˜ , ˜ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) Generalization of intuitionistic fuzzy ideals via (∂̃ , ℘̃) intuitionistic Q interval-valued fuzzy ideals using regular ordered ternary semigroups Ayman Hazaymeh1, Abdallah Al-Husban2,3, M.Palanikumar4 1Department of Mathematics, Faculty of Science, Jadara University, P.O. Box 733, Irbid 21110, Jordan. 2Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600 Irbid, Jordan. 3Jadara Research Center, Jadara University, Irbid 21110, Jordan. 4Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai-602105, India. E-mails:1aymanha@ jadara.edu.jo, 3dralhosban@inu.edu.jo, 4palanimaths86@gmail.com, ∗Corresponding author: M.Palanikumar. Received: 25-10-2024; Revised: 02-12-2024; Accepted: 18-12-2024 Abstract This paper introduces the notion of ∂̃ , ℘̃ intuitionistic Q interval-valued fuzzy subsemigroup (IQVFSS), fuzzy left ideal (IQVFLI), fuzzy right ideal (IQVFRI), fuzzy lateral ideal (IQVFLATI), fuzzy ideal (IQVFI), and fuzzy bi-ideal (IQVFBI) of an ordered semigroups. Let ∂̃ , ℘̃-IQVFI is a new extension of IQVFI over ternary semigroups Z . The subset ~ = [< =] represents a (∂̃ , ℘̃) − IQV F SS[IQV F LI, IQV FRI, IQV F LAT I, IQV FBI] of Z if and only if every level subset ~t is an SS [LI, RI, LAIQV F, T BI] of Z for every t ∈ (∂̃ , ℘̃]. A few examples can be presented to demonstrate our results. Keywords: IQVFSS, IQVFLI, IQVFRI, IQVFLATI, IQVFBI. 1 Introduction D. H. Lehmer initially introduced triplexes, which are ternary algebraic systems, in 1932.1 Triplexes, ternary algebraic systems that prove to be commutative ternary groups, are the subject of his investigation. The concept of a semiring was initially put out by Vandiver in 1934. In 1962, Hestenes2 used the idea of ternary algebra to matrices and linear transformation. The fuzzy set (FS) theory, first presented by Zadeh,3 is the most effective approach to dealing with ambiguity and uncertainty. If an element in an FS has a single value inside the interval, it is regarded as a member degree (MD). However, the degree of non-membership degree (NMD) could not always be equal to one minus the MD, uncertain theories, such as FS,3 intuitionistic FS (IFS),4 Pythagorean FS (PFS),5 and spherical FS (SFS).6 An FS is made up of sets of various grades, such as MG, which range from 0 to 1. MG is the classification for IFS regardless of the assertion made by Atanassov4 that NMG can only be worth 1. Using PFS logic, Yager5 built the generalized MG and NMG, which has a maximum value of 1 and is based on the square of the MGs and NMGs. The neutral condition, which is neither positive nor negative, cannot be adequately described by these concepts. The practical applications of FS extensions were discussed by Al-Husband et al.7-.10 He investigated their characteristics in a manner similar to that of set theory. Rosenfeld11 created fuzzy subgroups and listed some of their characteristics in 1971. Fuzzy semigroups were first presented by Kuroki12 as an expansion of classical semigroups. Some fuzzy semigroup characterisation was developed by Mordeson.13 Sen et al. supplied the i-semigroups’ characteristics.14, 15 Kehayopula looked at the ordered i-semigroup.16 Somsak Lekkoksung used ordered semigroups to investigate Q-fuzzy ideals.1718 Kehayopula et al. started the research on fuzzy ordered semigroups. Initial proposals for the (∂̃ , ℘̃) fuzzy bi-ideal and FSS were made by Muhammad Khan et al.19 Numerous scholars have recently examined the idea of IFS, NSS, and its characterisation20-.24 