋(ã) + 𝜋(𝑏)} + (1 −

1

2
{𝜋(0) + 𝜋(𝑏)}) 

=
1

2
Ώ(ã) + 1 −

1

2
(1) 

=
1

2
Ώ(ã) +

1

2
 

=
1

2
(Ώ(ã) + 1) ≤ Ώ(ã), ∀ã ∈ 𝑋 

∴ 𝜋𝑛(0) =
1

2
(Ώ(0) + 1) = 1 

∴ 𝜋𝑛 is normal ⇒𝜋𝑛 ∈ 𝜗 

Also 𝜋𝑛(ã) < Ώ(ã), ∀ã ∈ ᾎ. 

The contradiction the fact  that of Ώ is normal. 

⇒Ώ(ã) = 0, ∀ã ∈ ᾎ. 

Theorem: 3.12 

 Let Ώ is a 𝐷𝐹𝑇𝐼 of 𝑋, then so is Ώ𝑠 and 𝑁Ώ
𝑠 = 𝑁Ώ. 

Proof: 

 Let ã, ɓ ∈ ᾎ. 

Now,                  (i)       Ώ𝑠(0) = (Ώ(0))
𝑠
 

≤ (Ώ(ã))
𝑠
 

= Ώ𝑠(ã). 



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⇒      Ώ𝑠(0) ≥ Ώ𝑠(ã) 

      (ii) Ώ𝑠(ã ∗ ĉ) = (Ώ(ã ∗ ĉ))
𝑠
 

≤ (𝑚𝑎𝑥{Ώ((ã ∗ ɓ) ∗ ĉ), Ώ(ɓ)})
𝑠
 

= 𝑚𝑎𝑥 {(Ώ((ã ∗ ɓ) ∗ ĉ))
𝑠

, (Ώ(ɓ))𝑠} 

= 𝑚𝑎𝑥{Ώ𝑠((ã ∗ ɓ) ∗ ĉ), Ώ𝑠(ɓ) } 

⇒ Ώ𝑠(ã ∗ ĉ) ≤ 𝑚𝑎𝑥{Ώ𝑠((ã ∗ ɓ) ∗ ĉ), Ώ𝑠(ɓ) } 

∴ Ώ𝑠 is a 𝐷𝐹𝑇𝐼. 

𝑁Ώ
𝑠 = {ã ∈ 𝐽/Ώ𝑠(ã) = Ώ𝑠(0)} 

= {ã ∈ 𝐽/Ώ(ã) = Ώ(0)} 

⇒𝑁Ώ
𝑠 = 𝑁Ώ. 

III. Conclusion 

Hence we have to this paper discussed about the 𝑁𝐷𝐹𝑇𝐼 and Poset under the set of inclusion principle 

over T-Algebra. This idea can be further extended to normalization of intuitionistic FS, normalization 

of interval valued FS, and normalization of bipolar FSs for new findings in future studies.  

References 

[1] M. Abu Ayub Ansari and M. Chandramouleeswaran, T-Fuzzy 𝛽 −subalgebras of 𝛽 −algebras, International J. of 

Maths. Sci. and Engg. Appls. (IJMSEA), 8 (2014), no. 1, 177-187. 

[2] A. Prasanna, M. Premkumar and A. Solairaju, Normalization of fuzzy B-Ideals in B-Algebra, International Journal 

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[3] A. Prasanna, M. Premkumar and S. Ismail Mohideen, Normalization of fuzzy BG-Ideals in BG-Algebra, International 

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[4] PriyaT and Ramachandran T, Normalization of Fuzzy PS-ideals and Fuzzy PS-sub algebras of PS-algebras, Research 

journal’s Journal of Mathematics, 1,4(2014),1-12. 

[5] K. Rajam and M. Chadramouleeswaran, L-Fuzzy T-ideals in 𝛽 −Algebras, Applied Mathematical Sciences, 9 (2015), 

no. 145, 7221-7228. https://doi.org/10.12988/ams.2015.59581. 

[6] P.M. Sithar Selvam and K.T. Nagalakshmi, A study on Normalization of fuzy PMS-Algebra, International journal of 

Trend in Reesrach and Development, 3 (2016), no.6, 49-55. 

[7] A. Tamilarasi and K. Megalai, Fuzzy Subalgebras and fuzzy T-ideals in TM-Algebras, Journal of Mathematics and 

Statistics, 7 (2011), no. 2, 107-111. https://doi.org/10.3844/jmssp.2011.107.111. 

[8] L.A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65)90241-x. 

https://doi.org/10.12988/ams.2015.59581
https://doi.org/10.3844/jmssp.2011.107.111
https://doi.org/10.1016/s0019-9958(65)90241-x