Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Ternary algebraic structure convey to new type of intuitionistic Q1 anti fuzzy ideals of an regular ordered ternary semigroups R. Balaji1, P.Srikanth Rao2, K Rajeshwar Reddy3, Aiyared Iampan4,∗ 1Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai-602105, India. 2B V Raju Institute of Technology, Narsapur Medak Dist, Telangana state-502313, India. 3Department of Mathematics, Malla Reddy College of Engineering and Technology, Maisammaguda, Medchal Malkangiri, Telangana State-500014. 4Department of Mathematics, School of Science, University of Phayao, 19 Moo 2, Tambon Mae Ka, Amphur Mueang, Phayao 56000, Thailand. E-mails:1balaji 2410@yahoo.co.in, 2srikanthrao.p@bvrit.ac.in, 3kattarajeshwarreddy@gmail.com, 4aiyared.ia@up.ac.th, ∗Corresponding author: Aiyared Iampan. Received: 04-11-2024 Revised: 12-12-2024 Accepted: 01-01-2025. Abstract We examine some of the properties of these ordered ternary semigroups, such as intuitionistic Q1 anti fuzzy left ideal ,Q1 anti fuzzy right ideal, Q1 anti fuzzy lateral ideal, Q1 anti fuzzy ideal and Q1 anti fuzzy bi-ideal. We introduce the idea of TSS. The Q1 anti anti fuzzy ideal is extended in a new way over ternary semigroups B. Keywords: ternary semigroups, anti fuzzy ideals, anti fuzzy bi-ideals, Q1 anti fuzzy bi-ideals. 1 Introduction The ternary algebraic systems known as triplex structures were initially conceived in 1932 by D. H. Lehmer.1 Vandiver came up with the idea of the semiring in 1934. In 1962 Hestenes2 used linear transformation and matrices as examples to establish the concept of ternary algebra. In 1971, after describing those additive sub- groups of rings that are closed under the triple ring product, Lister talked about this algebraic system as a ternary ring. The fuzzy set (FS) theory, which was created by Zadeh,3 works best when it comes to handling ambiguity and uncertainty. An element with a single value inside the interval is called a member of an FS. The NMG might not always be equal to one minus the MG in practice, though, because of potential pushback. A growing number of hybrid fuzzy models are being created as FS theory develops quickly. As a result of the uncertainties, several theories of uncertainty have been created, such as Pythagorean FS (PFS),5 intuition- istic FS (IFS),4 and FS.3 An FS is composed of MG sets, or sets with grades ranging from 0 to 1. Although Atanassov4 asserts that non-membership grades (NMG) can only be worth 1, IFS is classified as MG. There are times when the sum of MGs and NMGs throughout a decision-making process can approach 1. The gen- eralized MG and NMG logic, which is based on the square of the MGs and NMGs and has a value of no more than 1, was developed by Yager5 using PFS logic. Since the neutral state is neither positive nor negative, it cannot be described by these theories. Palanikumar and colleagues have introduced an intuitionistic fuzzy normal subbisemiring of bisemiring.6 The idea of bisemiring was created by Palanikumar et al.7 employing bipolar-valued neutrosophic normal sets. Bi-ideals on ordered semigroups were examined by Hila and asso- ciates.8 Dutta T.K. and associates presented novel concepts based on ternary semiring prime ideals and prime radicals.9 Several prime and semiprime bi-ideals of the rings were discussed by Palanikumar and associates11 and others. The several ideals of semigroups, semirings, and ternary semirings were examined by Palanikumar et al.12, 22–25 https://internationalpubls.com 761 