An IFS with normal subbisemiring was introduced by Palanikumar et al.25 Hila et al.26 explored bi-ideals on ordered semigroups. Dutta T.K. et al. introduced novel concepts using prime ideals of ternary semirings.27 A number of prime bi- ideals of the rings have been studied by Palanikumar et al.28, 29 The several ideals of different algebraic https://internationalpubls.com 579 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) approach were examined by Palanikumar et al.30–34 The notion of various operators, including averaging, geometric, and its generalized forms, was explored by new researchers Hatamleh et al.35-,41 Bataihah,42 and Hazaymeh.43 We study ordered ternary semigroups based on (∂̃, ℘̃) ternary IFS and provide examples to show their properties. 2 Basic concepts Definition 2 .1. Let i and i 1 be subsets of Z . Then 1. (i] = {t ∈ Z | t 6 h for someh ∈ i}, 2. ii1 = {ab : a ∈ i, b ∈ i1}, 3. ia = {(b, c) ∈ Z ×Z | a 6 bc}. Definition 2.2. A fuzzy subset ð2 of an ordered semigroup Z is called a FRI(FLI) of Z if 1. a 6 b⇒ ð2(a) > ð2(b) for all a, b ∈ Z , 2. ð2(ab) > ð2(a) (resp. ð2(ab) > ð2(b)) for all a, b ∈ Z , Definition 2.3. If (Z ,+) is a commutative semigroup and ternary multiplication meets the following condi- tions, then 1. (fgh)ij = f(ghi)j = fg(hij), 2. (f + g)hi = fhi+ ghi, 3. f(g + h)i = fgi+ fhi, 4. fg(h+ i) = fgh+ fgi for all f, g, h, i, j ∈ Z Definition 2.4. The subset K of Z is called a 1. SS if υ1υ2υ3 ∈ K for all υ1, υ2, υ3 ∈ K. 2. right (lateral, left) ideal if is1s2 ∈ K(s1is2 ∈ K, s1s2i ∈ K) for all s1, s2 ∈ Z and i ∈ K. Corollary 2.5. If Z is regular if and only if RI i, LATIF i1 and LI i2 of Z , then (ifi1fi2] = (i∗i1∗i2]. 3 (∂̃, ℘̃) ternary intuitionistic Q interval-valued fuzzy ideals Here, Z represents an ordered ternary semigroup. Assuming (∂̃, ℘̃) ∈ [0, 1] and 0 6 ∂̃ ≺ ℘̃ 6 1, both (∂̃, ℘̃) are arbitrary fixed points. Definition 3.1. An IVFS N and Q be any set, then the pair N ×Q is called an IQVFS. Let N = [<̃N , =̃N ] of Z is called a (∂̃, ℘̃) IQVFSS of Z if 1. ð1 6 ð3 ⇒ <̃(ð1) > <̃(ð3), 2. max{<̃(ð1ð2ð3, ǎ), ∂̃} > min{<̃(ð1, ǎ), <̃(ð2, ǎ), <̃(ð3, ǎ), ℘̃}, 3. min{=̃(ð1ð2ð3, ǎ), ∂̃} 6 max{=̃(ð1, ǎ), =̃(ð2, ǎ), =̃(ð3, ǎ), ℘̃} for all ð1,ð2,ð3 ∈ Z and ǎ ∈ Q. Example 3.2. Let Z = {a, b, c, d} with the following Cayley table: https://internationalpubls.com 580 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) ∗ a b c d a l l l l b l m n o c l n n n d l n n n ∗ a b c d l a a a a m a b c d n a c c c o a c c c 6: = {(a, a), (a, b), (a, c), (a, d), (b, b), (b, c), (b, d), (c, c), (d, c), (d, d)}. Define the mapping N = [<̃N , =̃N ] : Z ×Z ×Z → [0, 1]. <̃(κ, ǎ) =  [0.6, 0.65] if κ = a [0.4, 0.45] if κ = b [0.1, 0.15] if κ = c [0.2, 0.25] if κ = d =̃(κ, ǎ) =  [0.3, 0.35] if κ = a [0.35, 0.4] if κ = b [0.45, 0.5] if κ = c [0.4, 0.45] if κ = d Then N is a ([0.5, 0.55], [0.65, 0.7]) IQV FSS of Z . Definition 3.3. A IQVFS N of Z is called a (∂̃, ℘̃)-IQVFBI of Z if 1. If ð1 6 ð3, then <̃(ð1) > <̃(ð3) and =̃(ð1) 6 =̃(ð3), 2. max{<̃(ð1ð2ð3, ǎ), ∂̃} > min{<̃(ð1, ǎ), <̃(ð3, ǎ), ℘̃}, min{=̃(ð1ð2ð3, ǎ), ∂̃} 6 max{=̃(ð1, ǎ), =̃(ð3, ǎ), ℘̃}, 3. max{<̃(ð1ð2ð3ð4ð5, ǎ), ∂̃} > min{<̃(ð1, ǎ), <̃(ð5, ǎ), ℘̃}, min{=̃(ð1ð2ð3ð4ð5, ǎ), ∂̃} 6 max{=̃(ð1, ǎ), =̃(ð5, ǎ), ℘̃}, for ð1,ð2,ð3,ð4,ð5,∈ Z . 