Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 50 https://internationalpubls.com An operations and relations on the Cartesian Product of Interval Valued Intuitionistic Fuzzy Matrices R. Amutha1,2 and C. Ragavan1 1PG & Research Department of Mathematics, Sri Vidya Mandir Arts & Science College (Autonomous), Katteri – 636 902, Uthangarai, Tamilnadu, India 2Department of Mathematics, Periyar University, Salem – 636 011, Tamilnadu, India 1saraswathiamutha@gmail.com, 2ragavanshana@gmail.com Article History: Received: 12-11-2024 Revised:24-12-2024 Accepted:09-01-2025 Abstract: In this paper, we study an operations and relations on the cartesian product over interval valued intuitionistic fuzzy matrices are introduced and its some properties are explored. We prove some equality based on the operation and the relation over interval valued intuitionistic fuzzy matrices. Finally, we introducing some cartesian formulas ×1, ×2, ×3, ×4, ×5 in cartesian product of interval valued intuitionistic fuzzy matrices. Keywords: fuzzy sets, intuitionistic fuzzy sets, fuzzy matrix, cartesian product over intuitionistic fuzzy sets, operation, geometric interpretation, interval valued intuitionistic fuzzy set. 1. Introduction In 1986, the theory of intuitionistic fuzzy sets was presented initially by Atanassov [1] and a series of operations and concepts is defined in [2]. Later, the interval valued intuitionistic fuzzy sets [3] was presented by Atanassov in 1989, and it has achieved a tremendous amount of research and development in various fields. For example, Zeng and Hu studied the necessity operator and the possibility operator of interval valued intuitionistic fuzzy sets [4]. T. Muthuraji, S. Sriram, P. Murugadas have proposed Decomposition of Intuitionistic Fuzzy Matrices [5]. H. Bustince, E. Barrenechea, M. Pagola, J. Fernandez proposed Interval-valued fuzzy sets constructed from matrices [17]. B. Chetia and P. K. Das. introduced Some Results of Intuitionistic Fuzzy Soft Matrix Theory [9]. Jian-qiang Wang, Rong- rong Nie, Hong-yu Zhang, Xiao-hong Chen initiated Intuitionistic fuzzy multi-criteria decision- making method based on evidential reasoning [12]. S. Senthilkumar, Eswari Prem and C. Ragavan have explored the concept of Intuitionistic fuzzy translation of anti-intuitionistic fuzzy T-ideals of subtraction BCK/BCI-algebras [20]. D. Pandey and Kamesh Kumar have achieved their results in the Interval Valued Intuitionistic Fuzzy Sets in Medical Diagnosis [22]. For distance measures, further research was done by Khalid and Abbas [23]. Atanassov defined five versions of Cartesian products of two IFSs. The sixth cartesian product over IFSs was defined by Velin Andonov in 2008 [24]. The seventh, eighth and ninth cartesian products over IFSs were defined by Annie Varghese and Sunny Kuriakose in 2012 [25], they also defined the tenth and eleventh Cartesian products over IFSs in the same year [26], that is to say ×7, ×8, ×9, ×10 and ×11, respectively. And the corresponding equations for some operations and relations over IFSs were proved. According to the comparison of interval valued fuzzy sets and intuitionistic fuzzy set, Atanassov introduced and proposed five kinds of Cartesian products of two interval valued intuitionistic fuzzy sets. As we all know, there are eleven mailto:saraswathiamutha@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 51 https://internationalpubls.com kinds of Cartesian products of IFSs. However, there are only five types of Cartesian products of interval valued intuitionistic fuzzy sets. Finally, we prove that