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 2 (∝1,∝2) intuitionistic Q1 anti fuzzy ideals Here B denotes the ordered ternary semigroup and (∝1,∝2) ∈ [0, 1] be such that 0 6∝1<∝26 1 both (∝1,∝2) are arbitrary fixed. Definition 2.1. A IFS A = [tA,>A] of B and Q1 is a any non-empty set of A, the pair (A,Q1) Q1IFS is called a (∝1,∝2) IQ1TFSS of B if 1. If ] 6 ø, then t (]) 6t (ø) and >(]) > >(ø), 2. min{t (]∂ø, ι),∝1} 6 max{t (], ι),t (∂, ι),t (ø, ι),∝2} 3. max{>(]∂ø, ι),∝1} > min{>(], ι),>(∂, ι),>(ø, ι),∝2} for all ], ∂, ø ∈ B and q ∈ Q1. Example 2.2. Let B = {øl, øm, øn, øo} with the Cayley table: · øl øm øn øo øl t t t t øm t o p c øn t p p p øo t p p p · øl øm øn øo t øl øl øl øl o øl øm øn øo p øl øn øn øn c øl øn øn øn 6: = {(øl, øl), (øl, øm), (øl, øn), (øl, øo), (øm, øm), (øm, øn), (øm, øo), (øn, øn), (øo, øn), (øo, øo)}. Define A = [tA,>A] : B×B×B→ [0, 1]. t (ø, ι) =  0.26 if ø = øl 0.33 if ø = øm 0.43 if ø = øn 0.38 if ø = øo >(ø, ι) =  0.58 if ø = øl 0.38 if ø = øm 0.08 if ø = øn 0.18 if ø = øo Then A is a (0.48, 0.63) IQ1TFSS of B. Definition 2.3. A intuitionistic Q1 subset A of B is called a (∝1,∝2)-IQ1AFBI of B if 1. If ] 6 ø, then t (]) 6t (ø) and >(]) > >(ø), 2. min{t (]∂1ø, ι),∝1} 6 max{t (], ι),t (ø, ι),∝2}, max{>(]∂1ø, ι),∝1} > min{>(], ι),>(ø, ι),∝2}, 3. min{t (]∂1ø∂2ε, ι),∝1} 6 max{t (], ι),t (ε, ι),∝2}, max{>(]∂1ø∂2ε, ι),∝1} > min{>(], ι),>(ε, ι),∝2}, for ], ø, ε, ∂1, ∂2 ∈ B and q ∈ Q1. Example 2.4. Let B = {øl, øm, øn, øo} with Cayley table: · øl øm øn øo øl t t t t øm t o p c øn t p p p øo t p p p · øl øm øn øo t øl øl øl øl o øl øm øn øo p øl øn øn øn c øl øn øn øn · øl øm øn øo t t t t t o t o p d p t p p p c t o p c · øl øm øn øo t øl øl øl øl o øl øm øn øo p øl øn øn øn c øl øm øn øo https://internationalpubls.com 762 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 6: = {(øl, øl), (øl, øm), (øl, øn), (øl, øo), (øm, øm), (øm, øn), (øm, øo), (øn, øn), (øo, øn), (øo, øo)}. Define i = [t,>] : B×B×B→ [0, 1] t (ø, ι) =  0.32 if ø = øl 0.37 if ø = øm 0.47 if ø = øn 0.42 if ø = øo >(ø, ι) =  0.49 if ø = øl 0.30 if ø = øm 0.02 if ø = øn 0.11 if ø = øo Then i is a (0.35, 0.50) IQ1AFBI of B. Theorem 2.5. A non-empty subset i∝1 is a t∝1 is a (∝1,∝2)-IQ1TFSS (IQ1AFLI,IQ1AFLATI, IQ1AFRI, IQ1AFBI) of B. Then the lower level set t∝1 is an TSS (TLI,TLATI, TRI, TBI) of B, where t∝1= {] ∈ B| t (], ι) ≺∝1} and >∝1 = {] ∈ B|>(], ι) �∝1}. Proof. Suppose that i∝1 is a (∝1,∝2)-IQ1TFSS of B. Let ], ∂, ø ∈ B such that ], ∂, ø ∈t∝1 . Then t (], ι) ≺∝1,t (∂, ι) ≺∝1,t (ø, ι) ≺∝1. Therefore min{t (]∂ø, ι),∝1} 6 max{t (], ι),t (∂, ι),t (ø, ι),∝2} ≺ max{∝1,∝1,∝1,∝2} =∝2. Hence t (]∂ø, ι) ≺∝1. It shows that ]∂ø ∈t∝1 . Therefore t∝1 is a TSS of B. Let ], ∂, ø ∈ B such that ], ∂, ø ∈ >∝1 . Then >(], ι) �∝1,>(∂, ι) �∝1 >(ø, ι) �∝1. Therefore max{>(]∂ø, ι),∝1} > min{>(], ι),>(∂, ι),>(ø, ι),∝2} � min{∝1,∝1,∝1,∝2} =∝1. Hence >(]∂ø, ι) �∝1. It shows that ]∂ø ∈ >∝1 . Therefore >∝1 is a TSS of B. Therefore i∝1 is a TSS of B. Theorem 2.6. A non-empty subset ` of B is a SS [TLI, TLATI, TRI, TBI] of B if and only if the IQ1FS i = [t,>] of B is defined as t (], ι) = { 6∝2 for all ] ∈ (`] ∝1 for all ] /∈ (`] >(], ι) = { >∝2 for all ] ∈ (`] ∝1 for all ] /∈ (`] is a (∝1,∝2)IQ1TFSS[IQ1AFLI, IQ1AFLATI, IQ1AFRI, IQ1AFBI] of B. Proof. Suppose that ` is an TSS of B. Let ], ∂, ø ∈ B be such that ], ∂, ø ∈ (`] then ]∂ø ∈ (`]. Hence t (]∂ø, ι) 6∝2 and >(]∂ø, ι) >∝2. Thus, min{t (]∂ø, ι),∝1} 6∝2= max{t (], ι),t (∂, ι),t (ø, ι),∝2} and max{>(]∂ø, ι),∝1} >∝2= min{>(], ι),>(∂, ι),>(ø, ι),∝2}. If ] /∈ (`] or ∂ /∈ (`] or ø /∈ (`], then max{t (], ι),t (∂, ι),t (ø, ι),∝2} =∝1 and min{>(], ι),>(∂, ι),>(ø, ι),∝2} =∝2. That is min{t (]∂ø, ι),∝1} 6 max{t (], ι),t (∂, ι),t (ø, ι),∝2} and max{>(]∂ø, ι),∝1} > min{>(], ι),>(∂, ι),>(ø, ι),∝2}. Therefore i is a (∝1,∝2) IQ1TFSS of B. Conversely assume that i = [t,>] is a (∝1,∝2)-IQ1TFSS of B. Let ]∂ø ∈ (`]. Then t (], ι) 6∝2,t (∂, ι) 6∝2,t (ø, ι) 6∝2 and >(], ι) >∝2,>(∂, ι) >∝2,>(ø, ι) >∝2. Now i = [t,>] is a (∝1,∝2)- IQ1TFSS of B. Therefore min{t (]∂ø, ι),∝1} 6 max{t (], ι),t (∂, ι),t (ø, ι),∝2} 6 max{∝2,∝2,∝2 ,∝2} =∝2 and max{>(]∂ø, ι),∝1} > min{>(], ι),>(∂, ι),>(ø, ι),∝2} > min{∝2,∝2,∝2,∝2} =∝2 . It follows that ]∂ø ∈ (`] . Therefore ` is a TSS of B. Theorem 2.7. A subset i = [t,>] is a (∝1,∝2)−IQ1TFSS[IQ1AFLI, IQ1AFLATI, IQ1AFRI, IQ1AFBI] of B if and only if each non-empty level subset it is a TSS [TLI,TLATI,TRI,TBI] of B for all t ∈ (∝1,∝2] . Proof. Assume that it is a TSS of B for each t ∈ [0, 1]. Let t = max{t (]a, ι),t (]b, ι),t (]c, ι)}. Then ]a, ]b, ]c ∈tt for each ]a, ]b, ]c ∈ B. Thus min{t (]∂ø, ι),∝1} 6 t = max{t (]a, ι),t (]b, ι),t (]c, ι),∝2}. Let t = min{>(]a, ι),>(]b, ι),>(]c, ι)}. Then ]a, ]b, ]c ∈ >t for each ]a, ]b, ]c ∈ B. Thus max{>(]∂ø, ι),∝1} > t = min{>(]a, ι),>(]b, ι),>(]c, ι),∝2}. This shows that it is IQ1TFSS of B. Conversely, assume that it is a IQ1TFSS of B. For each t ∈ [0, 1] and ]a, ]b, ]c ∈tt. We have t (]a, ι) 6 t,t (]b, ι) 6 t,t (]c, ι) 6 t. Since t is a TSS of B, min{t (]a]b]c, ι),∝1} 6 max{t (]a, ι),t (]b, ι),t (]c, ι),∝2} 6 t. This implies that ]a]b]c ∈tt. We have >(]a, ι) > t,>(]b, ι) > t,>(]c, ι) > t. Since > is a TSS of B,max{>(]a]b]c, ι),∝1} > min{>(]a, ι),>(]b, ι),>(]c, ι),∝2} > t. This implies that ]a]b]c ∈ >t. Therefore it is a TSS of B for each t ∈ (∝1,∝2]. https://internationalpubls.com 763 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Example 2.8. Every IQ1TFSS i of B is a (∝1,∝2)-IQ1TFSS of B, but converse is not true. For the Example 2.2, we define subset i by t (ø, ι) =  0.19 if ø = øl 0.24 if ø = øm 0.34 if ø = øn 0.29 if ø = øo >(ø, ι) =  0.34 if ø = øl 0.27 if ø = øm 0.17 if ø = øn 0.22 if ø = øo Then i is a (0.25, 0.39)-IQ1TFSS of B, but not a IQ1TFSS. Since t (øo∂øo, ι) = 0.34 66 max{t (øo, q),t (øo, q)} = 0.29 and >(øo∂øo, ι) = 0.17 6> min{>(øo, q),>(øo, q)} = 0.22. Example 2.9. Every IQ1AFBI i = [t,>] of B is a (∝1,∝2)-IQ1AFBI of B, but converse need not be true. For the Example 2.4, we define subset i by, t (ø, ι) =  0.08 if ø = øl 0.23 if ø = øm 0.33 if ø = øn 0.28 if ø = øo >(ø, ι) =  0.42 if ø = øl 0.23 if ø = øm 0.01 if ø = øn 0.04 if ø = øo Then i is a (0.18, 0.43)IQ1AFBI, but not a IQ1AFBI. Since t (øo∂1øo∂2øo, ι) =t (øn, ι) = 0.33 66 max{t (øo, ι),t (øo, ι)} = 0.28 and >(øo∂1øo∂2øo, ι) = >(øn, ι) = 0.01 6> min{t (øo, ι),t (øo, ι)} = 0.04. Definition 2.10. If ` is the characteristic function of `, then (`)∝2 ∝1 is defined as (T ` )∝2 ∝1 (], ι) = { ∝2 if ] ∈ (`] ∝1 if ] /∈ (`] (F ` )∝2 ∝1 (], ι) = { ∝1 if ] ∈ (`] ∝2 if ] /∈ (`] Theorem 2.11. A non empty subset ` of B is a TSS [TLI, TLATI, TRI, TBI] of B if and only if subset  (`] is a (∝1,∝2)-IQ1TFSS[IQ1AFLI, IQ1AFLATI, IQ1AFRI, IQ1AFBI] of B. Proof. Assume that ` is a TSS of B. Then  (`] is a IQ1TFSS of B and hence  (`] is an (∝1,∝2)-IQ1TFSS of B. Conversely, Let  (`] is an (∝1,∝2)-IQ1TFSS of B. Let ], ∂, ø ∈ B be such that ], ∂, ø ∈ (`]. Then T (`] (], ι) =∝2,  T (`] (∂, ι) =∝2,  T (`] (ø, ι) =∝2. Since T (`] is a (∝1,∝2)IQ1TFSS. Consider min{T (`] (]∂ø, ι),∝1} 6 max{T (`] (], ι), T (`] (∂, ι), T (`] (ø, ι),∝2} = max{∝2,∝2,∝2,∝2} =∝2 