4 Level set concepts Theorem 4.1. A subset ~∂̃ is a <̃∂ is a (∂̃, ℘̃)-IQVFSS (IQVFLI,IQVFLATI, IQVFRI, IQVFBI) of Z . Then the lower level set <̃∂ is an SS (LI,LATIF, RI, TBI) of Z , where <̃∂ = {ð1 ∈ Z |<̃(ð1, ǎ) � ∂̃} and =̃∂ = {ð1 ∈ Z |<̃(ð1, ǎ) ≺ ∂̃}. Proof. Suppose that ~∂̃ is a (∂̃, ℘̃)-IQVFSS of Z . Let ð1,ð2,ð3 ∈ Z such that ð1,ð2,ð3 ∈ <̃∂ . Then <̃(ð1, ǎ) � ∂̃, <̃(ð2, ǎ) � ∂̃, <̃(ð3, ǎ) � ∂̃. Therefore max{<̃(ð1ð2ð3, ǎ), ∂̃} > min{<̃(ð1, ǎ), <̃(ð2, ǎ), <̃(ð3, ǎ), ℘̃} � min{∂̃, ∂̃, ∂̃, ℘̃} = ∂̃. Hence <̃(ð1ð2ð3, ǎ) � ∂̃. It shows that ð1ð2ð3 ∈ <̃∂ . Therefore <̃∂ is a SS of Z . Let ð1,ð2,ð3 ∈ Z such that ð1,ð2,ð3 ∈ =̃∂ . Then =̃(ð1, ǎ) ≺ ∂̃, =̃(ð2, ǎ) ≺ ∂̃=̃(ð3, ǎ) ≺ ∂̃. Therefore min{=̃(ð1ð2ð3, ǎ), ∂̃} 6 max{=̃(ð1, ǎ), =̃(ð2, ǎ), =̃(ð3, ǎ), ℘̃} ≺ max{∂̃, ∂̃, ∂̃, ℘̃} = ℘̃. Hence =̃(ð1ð2ð3, ǎ) ≺ ∂̃. It shows that ð1ð2ð3 ∈ =̃∂ . Therefore =̃∂ is a SS of Z . Therefore ~∂̃ is a SS of Z . Theorem 4.2. A subset i of Z is a SS [LI, LATIF,RI, TBI] of Z if and only if the IQVFS ~ = [<̃, =̃] of Z is defined as <̃(ð1, ǎ) = { > ℘̃ for all ð1 ∈ (i] ∂̃ for all ð1 /∈ (i] =̃(ð1, ǎ) = { 6 ℘̃ for all ð1 ∈ (i] ∂̃ for all ð1 /∈ (i] is a (∂̃, ℘̃)IQV FSS[IQV FLI, IQV FLATI, IQV FRI, IQV FBI] of Z . Proof. Suppose that i is an SS of Z . Let ð1,ð2,ð3 ∈ Z be such that ð1,ð2,ð3 ∈ (i] then ð1ð2ð3 ∈ (i]. Hence <̃(ð1ð2ð3, ǎ) > ℘̃ and =̃(ð1ð2ð3, ǎ) 6 ℘̃. Thus max{<̃(ð1ð2ð3, ǎ), ∂̃} > ℘̃ = min{<̃(ð1, ǎ), <̃(ð2, ǎ), <̃(ð3, ǎ), ℘̃} and min{=̃(ð1ð2ð3, ǎ), ∂̃} 6 ℘̃ = max{=̃(ð1, ǎ), =̃(ð2, ǎ), =̃(ð3, ǎ), ℘̃}. If ð1 /∈ (i] or ð2 /∈ (i] or ð3 /∈ (i], then min{<̃(ð1, ǎ), <̃(ð2, ǎ), <̃(ð3, ǎ), ℘̃} = ∂̃ and max{=̃(ð1, ǎ), =̃(ð2, ǎ), =̃(ð3, ǎ), ℘̃} = ℘̃. https://internationalpubls.com 581 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ , ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ , ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ , ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) That is max{<(ð1ð2ð3, ̌a), ∂̃} > min{<(ð1, ̌a), <(ð2, ̌a), <(ð3, ̌a), ℘̃} and min{=(ð1ð2ð3, ̌a), ∂̃} 6 max{=(ð1, ̌a), =(ð2, ̌a), =(ð3, ̌a), ℘̃}. Therefore ~ is a (∂̃, ℘̃) IQVFSS of Z . Conversely assume that ~ = [< =] is a (∂̃, ℘̃)-IQVFSS of Z . Let ð1ð2ð3 ∈ (i]. Then <(ð1, ̌a) > ℘̃, <(ð2, ̌a) > ℘̃, <(ð3, ̌a) > ℘̃ and =(ð1, ̌a) 6 ℘̃, =(ð2, ̌a) 6 ℘̃, =(ð3, ̌a) 6 ℘̃. Now ~ = [< =] is a (∂̃, ℘̃)-IQVFSS of Z . Therefore max{<(ð1ð2ð3, ̌a), ∂̃} > min{<(ð1, ̌a), <(ð2, ̌a), <(ð3, ̌a), ℘̃} > min{ ̃℘, ℘̃, ℘̃, ℘̃} = ℘̃ and min{=(ð1ð2ð3, ̌a), ∂̃} 6 max{=(ð1, ̌a), =(ð2, ̌a), =(ð3, ̌a), ℘̃} 6 max{ ̃℘, ℘̃, ℘̃, ℘̃} = ℘̃. It follows that ð1ð2ð3 ∈ (i] . Therefore i is a SS of Z . Theorem 4.3. A subset ~ = [< =] is a (∂̃, ℘̃) − IQV F SS[IQV F LI, IQV F LAT I, IQV FRI, IQV FBI] of Z if and only if each level subset ~t is a SS [LI,LAIQVF,RI,TBI] of Z for all t ∈ (∂̃, ℘̃] . Proof. Assume that ~t is a SS of Z for each t ∈ [0, 1]. Let t = min{<(ð1, ̌a), <(ð2, ̌a), <(ð3, ̌a)}. Then ð1, ð2, ð3 ∈ t = min{<(ð1, ̌a), <(ð2, ̌a), <(ð3, ̌a), ℘̃}. Let t = max{=(ð1, ̌a), =(ð2, ̌a), =(ð3, ̌a)}. Then ð1, ð2, ð3 ∈ =t for each ð1, ð2, ð3 ∈ Z . Thus min{=(ð1ð2ð3, ̌a), ∂̃} 6 t = max{=(ð1, ̌a), =(ð2, ̌a), =(ð3, ̌a), ℘̃}. This shows that ~t is IQVFSS of Z . Conversely, assume that ~t is a IQVFSS of Z . For each t ∈ [0, 1] and ð1, ð2, ð3 ∈ t, <(ð2, ̌a) > t, <(ð3, ̌a) > t. Since < is a SS of Z , max{<(ð1ð2ð3, ̌a), ∂̃} > min{<(ð1, ̌a), <(ð2, ̌a), <(ð3, ̌a), ℘̃} > t. This implies that ð1ð2ð3 ∈ min{<̃(d, q), <̃(d, q)} = 0.30 and =̃(dbd) = 0.42 66 max{=̃(d, q), =̃(d, q)} = 0.37. Definition 4.5. If ii is the characteristic function is defined as (i> i )℘̃ ∂̃ (ð1, ǎ) = { ℘̃ if ð1 ∈ (i] ∂̃ if ð1 /∈ (i] (iz i )℘̃ ∂̃ (ð1, ǎ) = { ∂̃ if ð1 ∈ (i] ℘̃ if ð1 /∈ (i] Theorem 4.6. A non empty subset i of Z is a SS [LI, LAIQV F,RI, TBI] of Z if and only if subset i (i] is a (∂̃, ℘̃)-IQV FSS[IQV FLI, IQV FLATI, IQV