interval valued intuitionistic fuzzy matrices is a closed algebraic system for all these operations as fuzzy sets of intuitionistic fuzzy sets, and interval valued intuitionistic fuzzy sets. Therefore, this paper generalizes the interval valued intuitionistic fuzzy set theory and provides some valuable conclusions for the field of application research of interval valued intuitionistic fuzzy matrices and it is also useful to the generalization of interval valued intuitionistic fuzzy reasoning. In this article, the intuitionistic fuzzy set approach using interval belief degrees is introduced to interval valued intuitionistic fuzzy matrices with interval belief structures. Then a fuzzy set analytical algorithm is developed to aggregate decision attributes of alternatives for multicriteria decision making problems with interval valued intuitionistic fuzzy matrices and incomplete decision information. A series of non-linear programming models are constructed based on criteria weights intervals, belief degrees intervals and fuzzy evidential reasoning analytical algorithm, then the genetic algorithm is employed to solve the non-linear models yielding the minimal and maximal fuzzy utilities of each alternative. With our proposed method, procedures involving arithmetic operations in aforementioned literature are not needed, thereby removing the limitations in those works. 2. Objectives • We introduced an operation and relations on the Cartesian product of interval valued intuitionistic fuzzy matrices and some properties of the interval valued intuitionistic fuzzy matrices are discussed. • An interval valued intuitionistic fuzzy matrices play an important role in the field of fuzzy system modelling. An interval valued intuitionistic fuzzy matrices are extension of the ordinary matrices. • Five new operations are introduced over extended intuitionistic fuzzy matrices set and over their simpler cases, such as interval valued intuitionistic fuzzy matrices, extended interval valued intuitionistic fuzzy matrices. • The concepts of an intuitionistic fuzzy set with elements being intuitionistic fuzzy matrix and of an extended interval valued intuitionistic fuzzy matrices with elements being predicates are introduced. • Some operations, relations and operators over these new types of matrices are defined. Some properties of these ideas are discussed and statements are expressed. 3. Methods In this section, several fundamental notions about interval valued intuitionistic fuzzy matrices are discussed. Fuzzy matrices play a vital role in scientific development, A possible application of the newly proposed interval valued intuitionistic fuzzy matrices of the student’s training is discussed. Definition 3.1. A Fuzzy matrix may be matrix that has its parts from [0, 1]. Consider a matrix 𝐴 = [𝑎𝑖𝑗]3×3 where 𝑎𝑖𝑗 ∈ [0,1], 1 ≤ 𝑗 ≤ 𝑛. Then A is a Fuzzy Matrix [FM]. Definition 3.2. An intuitionistic fuzzy set (IFS) A in E is defined as an object of the following form A = {〈𝑥, 𝜇𝐴(𝑥), 𝜈𝐴(𝑥)〉|𝑥 ∈ E} where the functions 𝜇𝐴 ∶ 𝐸 → [0,1] define the membership and the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 52 https://internationalpubls.com degree of non-membership of the element𝑥 ∈ E, respectively, and for every 𝑥 ∈ E, 0 ≤ 𝜇𝐴(𝑥) + 𝜈𝐴(𝑥) ≤ 1. obviously, each ordinary fuzzy set may be written as {〈𝑥, 𝜇𝐴(𝑥), 𝜈𝐴(𝑥)〉|𝑥 ∈ E}. Definition 3.3. The Cartesian products of two IFSs A and B are defined as follows, The Cartesian product “×1” then 𝐴 ×1 𝐵 ={〈〈𝑥, 𝑦〉, 𝜇𝐴(𝑥). 𝜇𝐵(𝑦), 𝜈𝐴(𝑥). 