as ∝1≺∝2, this implies that T (`] (]∂ø, ι) 6∝2. Thus ]∂ø ∈ (`]. Thus ]∂ø ∈ (`]. Let ], ∂, ø ∈ B be such that ], ∂, ø ∈ (`]. Then F (`] (], ι) =∝1,  F (`] (∂, ι) =∝1,  F (`] (ø, ι) =∝1. Since F (`] is a (∝1,∝2)IQ1TFSS. Consider max{F (`] (]∂ø, ι),∝1} > min{F (`] (], ι), F (`] (∂, ι), F (`] (ø, ι),∝2} = min{∝1,∝1,∝1,∝2} =∝1 as ∝1≺∝2, this implies that F (`] (]∂ø, ι) >∝1. Thus ]∂ø ∈ (`]. Thus ]∂ø ∈ (`]. Therefore ` is a TSS of B. https://internationalpubls.com 764 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Let ], ∂, ø ∈ B be such that ], ∂, ø /∈ (`]. Then T (`] (], ι) =∝1,  T (`] (∂, ι) =∝1,  T (`] (ø, ι) =∝1. Since T (`] is a (∝1,∝2)IQ1TFSS. min{T (`] (]∂ø, ι),∝1} 6 max{T (`] (], ι), T (`] (∂, ι), T (`] (ø, ι),∝2} = max{∝1,∝1,∝1,∝2} =∝2 as ∝1≺∝2, this implies that T (`] (]∂ø, ι) 6∝1. Thus ]∂ø 6∈ (`]. Let ], ∂, ø ∈ B be such that ], ∂, ø /∈ (`]. Then F (`] (], ι) =∝2,  F (`] (∂, ι) =∝2,  F (`] (ø, ι) =∝2. Since F (`] is a (∝1,∝2)IQ1TFSS. max{F (`] (]∂ø, ι),∝1} > min{F (`] (], ι), F (`] (∂, ι), F (`] (ø, ι),∝2} = min{∝2,∝2,∝2,∝2} =∝2 as ∝1≺∝2, this implies that F (`] (]∂ø, ι) >∝2. Thus ]∂ø 6∈ (`]. Therefore ` is a TSS of B. Definition 2.12. For three IQ1AFSs i, ∂ and ð of B. Their product i · ∂ · ð is defined as (iT · ∂T · ðT)(], ι) =  inf (r,s,t)∈`] {iT(r, ι)5 ∂T(s, ι)5 ðT(t, ι)} if `] 6= 0 1 otherwise (iF · ∂F · ðF)(], ι) =  sup (r,s,t)∈`] {iF(r, ι)4 ∂F(s, ι)4 ðF(t, ι)} if `] 6= 0 0 otherwise Definition 2.13. Let i be subset of B, we define the subset (t)∝2 ∝1 (], ι) = {t (], ι)5 ∝2}4 ∝1 and (>)∝2 ∝1 (], ι) = {>(], ι)4 ∝2}5 ∝1, for all ] ∈ B. Lemma 2.14. Let `, `1 and `2 be non-empty subsets of B. Then 1. ( (`] 5  (`1] 5  (`2] )∝2 ∝1 = ((`∪`1∪`2]) ∝2 ∝1 , 2. ( (`] 4  (`1] 4  (`2] )∝2 ∝1 = ((`∩`1∩`2]) ∝2 ∝1 , 3. ( (`] ·(`1]·(`2]) ∝2 ∝1 = ((``1`2]) ∝2 ∝1 . Proof. (iii) Let ] ∈ B. If ] ∈ (``1`2], then ((``1`2])(], ι) =∝2. Since ] 6 τ1τ2τ3 for some τ1 ∈ (`],τ2 ∈ (`1] and τ3 ∈ (`2]. We have (τ1, τ2, τ3) ∈ `] and `] 6= 0. (T (`] · T (`1] · T (`2] )(], ι) = inf ]=β1β2β3 max{T (`] (β1, ι),  T (`1] (β2, ι),  T (`2] (β3, ι)} 6 max{T (`] (τ1, ι),  T (`1] (τ2, ι),  T (`2] (τ3, ι)} =∝2 (F (`] · F (`1] · F (`2] )(], ι) = sup ]=β1β2β3 min{F (`] (β1, ι),  F (`1] (β2, ι),  F (`2] (β3, ι)} > min{F (`] (τ1, ι),  F (`1] (τ2, ι),  F (`2] (τ3, ι)} =∝1 https://internationalpubls.com 765 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) Therefore ( (`] ·  (`1] ·  (`2] )(], ι) = ((``1`2])(], ι). If ] /∈ (``1`2] then (T(``1`2])(], ι) =∝1 and (F(``1`2])(], ι) =∝2. Since ] 6 τ1τ2τ3 for some τ1 /∈ (`], τ2 /∈ (`1] and τ3 /∈ (`2]. We have (T (`] · T (`1] · T (`2] )(], ι) = inf ]=β1β2β3 max{T (`] (β1, ι),  T (`1] (β2, ι),  T (`2] (β3, ι)} 6 max{T (`] (τ1, ι),  T (`1] (τ2, ι),  T (`2] (τ3, ι)} =∝1 (F (`] · F (`1] · F (`2] )(], ι) = sup ]=β1β2β3 min{F (`] (β1, ι),  F (`1] (β2, ι),  F (`2] (β3, ι)} > min{F (`] (τ1, ι),  F (`1] (τ2, ι),  F (`2] (τ3, ι)} =∝2 Hence ( (`] ·  (`1] ·  (`2] )(], ι) = ((``1`2])(], ι). Theorem 2.15. For `,`2 ⊆ B and {`j |j ∈ J} be a family of subsets of B. Then (i) (`] ⊆ (`1] if and only if ((`]) ∝2 ∝1 6 ((`1]) ∝2 ∝1 . (ii) (∩j∈J (`j ]) ∝2 ∝1 = (∩j∈J (`j ]) ∝2 ∝1 . (iii) (∪j∈J (`j ]) ∝2 ∝1 = (∪j∈J (`j ]) ∝2 ∝1 . Theorem 2.16. Let ` be an (∝1,∝2)IQ1AFRI, `1 be an (∝1,∝2)IQ1AFLATI and `2 be an (∝1,∝2)IQ1AFLI of B then ((` · `1 · `2])∝2 ∝1 ⊆ (` ∩ `1 ∩ `2]∝2 ∝1 . Proof. Let ` = [t`,>`] be an (∝1,∝2)IQ1AFRI, `1 = [t`1 ,>`1 ] be an (∝1,∝2)IQ1AFLATI and `2 = [t`2 ,>`2 ] be an (∝1,∝2)IQ1AFLI of B. Let (], ∂, ø) ∈ Xε. If Xε 6= ∅, then ε 6 ]∂ø. Thus t` (ε, ι) 6t` (]∂ø, ι) 6t` (], ι) and >`(ε, ι) > >`(]∂ø, ι) > >`(], ι). Similarly t`1 (ε, ι) 6t`1 (]∂ø, ι) 6t`1 (∂, ι) and >`1(ε, ι) > >`1(]∂ø, ι) > >`1(∂, ι). Similarly, t`2 (ε, ι) 6t`2 (]∂ø, ι) 6t`2 (ø, ι) and >`2 (ε, ι) > >`2 (]∂ø, ι) > >`2 (ø, ι). We have (t(`·`1·`2]) ∝2 ∝1 (ε, ι) = (t(`·`1·`2] (ε, ι)5 ∝2)4 ∝1 = [ [ inf ε6]∂ø {t` (], ι)5 t`1 (∂, ι)5 t`2 (ø, ι)}5 ∝2] ] 4 ∝1 = [ inf ε6]∂ø {t` (], ι)5 t`1 (∂, ι)5 t`2 (ø, ι)}5 ∝2 5 ∝2 5 ∝2 5 ∝2 ] 4 ∝1 = [ inf ε6]∂ø {(t` (], ι)5 ∝2)5 (t`1 (∂, ι)5 ∝2)5 (t`2 (ø, ι)5 ∝2)}5 ∝2 ] 4 ∝1 > ({(t` (ε, ι)4 ∝1)5 (t`1 (ε, ι)4 ∝1)5 (t`2 (ε, ι)4 ∝1)}5 ∝2)4 ∝1 = {((t` (ε, ι)5 t`1 (ε, ι)5 t`2 (ε, ι))4 ∝1)5 ∝2}4 ∝1 = {((t` 5 t`1 5 t`2)(ε, ι)5 ∝2}4 ∝1 = (t`∩`1∩`2 )∝2 ∝1 (ε, ι) https://internationalpubls.com 766 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) (>(`·`1·`2]) ∝2 ∝1 (ε, ι) = (>(`·`1·`2](ε, ι)4 ∝2)5 ∝1 = [ [ sup ε6]∂ø {>`(], ι)4>`1 (∂, ι)4>`2 (ø, ι)}4 ∝2] ] 5 ∝1 = [ sup ε6]∂ø {>`(], ι)4>`1 (∂, ι)4>`2 (ø, ι)}4 ∝2 4 ∝2 4 ∝2 4 ∝2 ] 5 ∝1 = [ sup ε6]∂ø {(>`(], ι)4 ∝2)4 (>`1 (∂, ι)4 ∝2)4 (>`2 (ø, ι)4 ∝2)}4 ∝2 ] 5 ∝1 6 ({(>`(ε, ι)5 ∝1)4 (>`1 (ε, ι)5 ∝1)4 (>`2 (ε, ι)5 ∝1)}4 ∝2)5 ∝1 = {((>`(ε, ι)4>`1 (ε, ι)4>`2 (ε, ι))5 ∝1)4 ∝2}5 ∝1 = {((>` 4>`1 4>`2 )(ε, ι)4 ∝2}5 ∝1 = (>`∪`1∪`2 )∝2 ∝1 (ε, ι) Let ], ∂, ø /∈ Xε. If Xε = ∅, then (t` ·`1· t`2 )(ε, ι) = 1 and (>` · `1 · >`2 )(ε, ι) = 0 such that ε 6 ]∂ø. (t(`·`1·`2]) ∝2 ∝1 (ε, ι) = (t(`·`1·`2] (ε, ι)5 ∝2)4 ∝1 = 14 ∝1 > (t`∩`1∩`2 (ε, ι)5 ∝2)4 ∝1 = (t`∩`1∩`2 (ε, ι)5 ∝2) (>(`·`1·`2]) ∝2 ∝1 (ε, ι) = (>(`·`1·`2](ε, ι)4 ∝2)5 ∝1 = 05 ∝1 =∝1 6 (>`∪`1∪`2(ε, ι)4 ∝2)5 ∝1 = (>`∪`1∪`2(ε, ι)4 ∝2) Therefore ((` · `1·`2])∝2 ∝1 ⊆ ((` ∩ `1 ∩ `2])∝2 ∝1 . Theorem 2.17. An ordered -semigroup B is regular, ` be an (∝1,∝2)IQ1AFRI, `1 be an (∝1,∝2)IQ1AFLATI and `2 be an (∝1,∝2)IQ1AFLI of B if and only if ((` · `1 · `2])∝2 ∝1 = ((` ∩ `1 ∩ `2])∝2 ∝1 . Proof. Let B be an ordered -regular ternary semigroup and ` be an (∝1,∝2)IQ1AFRI, `1 be an (∝1,∝2 )IQ1AFLATI and `2 be an (∝1,∝2)IQ1AFLI of B. Let (], ø) ∈ Xε. If Xε 6= ∅, then ε 6 ]∂ø. Thus t` (ε, ι) 6t` (]∂ø, ι) 6t` (], ι) and >`(ε, ι) > >`(]∂ø, ι) > >`(], ι). Similarly t`1 (ε, ι) 6t`1 (]∂ø, ι) 6t`1 (∂, ι) and >`1 (ε, ι) > >`1 (]∂ø, ι) > >`1 (∂, ι). Similarly, t`2 (ε, ι) 6t`2 (]∂ø, ι) 6t`2 (ø, ι) and >`2 (ε, ι) > >`2 (]∂ø, ι) > >`2 (ø, ι). For ε ∈ B, there exists x ∈ B such that ε 6 εζ1εζ2εζ3ε. Then ε, (ζ1εζ2εζ3), ε ∈ Xε. We have (t(`·`1·`2]) ∝2 ∝1 (ε, ι) = (t(`·`1·`2] (ε, ι)5 ∝2)4 ∝1 = [ [ inf ε6εζ1εζ2εζ3ε {t` (], ι)5 t`1 (∂, ι)5 t`2 (ø, ι)}5 ∝2] ] 4 ∝1 = [ inf ε6εζ1εζ2εζ3ε {t` (], ι)5 t`1 (∂, ι)5 t`2 (ø, ι)}5 ∝2 5 ∝2 5 ∝2 5 ∝2 ] 4 ∝1 = [ inf ε6εζ1εζ2εζ3ε {(t` (], ι)5 ∝2)5 (t`1 (∂, ι)5 ∝2)5 (t`2 (ø, ι)5 ∝2)}5 ∝2 ] 4 ∝1 6 ({(t` (ε, ι)4 ∝1)5 (t`1 (ζ1εζ2εζ3)4 ∝1)5 (t`2 (ε, ι)4 ∝1)}5 ∝2)4 ∝1 6 ({(t` (ε, ι)4 ∝1)5 (t`1 (ε, ι)4 ∝1)5 (t`2 (ε, ι)4 ∝1)}5 ∝2)4 ∝1 = {((t` (ε, ι)5 t`1 (ε, ι)5 t`2 (ε, ι))4 ∝1)5 ∝2}4 ∝1 = {((t` 5 t`1 5 t`2 )(ε, ι)5 ∝2}4 ∝1 = (t`∩`1∩`2)∝2 ∝1 (ε, ι) https://internationalpubls.com 767 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) (>(`·`1·`2]) ∝2 ∝1 (ε, ι) = (>(`·`1·`2](ε, ι)4 ∝2)5 ∝1 = [ [ sup ε6εζ1εζ2εζ3ε {>`(], ι)4>`1 (∂, ι)4>`2 (ø, ι)}4 ∝2] ] 5 ∝1 = [ sup ε6εζ1εζ2εζ3ε {>`(], ι)4>`1 (∂, ι)4>`2 (ø, ι)}4 ∝2 4 ∝2 4 ∝2 4 ∝2 ] 5 ∝1 = [ sup ε6εζ1εζ2εζ3ε {(>`(], ι)4 ∝2)4 (>`1 (∂, ι)4 ∝2)4 (>`2 (ø, ι)4 ∝2)}4 ∝2 ] 5 ∝1 > ({(>`(ε, ι)5 ∝1)4 (>`1 (ζ1εζ2εζ3)5 ∝1)4 (>`2 (ε, ι)5 ∝1)}4 ∝2)5 ∝1 > ({(>`(ε, ι)5 ∝1)4 (>`1 (ε, ι)5 ∝1)4 (>`2 (ε, ι)5 ∝1)}4 ∝2)5 ∝1 = {((>`(ε, ι)4>`1 (ε, ι)4>`2 (ε, ι))5 ∝1)4 ∝2}5 ∝1 = {((>` 4>`1 4>`2 )(ε, ι)4 ∝2}5 ∝1 = (>`∪`1∪`2)∝2 ∝1 (ε, ι) Thus ((` · `1 · `2])∝2 ∝1 ⊇ ((` ∩ `1 ∩ `2])∝2 ∝1 and by Theorem 2.16. Hence ((` · `1 · `2])∝2 ∝1 = ((` ∩ `1 ∩ `2])∝2 ∝1 . Conversely assume that ((` · `1 · `2])∝2 ∝1 = ((` ∩ `1 ∩ `2])∝2 ∝1 . Let ` = (t`,>`) be an (∝1,∝2)IQ1AFRI, `1 = (t`1 ,>`1 ) be an (∝1,∝2)IQ1AFLATI and `2 = (t`2 ,Ξ`2 ,>`2 ) be an (∝1,∝2)IQ1AFLI of B. Then by Theorem 2.11, ` is a (∝1,∝2)IQ1AFRI, `1 is a (∝1,∝2)IQ1AFLATI and `2 be a (∝1,∝2)IQ1AFLI of B. By Lemma 2.14 and Theorem 2.15, ((`∩`1∩`2]) ∝2 ∝1 = (`∩`1 ∩`2 )∝2 ∝1 = (` ·`1 ·`2 )∝2 ∝1 = ((`·`1·`2]) ∝2 ∝1 . This implies (` ∩ `1 ∩ `2]∝2 ∝1 = ((` · `1 · `2])∝2 ∝1 . Hence by Corollary ??, B is regular. Theorem 2.18. An TSS B is regular, ` be an (∝1,∝2)IQ1AFBI, `1 be an (∝1,∝2)IQ1AFLATI and `2 be an (∝1,∝2)IQ1AFLI of B if and only if ((` · `1 · `2])∝2 ∝1 = ((` ∩ `1 ∩ `2])∝2 ∝1 . Proof. Let B be an TSS and ` be an (∝1,∝2)IQ1AFBI and `2 be an (∝1,∝2)IQ1AFLI of B. Let (], ø) ∈ Xε. If Xε 6= ∅, then ε 6 ]∂ø. Thus t` (ε, ι) 6t` (]∂ø, ι) 6t` (], ι) and >`(ε, ι) > >`(]∂ø, ι) > >`(], ι). Similarly t`1 (ε, ι) 6t`1 (]∂ø, ι) 6t`1 (∂, ι) and >`1(ε, ι) > >`1(]∂ø, ι) > >`1(∂, ι). Similarly, t`2 (ε, ι) 6t`2 (]∂ø, ι) 6t`2 (ø, ι) and >`2 (ε, ι) > >`2 (]∂ø, ι) > >`2 (ø, ι). For ε ∈ B, there exists x ∈ B such that ε 6 εζ1εζ2εζ3εζ4εζ5ε. Then ε 6 (εζ1εζ2ε), (ζ3εζ4εζ5), ε ∈ Xε. We have (t(`·`1·`2]) ∝2 ∝1 (ε, ι) = (t(`·`1·`2] (ε, ι)5 ∝2)4 ∝1 = [ [ inf ε6εζ1εζ2εζ3εζ4εζ5ε {t` (], ι)5 t`1 (∂, ι)5 t`2 (ø, ι)}5 ∝2] ] 4 ∝1 = [ inf ε6εζ1εζ2εζ3εζ4εζ5ε {t` (], ι)5 t`1 (∂, ι)5 t`2 (ø, ι)}5 ∝2 5 ∝2 5 ∝2 5 ∝2 ] 4 ∝1 = [ inf ε6εζ1εζ2εζ3εζ4εζ5ε {(t` (], ι)5 ∝2)5 (t`1 (∂, ι)5 ∝2)5 (t`2 (ø, ι)5 ∝2)}5 ∝2 ] 4 ∝1 6 ({(t` (εζ1εζ2ε, ι)4 ∝1)5 (t`1 (ζ3εζ4εζ5)4 ∝1)5 (t`2 (ε, ι)4 ∝1)}5 ∝2)4 ∝1 6 ({(t` (ε, ι)4 ∝1)5 (t`1 (ε, ι)4 ∝1)5 (t`2 (ε, ι)4 ∝1)}5 ∝2)4 ∝1 = {((t` (ε, ι)5 t`1 (ε, ι)5 t`2 (ε, ι))4 ∝1)5 ∝2}4 ∝1 = {((t` 5 t`1 5 t`2 )(ε, ι)5 ∝2}4 ∝1 = (t`∩`1∩`2)∝2 ∝1 (ε, ι) https://internationalpubls.com 768 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) (>(`·`1·`2]) ∝2 ∝1 (ε, ι) = (>(`·`1·`2](ε, ι)4 ∝2)5 ∝1 = [ [ sup ε6εζ1εζ2εζ3εζ4εζ5ε {>`(], ι)4>`1 (∂, ι)4>`2 (ø, ι)}4 ∝2] ] 5 ∝1 = [ sup ε6εζ1εζ2εζ3εζ4εζ5ε {>`(], ι)4>`1 (∂, ι)4>`2 (ø, ι)}4 ∝2 4 ∝2 4 ∝2 4 ∝2 ] 5 ∝1 = [ sup ε6εζ1εζ2εζ3εζ4εζ5ε {(>`(], ι)4 ∝2)4 (>`1 (∂, ι)4 ∝2)4 (>`2 (ø, ι)4 ∝2)}4 ∝2 ] 5 ∝1 > ({(>`(εζ1εζ2ε, ι)5 ∝1)4 (>`1 (ζ3εζ4εζ5)5 ∝1)4 (>`2 (ε, ι)5 ∝1)}4 ∝2)5 ∝1 > ({(>`(ε, ι)5 ∝1)4 (>`1 (ε, ι)5 ∝1)4 (>`2 (ε, ι)5 ∝1)}4 ∝2)5 ∝1 = {((>`(ε, ι)4>`1 (ε, ι)4>`2 (ε, ι))5 ∝1)4 ∝2}5 ∝1 = {((>` 4>`1 4>`2 )(ε, ι)4 ∝2}5 ∝1 = (>`∪`1∪`2)∝2 ∝1 (ε, ι) Thus, ((`·`1 ·`2])∝2 ∝1 ⊇ ((`∩`1∩`2])∝2 ∝1 and by Theorem 2.16 and hence ((`·`1 ·`2])∝2 ∝1 = ((`∩`1∩`2])∝2 ∝1 . Conversely assume that ((` · `1 · `2])∝2 ∝1 = ((` ∩ `1 ∩ `2])∝2 ∝1 . Let ` = (t`,>`) be an (∝1,∝2)IQ1AFBI, `1 = (t`1 ,Ξ`1 ,>`1) be an (∝1,∝2)IQ1AFLATI and `2 = (t`2 ,>`2) be an (∝1,∝2)IQ1AFLI of B. Then by Theorem 2.11, ` is a (∝1,∝2)IQ1AFBI, `1 is a (∝1,∝2)IQ1AFLATI and `2 be a (∝1,∝2)IQ1AFLI of B. By Lemma 2.14 and Theorem 2.15, ((`∩`1∩`2]) ∝2 ∝1 = (`∩`1 ∩`2 )∝2 ∝1 = (` ·`1 ·`2 )∝2 ∝1 = ((`·`1·`2]) ∝2 ∝1 . This implies (` ∩ `1 ∩ `2]∝2 ∝1 = ((` · `1 · `2])∝2 ∝1 . Hence by Corollary ??, B is regular. Acknowledgment. This research was supported by University of Phayao and Thailand Science Research and Innovation Fund (Fundamental Fund 2025, Grant No. 5027/2567). Conflicts of Interest The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] Lehmer D. H., A ternary analogue of abelian groups. American Journal of Mathematics, (1932), 329-338. [2] Hestenes M.R. A ternary algebra with applications to matrices and linear