FRI, IQV FBI] of Z . Proof. Assume that i is a SS of Z . Then i (i] is a IQVFSS of Z and hence i (i] is an (∂̃, ℘̃)-IQVFSS of Z . Conversely, Let i (i] is an (∂̃, ℘̃)-IQVFSS of Z . Let ð1,ð2,ð3 ∈ Z be such that ð1,ð2,ð3 ∈ (i]. Then i> (i] (ð1, ǎ) = ℘̃,i> (i] (ð2, ǎ) = ℘̃,i> (i] (ð3, ǎ) = ℘̃. Since i> (i] is a (∂̃, ℘̃)IQVFSS. Consider max{i> (i] (ð1ð2ð3, ǎ), ∂̃} > min{i> (i] (ð1, ǎ),i> (i] (ð2, ǎ),i> (i] (ð3, ǎ), ℘̃} = min{℘̃, ℘̃, ℘̃, ℘̃} = ℘̃ https://internationalpubls.com 582 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) as ∂̃ ≺ ℘̃, this implies that i> (i] (ð1ð2ð3, ǎ) > ℘̃. Thus ð1ð2ð3 ∈ (i]. Thus ð1ð2ð3 ∈ (i]. Let ð1,ð2,ð3 ∈ Z be such that ð1,ð2,ð3 ∈ (i]. Then iz (i] (ð1, ǎ) = ∂̃,iz (i] (ð2, ǎ) = ∂̃,iz (i] (ð3, ǎ) = ∂̃. Since iz (i] is a (∂̃, ℘̃)IQVFSS. Consider min{iz (i] (ð1ð2ð3, ǎ), ∂̃} 6 max{iz (i] (ð1, ǎ),iz (i] (ð2, ǎ),iz (i] (ð3, ǎ), ℘̃} = max{∂̃, ∂̃, ∂̃, ℘̃} = ℘̃ as ∂̃ ≺ ℘̃, this implies that iz (i] (ð1ð2ð3, ǎ) 6 ∂̃. Thus ð1ð2ð3 ∈ (i]. Thus ð1ð2ð3 ∈ (i]. Therefore i is a SS of Z . Let ð1,ð2,ð3 ∈ Z be such that ð1,ð2,ð3 /∈ (i]. Then i> (i] (ð1, ǎ) = ∂̃,i> (i] (ð2, ǎ) = ∂̃,i> (i] (ð3, ǎ) = ∂̃. Since i> (i] is a (∂̃, ℘̃)IQVFSS. max{i> (i] (ð1ð2ð3, ǎ), ∂̃} > min{i> (i] (ð1, ǎ),i> (i] (ð2, ǎ),i> (i] (ð3, ǎ), ℘̃} = min{∂̃, ∂̃, ∂̃, ℘̃} = ∂̃ as ∂̃ ≺ ℘̃, this implies that i> (i] (ð1ð2ð3, ǎ) > ∂̃. Thus ð1ð2ð3 6∈ (i]. Let ð1,ð2,ð3 ∈ Z be such that ð1,ð2,ð3 /∈ (i]. Then iz (i] (ð1, ǎ) = ℘̃,iz (i] (ð2, ǎ) = ℘̃,iz (i] (ð3, ǎ) = ℘̃. Since iz (i] is a (∂̃, ℘̃)IQVFSS. min{iz (i] (ð1ð2ð3, ǎ), ∂̃} 6 max{iz (i] (ð1, ǎ),iz (i] (ð2, ǎ),iz (i] (ð3, ǎ), ℘̃} = max{℘̃, ℘̃, ℘̃, ℘̃} = ℘̃ as ∂̃ ≺ ℘̃, this implies that iz (i] (ð1ð2ð3, ǎ) 6 ℘̃. Thus ð1ð2ð3 6∈ (i]. Therefore i is a SS of Z . Definition 4.7. For three IQVFSs ~, ð2 and κ of Z , their product ~ ∗ ð2 ∗ κ is defined as (~> ∗ ð>2 ∗ κ>)(ð, ǎ) =  sup (r,s,t)∈ið {~>(r)Oð>2 (s)Oκ>(t)} if ið 6= 0 [0, 0] otherwise (~z ∗ ðz2 ∗ κz)(ð, ǎ) =  inf (r,s,t)∈ið {~z(r) M ðz2 (s) M κz(t)} if ið 6= 0 [1, 1] otherwise Definition 4.8. Let ~ be subset of Z , we define the subset (<̃)℘∂ (ð, ǎ) = {<̃(ð, ǎ)O℘̃} M ∂̃, (=̃)℘∂ (ð, ǎ) = {=̃(ð, ǎ) M ℘̃}O∂̃, for all ð ∈ Z . Lemma 4.9. Let i, i1 and i2 be subsets of Z . Then 1. (i (i] Oi (i1] Oi (i2] )℘̃ ∂̃ = (i(ifi1fi2]) ℘̃ ∂̃ , 2. (i (i] M i (i1] M i (i2] )℘̃ ∂̃ = (i(igi1gi2]) ℘̃ ∂̃ , 3. (i (i] ∗i(i1]∗i(i2]) ℘̃ ∂̃ = (i(ii1i2]) ℘̃ ∂̃ . Proof. (3) Let ð1 ∈ Z . If ð1 ∈ (ii1i2], then (i(ii1i2])(ð1, ǎ) = ℘̃. Since ð1 6 abc , a ∈ (i],b ∈ (i1] and c ∈ (i2]. We have (a, b, c) ∈ ið1 and ið1 6= 0. https://internationalpubls.com 583 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) (i> (i] ∗i> (i1] ∗i> (i2] )(ð1, ǎ) = sup ð1=xyz min{i> (i] (x, ǎ),i> (i1] (y, ǎ),i> (i2] (z, ǎ)} > min{i> (i] (a, ǎ),i> (i1] (b, ǎ),i> (i2] (c, ǎ)} = ℘̃ (iz (i] ∗iz (i1] )(ð1, ǎ) ∗iz (i2] )(ð1, ǎ) = inf ð1=xyz max{iz (i] (x, ǎ),iz (i1] (y, ǎ),iz (i2] (z, ǎ)} 6 max{iz (i] (a, ǎ),iz (i1] (b, ǎ),iz (i2] (c, ǎ)} = ∂̃ Therefore (i (i] ∗i (i1] ∗i (i2] )(ð1, ǎ) = (i(ii1i2])(ð1, ǎ). If ð1 /∈ (ii1i2] then (i>(ii1i2] )(ð1, ǎ) = ∂̃ and (iz (ii1i2] )(ð1, ǎ) = ℘̃. Since ð1 6 abc , a /∈ (i], b /∈ (i1] and c /∈ (i2]. We have (i> (i] ∗i> (i1] ∗i> (i2] )(ð1, ǎ) = sup ð1=xyz min{i> (i] (x, ǎ),i> (i1] (y, ǎ),i> (i2] (z, ǎ)} > min{i> (i] (a, ǎ),i> (i1] (b, ǎ),i> (i2] (c, ǎ)} = ∂̃ (iz (i] ∗iz (i1] ∗iz (i2] )(ð1, ǎ) = inf ð1=xyz max{iz (i] (x, ǎ),iz (i1] (y, ǎ),iz (i2] (z, ǎ)} 6 max{iz (i] (a, ǎ),iz (i1] (b, ǎ),iz (i2] (c, ǎ)} = ℘̃ Hence (i (i] ∗i (i1] ∗i (i2] )(ð1, ǎ) = (i(ii1i2])(ð1, ǎ). Theorem 4.10. Let {ii|i ∈ I} be a family of subsets of Z and i,i2 ⊆ Z . Then (1) (i] ⊆ (i2] if and only if (i(i]) ℘̃ ∂̃ 6 (i(i2]) ℘̃ ∂̃ (2) (fi∈Ii(ii]) ℘̃ ∂̃ = (ifi∈I(ii]) ℘̃ ∂̃ (3) (gi∈Ii(ii]) ℘̃ ∂̃ = (igi∈I(ii]) ℘̃ ∂̃ . 5 Regular ordered ternary semigroups Theorem 5.1. If i is a (∂̃, ℘̃)-IQVFLI[IQVFSS,IQVFLATI,IQVFRI] of Z , then (i)℘̃ ∂̃ is a IQVFLI[IQVFSS, IQVFLATI,IQVFRI] of Z . Theorem 5.2. Let i be an (∂̃, ℘̃)IQVFRI, i1 be an (∂̃, ℘̃)IQVFLATI and i2 be an (∂̃, ℘̃)IQVFLI of Z . Then ((i ∗ i1 ∗ i2])℘̃ ∂̃ ⊆ (if i1 f i2]℘̃ ∂̃ . Proof. Let i = [<̃i, =̃i] be an (∂̃, ℘̃)IQVFRI, i1 = [<̃i1 , =̃i1 ] be an (∂̃, ℘̃)IQVFLATI and i2 = [<̃i2 , =̃i2 ] be an (∂̃, ℘̃)IQVFLI of Z . Let (ð1,ð2,ð3, ǎ) ∈ I}. If I} 6= ∅, then } 6 ð1ð2ð3. Thus <̃i(}, ǎ) > <̃i(ð1ð2ð3) > <̃i(ð1, ǎ) and =̃i(}, ǎ) 6 =̃i(ð1ð2ð3) 6 =̃i(ð1, ǎ). Similarly <̃i1 (}, ǎ) > <̃i1 (ð1ð2ð3) > <̃i1 (ð2, ǎ) and =̃i1 (}, ǎ) 6 =̃i1 (ð1ð2ð3) 6 =̃i1 (ð2, ǎ). https://internationalpubls.com 584 <̃ <̃ <̃ =̃ =̃ =̃ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) Similarly, i2 (}, ̌a) > i2 (ð1ð2ð3) > i2 (ð3, ̌a) and i2 (}, ̌a) 6 i2 (ð1ð2ð3) 6 i2 (ð3, ̌a). We have (<̃(i∗i1∗i2]) ℘̃ ∂̃ (}, ǎ) = (<̃(i∗i1∗i2](}, ǎ)O℘̃) M ∂̃ = [ [ sup }6ð1ð2ð3 {<̃i(ð1, ǎ)O<̃i1(ð2, ǎ)O<̃i2(ð3, ǎ)}O℘̃] ] M ∂̃ = [ sup }6ð1ð2ð3 {<̃i(ð1, ǎ)O<̃i1(ð2, ǎ)O<̃i2(ð3, ǎ)}O℘̃O℘̃O℘̃O℘̃ ] M ∂̃ = [ sup }6ð1ð2ð3 {(<̃i(ð1, ǎ)O℘̃)O(<̃i1(ð2, ǎ)O℘̃)O(<̃i2(ð3, ǎ)O℘̃)}O℘̃ ] M ∂̃ 6 ({(<̃i(}, ǎ) M ∂̃)O(<̃i1 (}, ǎ) M ∂̃)O(<̃i2 (}, ǎ) M ∂̃)}O℘̃) M ∂̃ = {((<̃i(}, ǎ)O<̃i1(}, ǎ)O<̃i2(}, ǎ)) M ∂̃)O℘̃} M ∂̃ = {((<̃iO<̃i1 O<̃i2 )(}, ǎ)O℘̃} M ∂̃ = (<̃ifi1fi2 )℘̃ ∂̃ (}, ǎ) (=̃(i∗i1∗i2]) ℘̃ ∂̃ (}, ǎ) = (=̃(i∗i1∗i2](}, ǎ) M ℘̃)O∂̃ = [ [ inf }6ð1ð2ð3 {=̃i(ð1, ǎ) M =̃i1 (ð2, ǎ) M =̃i2 (ð3, ǎ)} M ℘̃] ] O∂̃ = [ inf }6ð1ð2ð3 {=̃i(ð1, ǎ) M =̃i1 (ð2, ǎ) M =̃i2 (ð3, ǎ)} M ℘̃ M ℘̃ M ℘̃ M ℘̃ ] O∂̃ = [ inf }6ð1ð2ð3 {(=̃i(ð1, ǎ) M ℘̃) M (=̃i1 (ð2, ǎ) M ℘̃) M (=̃i2 (ð3, ǎ) M ℘̃)} M ℘̃ ] O∂̃ > ({(=̃i(}, ǎ)O∂̃) M (=̃i1 (}, ǎ)O∂̃) M (=̃i2 (}, ǎ)O∂̃)} M ℘̃)O∂̃ = {((=̃i(}, ǎ) M =̃i1(}, ǎ) M =̃i2(}, ǎ))O∂̃) M ℘̃}O∂̃ = {((=̃i M =̃i1 M =̃i2 )(}, ǎ) M ℘̃}O∂̃ = (=̃igi1gi2 )℘̃ ∂̃ (}, ǎ) Let ð1,ð2,ð3 /∈ I}. If I} = ∅, then (<̃i ∗ i1 ∗ <̃i2 )(}, ǎ) = 0 and (=̃i ∗ i1 ∗ =̃i2 )(}, ǎ) = 1 such that } 6 ð1ð2ð3. (<̃(i∗i1∗i2]) ℘̃ ∂̃ (}, ǎ) = (<̃(i∗i1∗i2](}, ǎ)O℘̃) M ∂̃ = 0 M ∂̃ 6 (<̃ifi1fi2 (}, ǎ)O℘̃) M ∂̃ = (<̃ifi1fi2(}, ǎ)O℘̃) (=̃(i∗i1∗i2]) ℘̃ ∂̃ (}, ǎ) = (=̃(i∗i1∗i2](}, ǎ) M ℘̃)O∂̃ = 1O∂̃ = ∂̃ > (=̃igi1gi2 (}, ǎ) M ℘̃)O∂̃ = (=̃igi1gi2(}, ǎ) M ℘̃) Therefore ((i ∗ i1∗i2])℘̃ ∂̃ ⊆ ((if i1 f i2])℘̃ ∂̃ . Theorem 5.3. Let Z is regular, i be an (∂̃, ℘̃)IQVFRI, i1 be an (∂̃, ℘̃) IQVFLATI and i2 be an (∂̃, ℘̃)IQVFLI of Z if and only if ((i ∗ i1 ∗ i2])℘̃ ∂̃ = ((if i1 f i2])℘̃ ∂̃ . https://internationalpubls.com 585 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) Proof. Let Z regular and i be an (∂̃, ℘̃)IQVFRI, i1 be an (∂̃, ℘̃)IQVFLATI and i2 be an (∂̃, ℘̃)IQVFLI of Z . Let (ð1,ð3) ∈ I}. If I} 6= ∅, then } 6 ð1ð2ð3. Thus, <̃i(}, ǎ) > <̃i(ð1ð2ð3) > <̃i(ð1, ǎ) and =̃i(}, ǎ) 6 =̃i(ð1ð2ð3) 6 =̃i(ð1, ǎ). Similarly <̃i1(}, ǎ) > <̃i1(ð1ð2ð3) > <̃i1(ð2, ǎ) and =̃i1(}, ǎ) 6 =̃i1(ð1ð2ð3) 6 =̃i1(ð2, ǎ). Similarly, <̃i2 (}, ǎ) > <̃i2 (ð1ð2ð3) > <̃i2 (ð3, ǎ) and =̃i2 (}, ǎ) 6 =̃i2 (ð1ð2ð3) 6 =̃i2 (ð3, ǎ). For } ∈ Z , there exists x ∈ Z such that } 6 }z1}z2}z3}. Then }, (z1}z2}z3), } ∈ I}. We have (<̃(i∗i1∗i2]) ℘̃ ∂̃ (}, ǎ) = (<̃(i∗i1∗i2](}, ǎ)O℘̃) M ∂̃ = [ [ sup }6}z1}z2}z3} {<̃i(ð1, ǎ)O<̃i1(ð2, ǎ)O<̃i2(ð3, ǎ)}O℘̃] ] M ∂̃ = [ sup }6}z1}z2}z3} {<̃i(ð1, ǎ)O<̃i1(ð2, ǎ)O<̃i2(ð3, ǎ)}O℘̃O℘̃O℘̃O℘̃ ] M ∂̃ = [ sup }6}z1}z2}z3} {(<̃i(ð1, ǎ)O℘̃)O(<̃i1 (ð2, ǎ)O℘̃)O(<̃i2 (ð3, ǎ)O℘̃)}O℘̃ ] M ∂̃ > ({(<̃i(}, ǎ) M ∂̃)O(<̃i1(z1}z2}z3) M ∂̃)O(<̃i2(}, ǎ) M ∂̃)}O℘̃) M ∂̃ > ({(<̃i(}, ǎ) M ∂̃)O(<̃i1 (}, ǎ) M ∂̃)O(<̃i2 (}, ǎ) M ∂̃)}O℘̃) M ∂̃ = {((<̃i(}, ǎ)O<̃i1 (}, ǎ)O<̃i2 (}, ǎ)) M ∂̃)O℘̃} M ∂̃ = {((<̃iO<̃i1O<̃i2)(}, ǎ)O℘̃} M ∂̃ = (<̃ifi1fi2)℘̃ ∂̃ (}, ǎ) (=̃(i∗i1∗i2]) ℘̃ ∂̃ (}, ǎ) = (=̃(i∗i1∗i2](}, ǎ) M ℘̃)O∂̃ = [ [ inf }6}z1}z2}z3} {=̃i(ð1, ǎ) M =̃i1 (ð2, ǎ) M =̃i2 (ð3, ǎ)} M ℘̃] ] O∂̃ = [ inf }6}z1}z2}z3} {=̃i(ð1, ǎ) M =̃i1(ð2, ǎ) M =̃i2(ð3, ǎ)} M ℘̃ M ℘̃ M ℘̃ M ℘̃ ] O∂̃ = [ inf }6}z1}z2}z3} {(=̃i(ð1, ǎ) M ℘̃) M (=̃i1(ð2, ǎ) M ℘̃) M (=̃i2(ð3, ǎ) M ℘̃)} M ℘̃ ] O∂̃ 6 ({(=̃i(}, ǎ)O∂̃) M (=̃i1 (z1}z2}z3)O∂̃) M (=̃i2 (}, ǎ)O∂̃)} M ℘̃)O∂̃ 6 ({(=̃i(}, ǎ)O∂̃) M (=̃i1 (}, ǎ)O∂̃) M (=̃i2 (}, ǎ)O∂̃)} M ℘̃)O∂̃ = {((=̃i(}, ǎ) M =̃i1(}, ǎ) M =̃i2(}, ǎ))O∂̃) M ℘̃}O∂̃ = {((=̃i M =̃i1 M =̃i2 )(}, ǎ) M ℘̃}O∂̃ = (=̃igi1gi2 )℘̃ ∂̃ (}, ǎ) Thus, ((i ∗ i1 ∗ i2])℘̃ ∂̃ ⊇ ((if i1 f i2])℘̃ ∂̃ and by Theorem 5.2. Hence, ((i ∗ i1 ∗ i2])℘̃ ∂̃ = ((if i1 f i2])℘̃ ∂̃ . Conversely assume that ((i ∗ i1 ∗ i2])℘̃ ∂̃ = ((i f i1 f i2])℘̃ ∂̃ . Let i = (<̃i, =̃i) be an (∂̃, ℘̃)IQVFRI, i1 = (<̃i1 , =̃i1) be an (∂̃, ℘̃)IQVFLATI and i2 = (<̃i2 , =̃i2) be an (∂̃, ℘̃)IQVFLI of Z . Then by Theorem 4.6, ii is a (∂̃, ℘̃)IQVFRI, ii1 is a (∂̃, ℘̃)IQVFLATI and ii2 be a (∂̃, ℘̃)IQVFLI of Z . By Lemma 4.9 and Theorem 4.10, (i(ifi1fi2]) ℘̃ ∂̃ = (ii fii1 fii2 )℘̃ ∂̃ = (ii ∗ii1 ∗ii2 )℘̃ ∂̃ = (i(i∗i1∗i2]) ℘̃ ∂̃ . This implies (if i1 f i2]℘̃ ∂̃ = ((i ∗ i1 ∗ i2])℘̃ ∂̃ . Hence by Corollary 2.5, Z is regular. Theorem 5.4. Let Z is regular, i be an (∂̃, ℘̃)IQVFBI, i1 be an (∂̃, ℘̃)IQVFLATI and i2 be an (∂̃, ℘̃)IQVFLI of Z if and only if ((i ∗ i1 ∗ i2])℘̃ ∂̃ = ((if i1 f i2])℘̃ ∂̃ . Proof. Let Z be regular semigroup and i be an (∂̃, ℘̃)IQVFBI and i2 be an (∂̃, ℘̃)IQVFLI of Z . Let (ð1,ð3) ∈ I}. If I} 6= ∅, then } 6 ð1ð2ð3. Thus <̃i(}, ǎ) > <̃i(ð1ð2ð3) > <̃i(ð1, ǎ) and =̃i(}, ǎ) 6 =̃i(ð1ð2ð3) 6 =̃i(ð1, ǎ). https://internationalpubls.com 586 <̃ <̃ <̃ =̃ =̃ =̃ <̃ <̃ <̃ =̃ =̃ =̃ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) Similarly i1 (}, ̌a) > i1 (ð1ð2ð3) > i1 (ð2, ̌a) and i1 (}, ̌a) 6 i1 (ð1ð2ð3) 6 i1 (ð2, ̌a). Similarly, i2 (}, ̌a) > i2 (ð1ð2ð3) > i2 (ð3, ̌a) and i2 (}, ̌a) 6 i2 (ð1ð2ð3) 6 i2 (ð3, ̌a). For } ∈ Z , there exists x ∈ Z such that } 6 }z1}z2}z3}z4}z5}. Then } 6 (}z1}z2}, ̌a), (z3}z4}z5), } ∈ I}. We have (<̃(i∗i1∗i2]) ℘̃ ∂̃ (}, ǎ) = (<̃(i∗i1∗i2](}, ǎ)O℘̃) M ∂̃ = [ [ sup }6}z1}z2}z3}z4}z5} {<̃i(ð1, ǎ)O<̃i1(ð2, ǎ)O<̃i2 (ð3, ǎ)}O℘̃] ] M ∂̃ = [ sup }6}z1}z2}z3}z4}z5} {<̃i(ð1, ǎ)O<̃i1 (ð2, ǎ)O<̃i2 (ð3, ǎ)}O℘̃O℘̃O℘̃O℘̃ ] M ∂̃ = [ sup }6}z1}z2}z3}z4}z5} {(<̃i(ð1, ǎ)O℘̃)O(<̃i1 (ð2, ǎ)O℘̃)O(<̃i2 (ð3, ǎ)O℘̃)}O℘̃ ] M ∂̃ > ({(<̃i(}z1}z2}, ǎ) M ∂̃)O(<̃i1(z3}z4}z5) M ∂̃)O(<̃i2(}, ǎ) M ∂̃)}O℘̃) M ∂̃ > ({(<̃i(}, ǎ) M ∂̃)O(<̃i1 (}, ǎ) M ∂̃)O(<̃i2 (}, ǎ) M ∂̃)}O℘̃) M ∂̃ = {((<̃i(}, ǎ)O<̃i1 (}, ǎ)O<̃i2 (}, ǎ)) M ∂̃)O℘̃} M ∂̃ = {((<̃iO<̃i1O<̃i2)(}, ǎ)O℘̃} M ∂̃ = (<̃ifi1fi2)℘̃ ∂̃ (}, ǎ) (=̃(i∗i1∗i2]) ℘̃ ∂̃ (}, ǎ) = (=̃(i∗i1∗i2](}, ǎ) M ℘̃)O∂̃ = [ [ inf }6}z1}z2}z3}z4}z5} {=̃i(ð1, ǎ) M =̃i1 (ð2, ǎ) M =̃i2 (ð3, ǎ)} M ℘̃] ] O∂̃ = [ inf }6}z1}z2}z3}z4}z5} {=̃i(ð1, ǎ) M =̃i1 (ð2, ǎ) M =̃i2 (ð3, ǎ)} M ℘̃ M ℘̃ M ℘̃ M ℘̃ ] O∂̃ = [ inf }6}z1}z2}z3}z4}z5} {(=̃i(ð1, ǎ) M ℘̃) M (=̃i1(ð2, ǎ) M ℘̃) M (=̃i2(ð3, ǎ) M ℘̃)} M ℘̃ ] O∂̃ 6 ({(=̃i(}z1}z2}, ǎ)O∂̃) M (=̃i1 (z3}z4}z5)O∂̃) M (=̃i2 (}, ǎ)O∂̃)} M ℘̃)O∂̃ 6 ({(=̃i(}, ǎ)O∂̃) M (=̃i1 (}, ǎ)O∂̃) M (=̃i2 (}, ǎ)O∂̃)} M ℘̃)O∂̃ = {((=̃i(}, ǎ) M =̃i1(}, ǎ) M =̃i2(}, ǎ))O∂̃) M ℘̃}O∂̃ = {((=̃i M =̃i1 M =̃i2 )(}, ǎ) M ℘̃}O∂̃ = (=̃igi1gi2 )℘̃ ∂̃ (}, ǎ) Thus ((i∗i1∗i2])℘̃ ∂̃ ⊇ ((ifi1fi2])℘̃ ∂̃ and by Theorem 