transformations. Arch. Ration. Mech. Anal. 11(1962), 138 -194. [3] L. A. Zadeh, Fuzzy sets, Information and Control, 8, (1965), 338-353. [4] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1), (1986) 87-96. [5] R. R. Yager, Pythagorean membership grades in multi criteria decision-making, IEEE. Trans. Fuzzy Systems, 22, (2014), 958-965. [6] Palanikumar M, Arulmozhi K, On intuitionistic fuzzy normal subbisemirings of bisemirings, Nonlinear studies, 28(3), 2021, 717-721. [7] Palanikumar M, Selvi G, Ganeshsree Selvachandran and Tan S.L, New approach to bisemiring theory via the bipolar-valued neutrosophic normal sets, Neutrosophic Sets and Systems, 55, 427-450, 2023. [8] K. Hila and E. Pisha. On bi-ideals on ordered Γ-semigroups. Hacettepe Journal of Mathematics and Statistics, 40(6), (2011), 793-804. [9] Dutta T.K and Kar S, On Prime ideals and Prime radical of ternary Semirings, Bull.Cal. Math. Soc., 97(5), 2005, 445-454. https://internationalpubls.com 769 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) [10] Palanikumar, M.; Jana, C.; Shanqiti, O.A.; Pal. M. A novel method for generating the M-tri-basis of an ordered Gamma semigroup. Mathematics 2023, 11, 893 [11] Mohanraj, G; Palanikumar, M. On various prime and semiprime bi-ideals of Rings. Nonlinear studies. 2021, 27(3), 811-815. [12] Palanikumar, M; Arulmozhi, K. Jana.C and Pal.M & Shum.K.P. New approach towards different bi-base of ordered b-semiring.Asian-European Journal of Mathematics. 2023, 16(2), 1-26. [13] Shihadeh, A., Matarneh, K. A. M., Hatamleh, R., Al-Qadri, M. O., & Al-Husban, A, On The Two-Fold Fuzzy n-Refined Neutrosophic Rings For 2 ≤ 3. Neutrosophic Sets and Systems, 68, (2024), 8-25. [14] Abdallah Shihadeh, Khaled Ahmad Mohammad Matarneh, Raed Hatamleh, Randa Bashir Yousef Hi- jazeen, Mowafaq Omar Al-Qadri, Abdallah Al-Husban, An Example of Two-Fold Fuzzy Algebras Based On Neutrosophic Real Numbers, Neutrosophic Sets and Systems, 67, (2024), 169-178. [15] A. Rajalakshmi, Raed Hatamleh, Abdallah Al-Husban, K. Lenin Muthu Kumaran, M. S. Malchijah raj, Various (ζ1, ζ2) neutrosophic ideals of an ordered ternary semigroups. Communications on Applied Non- linear Analysis, 32 (3), (2025), 400-417. [16] Raed Hatamleh, Abdallah Al-Husban, N. Sundarakannan, M. S. Malchijah Raj, Complex cubic intu- itionistic fuzzy set applied to subbisemirings of bisemirings using homomorphism. Communications on Applied Non-linear Analysis, 32 (3), (2025), 418-435. [17] Abubaker, Ahmad A, Hatamleh, Raed, Matarneh, Khaled, Al-Husban, Abdallah, On the Numerica Solu- tions for Some Neutrosophic Singular Boundary Value Problems by Using (LPM) Polynomials, Interna- tional Journal of Neutrosophic Science, 25(2), (2024), 197-205. [18] A., Ahmad. , Hatamleh, Raed. , Matarneh, Khaled. , Al-Husban, Abdallah. On the Irreversible k- Threshold Conversion Number for Some Graph Products and Neutrosophic Graphs, International Journal of Neutrosophic Science, 25(2), (2025), 183-196. [19] Raed Hatamleh, Abdallah Al-Husban, K. Sundareswari, G.Balaj, M.Palanikumar, Complex Tangent Trigonometric Approach Applied to (α, β)-Rung Fuzzy Set using Weighted Averaging, Geometric Op- erators and its Extension. Communications on Applied Nonlinear Analysis, 32 (5), (2025), 133-144. [20] Raed Hatamleh, Abdallah Al-Husban, M. Palanikumar, K. Sundareswari, Different Weighted Opera- tors such as Generalized Averaging and Generalized Geometric based on Trigonometric q-rung Interval- Valued Approach, Communications on Applied Nonlinear Analysis, 32 (5), (2025), 91-101. [21] Abdallah Shihadeh, Raed Hatamleh, M.Palanikumar, Abdallah Al-Husban, New algebraic structures towards different ([, `) intuitionistic fuzzy ideals and it characterization of an ordered ternary semigroups. Communications on Applied Nonlinear Analysis, 32 (6), (2025), 568-578. [22] Palanikumar, M; Iampan, A; Manavalan, L.J. M-bi-base generator of ordered Γ-semigroups. ICIC Ex- press Letters Part B: Applications. 