5.2 and hence ((i∗i1∗i2])℘̃ ∂̃ = ((ifi1fi2])℘̃ ∂̃ . Conversely assume that ((i ∗ i1 ∗ i2])℘̃ ∂̃ = ((i f i1 f i2])℘̃ ∂̃ . Let i = (<̃i, =̃i) be an (∂̃, ℘̃)IQVFBI, i1 = (<̃i1 ,Ξi1 , =̃i1 ) be an (∂̃, ℘̃)IQVFLATI and i2 = (<̃i2 , =̃i2 ) be an (∂̃, ℘̃)IQVFLI of Z . Then by Theorem 4.6, ii is a (∂̃, ℘̃)IQVFBI, ii1 is a (∂̃, ℘̃)IQVFLATI and ii2 be a (∂̃, ℘̃)IQVFLI of Z . By Lemma 4.9 and Theorem 4.10, (i(ifi1fi2]) ℘̃ ∂̃ = (ii fii1 fii2)℘̃ ∂̃ = (ii ∗ii1 ∗ii2)℘̃ ∂̃ = (i(i∗i1∗i2]) ℘̃ ∂̃ . This implies (if i1 f i2]℘̃ ∂̃ = ((i ∗ i1 ∗ i2])℘̃ ∂̃ . Hence by Corollary 2.5, Z is regular. 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. https://internationalpubls.com 587 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) [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 Sys- tems, 22, (2014), 958-965. [6] S. Ashraf, S. Abdullah, T. Mahmood, F. Ghani and T. Mahmood, Spherical fuzzy sets and their applica- tions in multi-attribute decision making problems, Journal of Intelligent and Fuzzy Systems, 36, (2019), 2829-284. [7] 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. [8] . Abdallah Al-Husban & Abdul Razak Salleh 2015. Complex Fuzzy Hyperring Based on Complex Fuzzy Spaces. Proceedings of 2nd Innovation and Analytics Conference & Exhibition (IACE), 1691, AIP Pub- lishing 2015, 040009-040017. [9] Al-Husban, A., & Salleh, A. R. Complex fuzzy hypergroups based on complex fuzzy spaces. Interna- tional Journal of Pure and Applied Mathematics, 107(4), (2016), 949-958. [10] Al-Husban, A., Amourah, A., & Jaber, J. J. Bipolar complex fuzzy sets and their properties. Italian Journal of Pure and Applied Mathematics, 43, 2020, 754-761. [11] Rosenfeld, Fuzzu groups, J.Math. Anal. Appl.35 (1971) 512-517. [12] N. Kuroki, On fuzzy semigroups, Inform. Sci. 53 (1991), 203-236. [13] J. N. Mordeson, D. S. Malik, N. Kuroki, Fuzzy semigroups, springer-Verlag Berlin Heidelberg GmbH, 2003. [14] M. K. Sen, On i-semigroups, Proceedings of International conference on Algebra and its Application Decker publication, New yark, (1981), 301. [15] M. K. Sen and N. K Saha, On i-semigroup, I, Bull.Calcutta Math. Soc.,(1986), 78 180-186. [16] N.Kehayopula, On ordered i-semigroups, Scientiae Mathematicae Japonicae Online, e-2010, 37-43. [17] Somsak Lekkoksung, On Q-fuzzy ideals in ordered semigroups, International Journal of Pure and Ap- plied Mathematics,92(3) (2014), 369–379. [18] N.Kehayopula and Tsingelis, Fuzzy sets in ordered groupoids, semigroup Forum, 65, (2005) 128-132 . [19] F. M. Khan, N. H. Sarmin and A. Khan. Some new characterization of ordered semigroups in terms of (κ, θ)-fuzzy bi-ideals, International Journal of Algebra and Statistics, 1(1)(2012), 22–32. [20] Shihadeh, A., Matarneh, K. A. M., Hatamleh, R., Al-Qadri, M. O., & Al-Husban, A. (2024). On The Two-Fold Fuzzy n-Refined Neutrosophic Rings For 2= 3. Neutrosophic Sets and Systems, 68, 8-25. [21] Abdallah Shihadeh, Khaled Ahmad Mohammad Matarneh, Raed Hatamleh, Randa Bashir Yousef Hi- jazeen, Mowafaq Omar Al-Qadri, Abdallah Al-Husban.(2024). An Example of Two-Fold Fuzzy Algebras Based On Neutrosophic Real Numbers, Neutrosophic Sets and Systems, 67, 169-178. [22] Raed Hatamleh, Abdallah Al-Husban, N. Sundarakannan, M. S. Malchijah Raj. (2025). Complex cubic intuitionistic fuzzy set applied to subbisemirings of bisemirings using homomorphism, Communications on Applied Non-linear Analysis, 32 (3), 418-435. [23] Abubaker, Ahmad A, Hatamleh, Raed, Matarneh, Khaled, Al-Husban, Abdallah.