2022, 13(8), 795-802. [23] Mohanraj, G; Palanikumar, M. Characterization of various k-regular in b-semirings, AIP Conference Proceedings, 2019, 2112 (1), 020021. [24] Palanikumar, M; Shanqiti, O. Al; Jana, C; Pal, M. Novelty for different prime partial bi-ideals in non- commutative partial rings and its extension Mathematics, 2023, 11(6), 1309. [25] Palanikumar, M; Mohanraj, G; Iampan, A. Characterization of Different Prime Bi-Ideals and Its Gener- alization of Semirings, International Journal of Analysis and Applications, 2024, 22, 112–112. [26] Hatamleh, R., Zolotarev,V. A. (2015). On Model Representations of Non-Selfadjoint Operators with Infinitely Dimensional Imaginary Component, Journal of Mathematical Physics, Analysis, Geometry, 11(2), 174-186. [27] Hatamleh, R., Zolotarev, V. A. (2016). Triangular Models of Commutative Systems of Linear Operators Close to Unitary Operators, Ukrainian Mathematical Journal, 68(5),791-811. [28] Hatamleh, R., Zolotarev, V. A. (2014). On Two-Dimensional Model Representations of One Class of Commuting Operators, Ukrainian Mathematical Journal, 66(1), 122-144. https://internationalpubls.com 770 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) [29] R.Hatamleh,V.A. Zolotarev.(2017).On the Abstract Inverse Scattering Problem for Traces Class Pertuba- tions. Journal of Mathematical Physics, Analysis, Geometry, 13(1), (2017), 1-32. [30] Raed Hatamleh, On a Novel Topological Space Based on Partially Ordered Ring of Weak Fuzzy Complex Numbers and its Relation with the Partially ordered Neutrosophic Ring of Real Numbers, Neutrosophic Sets and Systems, 78, (2025), 578-590. [31] Abdallah Al-Husban & Abdul Razak Salleh, Complex fuzzy ring. Proceedings of 2nd International Con- ference on Computing, Mathematics and Statistics, IEEE, 2015, 241-245. [32] Abdallah Al-Husban & Abdul Razak Salleh, Complex Fuzzy Hyperring Based on Complex Fuzzy Spaces. Proceedings of 2nd Innovation and Analytics Conference & Exhibition (IACE). Vol. 1691. AIP Publishing 2015, 040009-040017. [33] Al-Husban, A., & Salleh, A. R. Complex fuzzy hyper groups based on complex fuzzy spaces. Interna- tional Journal of Pure and Applied Mathematics, 107(4), (2016), 949-958. [34] Alsarahead, M. O., & Al-Husban, A, Complex multi-fuzzy subgroups. Journal of discrete mathematical sciences and cryptography, 25(8), (2022), 2707-2716. [35] Al-Husban, A, Multi-fuzzy hyper groups. Italian Journal of Pure and Applied Mathematics, 46, (2021), 382-390. [36] Al-Husban, A, Fuzzy Soft Groups based on Fuzzy Space, Wseas Transactions on Mathematics. 21, (2021), 53-57. [37] Al-Husban,. Abdallah, Al-Sharoa, Doaa., Al-Kaseasbeh, Mohammad., & Mahmood, R.M.S, Structures of fibers of groups actions on graphs. Wseas Transactions on Mathematics, 7, (2022), 650-658. [38] Hatamleh, R., Heilat, A. S., Palanikumar, M., & Al-Husban, A. Different operators via weighted av- eraging and geometric approach using trigonometric neutrosophic interval-valued set and its extension, Neutrosophic Sets and Systems, 80, 2025, 194-213. https://internationalpubls.com 771 1 Introduction 2 (1, 2) intuitionistic Q1 anti fuzzy ideals