(2024). On the Numer- ica Solutions for Some Neutrosophic Singular Boundary Value Problems by Using (LPM) Polynomials, International Journal of Neutrosophic Science, 25(2), 197-205. [24] A., Ahmad. , Hatamleh, Raed. , Matarneh, Khaled. , Al-Husban, Abdallah. On the Irreversible k- Threshold Conversion Number for Some Graph Products and Neutrosophic Graphs. (2025). International Journal of Neutrosophic Science, 25(2), 183-196 [25] Palanikumar M, Arulmozhi K, On intuitionistic fuzzy normal subbisemirings of bisemirings, Nonlinear studies, 28(3), 2021, 717-721. https://internationalpubls.com 588 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) [26] K. Hila and E. Pisha. On bi-ideals on ordered i-semigroups. Hacettepe Journal of Mathematics and Statistics, 40(6), (2011), 793-804. [27] Dutta T.K and Kar S, On Prime ideals and Prime radical of ternary Semirings, Bull.Cal. Math. Soc., 97(5), 2005, 445-454. [28] 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 [29] Mohanraj, G; Palanikumar, M. On various prime and semiprime bi-ideals of Rings. Nonlinear studies. 2021, 27(3), 811-815. [30] 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. [31] Palanikumar, M; Iampan, A; Manavalan, L.J. M-bi-base generator of ordered i-semigroups. ICIC Ex- press Letters Part B: Applications. 2022, 13(8), 795-802. [32] Mohanraj, G; Palanikumar, M. Characterization of various k-regular in b-semirings, AIP Conference Proceedings, 2019, 2112 (1), 020021. [33] 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. [34] 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. [35] Rajalakshmi, Raed Hatamleh, Abdallah Al-Husban, K. Lenin Muthu Kumaran, M. S. Malchijah raj. (2025). Various (δ1, δ2) neutrosophic ideals of an ordered ternary semigroups, Communications on Ap- plied Non-linear Analysis, 32 (3), 400-417. [36] Raed Hatamleh, Abdallah Al-Husban, K. Sundareswari, G.Balaj, M.Palanikumar (2025). Complex Tan- gent Trigonometric Approach Applied to (λ, µ)-rung Fuzzy Set using Weighted Averaging, Geometric Operators and its Extension, Communications on Applied Non-linear Analysis, 32 (5), 133-144. [37] Raed Hatamleh, Abdallah Al-Husban, M.Palanikumar, K. Sundareswari (2025). Different Weighted Operators such as Generalized Averaging and Generalized Geometric based on Trigonometric q-rung Interval-Valued Approach, Communications on Applied Non-linear Analysis, 32 (5), 91-101. [38] 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. [39] Hatamleh, R. (2003). On the Form of Correlation Function for a Class of Non stationary Field with a Zero Spectrum. Rocky Mountain Journal of Mathematics, 33(1), 1-13. [40] Hatamleh, R., Zolotarev, V. A. (2014). On Two-Dimensional Model Representations of One Class of Commuting Operators, Ukrainian Mathematical Journal, 66(1), 122-144. [41] Hatamleh, R., Zolotarev,V. A. (2015). On Model Representations of Non-Self adjoint Operators with Infinitely Dimensional Imaginary Component. Journal of Mathematical Physics, Analysis, Geometry, 11(2), 174-186. [42] Bataihah, A and Hazaymeh, A. (2025). Neutrosophic fuzzy metric spaces and fixed points results with integral contraction type . International Journal of Neutrosophic Science, 25(3), 561-572. [43] Hazaymeh, Ayman A, and Anwar Bataihah. 2025, Neutrosophic Fuzzy Metric Spaces and Fixed Points for Contractions of Nonlinear Type, Neutrosophic Sets and Systems 77: 96-112. https://internationalpubls.com 589 1 Introduction 2 Basic concepts 3 (, ) ternary intuitionistic Q interval-valued fuzzy ideals 4 Level set concepts 5 Regular ordered ternary semigroups