𝜈𝐵(𝑦)〉│𝑥 ∈ 𝐸1&𝑦 ∈ 𝐸2}. Definition 3.4. The Cartesian products of two IFSs A and B are defined as follows. The Cartesian product “×2” then 𝐴 ×2 𝐵 ={〈〈𝑥, 𝑦〉, 𝜇𝐴(𝑥) + 𝜇𝐵(𝑦) − 𝜇𝐴(𝑥). 𝜇𝐵(𝑦), 𝜈𝐴(𝑥). 𝜈𝐵(𝑦)〉│𝑥 ∈ 𝐸1&𝑦 ∈ 𝐸2}. Definition 3.5. The Cartesian products of two IFSs A and B are defined as follows. The Cartesian product “×3” then 𝐴 ×3 𝐵 ={〈〈𝑥, 𝑦〉, 𝜇𝐴(𝑥). 𝜇𝐵(𝑦), 𝜈𝐴(𝑥) + 𝜈𝐵(𝑦) − 𝜈𝐴(𝑥). 𝜈𝐵(𝑦)〉│𝑥 ∈ 𝐸1&𝑦 ∈ 𝐸2}. Definition 3.6. The Cartesian products of two IFSs A and B are defined as follows. The Cartesian product “×4” then 𝐴 ×4 𝐵 ={〈〈𝑥, 𝑦〉,𝑚𝑖𝑛 (𝜇𝐴(𝑥). 𝜇𝐵(𝑦)),𝑚𝑎𝑥(𝜈𝐴(𝑥). 𝜈𝐵(𝑦))〉│𝑥 ∈ 𝐸1&𝑦 ∈ 𝐸2}. Definition 3.7. The Cartesian products of two IFSs A and B are defined as follows, The Cartesian product “×5” then 𝐴 ×5 𝐵 ={〈〈𝑥, 𝑦〉,𝑚𝑎𝑥 (𝜇𝐴(𝑥). 𝜇𝐵(𝑦)),𝑚𝑖𝑛(𝜈𝐴(𝑥). 𝜈𝐵(𝑦))〉│𝑥 ∈ 𝐸1&𝑦 ∈ 𝐸2}. Definition 3.8. An interval valued fuzzy set (IVFS) A (over a basic set E) is specified by a function 𝑀𝐴 ∶ 𝐸 → 𝐼𝑁𝑇 ([0,1]), where 𝐼𝑁𝑇 (𝑋[0,1]) is the set of all intervals within [0,1], for all 𝑥 ∈ 𝐸, 𝑀𝐴(𝑥) is an interval [𝑎, 𝑏], 0 ≤ 𝑎 ≤ 𝑏 ≤ 1. Definition 3.9. An interval valued intuitionistic fuzzy set A is defined by the membership function 𝑀𝐴 ∶ 𝐸 → 𝐼𝑁𝑇 ([0,1]), The non-membership function 𝑁𝐴 ∶ 𝐸 → 𝐼𝑁𝑇 ([0,1]), Where 𝐼𝑁𝑇([0,1]) is the set of all subsets of the unit interval. 4. Results Main Theorem of Cartesian Product of Interval Valued Intuitionistic Fuzzy Matrices Theorem 4.1. If Ạ ×1 Ḅ are an interval valued intuitionistic fuzzy matrices, then Πα,β(Ạ ×1 Ḅ) and Ωα,β(Ạ ×1 Ḅ) is also an interval valued intuitionistic fuzzy matrices. Proof. Let us consider Ạ have 3×4 matrix and Ḅ have 4×3 matrix both are an interval valued intuitionistic fuzzy matrices. Then Πα,β(Ạ ×1 Ḅ) = [ (min [𝛼, 𝜇𝐴 𝐿 . 𝜇𝑩 𝐿 ], min [𝛼, 𝜇𝐴 𝑈. 𝜇𝐵 𝑈])], [(max [β, 𝐴 𝐿 . 𝑩 𝐿 ], max [β, 𝐴 𝑈 . 𝐵 𝑈])] and Ωα,β(Ạ ×1 Ḅ) = [ (max [𝛼, 𝜇𝐴 𝐿 . 𝜇𝑩 𝐿 ], max [𝛼, 𝜇𝐴 𝑈. 𝜇𝐵 𝑈])], [(min [β, 𝐴 𝐿 . 𝑩 𝐿 ], min [β, 𝐴 𝑈 . 𝐵 𝑈])]. Ạ ×1 Ḅ = ( 𝜏11 𝜏12 𝜏13 𝜏21 𝜏22 𝜏23 𝜏31 𝜏32 𝜏33 ) and Ạ ×1 Ḅ = ( 𝜔11 𝜔12 𝜔13 𝜔21 𝜔22 𝜔23 𝜔31 𝜔32 𝜔33 ) Πα,β(Ạ×1Ḅ)= ( [𝜖11 𝜃11 𝜎11 𝜋11] [𝜖12 𝜃12 𝜎12 𝜋12] [𝜖13 𝜃13 𝜎13 𝜋13] [𝜖21 𝜃21 𝜎21 𝜋21] [𝜖22 𝜃22 𝜎22 𝜋22] [𝜖23 𝜃23 𝜎23 𝜋23] [𝜖31 𝜃31 𝜎31 𝜋31] [𝜖32 𝜃32 𝜎32 𝜋32] [𝜖33 𝜃33 𝜎33 𝜋33] [𝜖14 𝜃14 𝜎14 𝜋14] [𝜖24 𝜃24 𝜎24 𝜋24] [𝜖34 𝜃34 𝜎34 𝜋34] ) ×1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 53 https://internationalpubls.com ( [11 𝜌11  11  11] [12 𝜌12  12  12] [13 𝜌13  13  13] [21 𝜌21  21  21] [22 𝜌22  22  22] [23 𝜌23  23  23] [31 𝜌31  31  31] [32 𝜌32  32  32] [33 𝜌33  33  33] [41 𝜌41  41  41] [42 𝜌42  42  42] [43 𝜌43  43  43]) 𝜏11 = [min (α, 𝜖11 ∙ 11), min (α, 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 ∙ 11), max (β, 𝜋11 ∙ 11)] +[min (α, 𝜖12 ∙  21 ), min (α, 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 ∙ 21), max (β, 𝜋12 ∙ 21)] +[min (α, 𝜖13 ∙ 31), min (α, 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 ∙ 31), max (β, 𝜋13 ∙ 31)] +[min (α, 𝜖14 ∙ 41), min (α, 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 ∙  41 ), max (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max ([min (α, 𝜖11 ∙ 11), min (α, 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 ∙ 11), max (β, 𝜋11 ∙ 11)], [min (α, 𝜖12 ∙ 21), min (α, 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 ∙ 21), max (β, 𝜋12 ∙ 21)]) +[min (α, 𝜖13 ∙ 31), min (α, 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 ∙ 31), max (β, 𝜋13 ∙ 31)] +[min (α, 𝜖14 ∙ 41), min (α, 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 ∙ 41), max (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max (max ([min (α, 𝜖11 ∙ 11), min (α, 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 ∙ 11), max (β, 𝜋11 ∙ 11)],[min (α, 𝜖12 ∙ 21), min (α, 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 ∙ 21), max (β, 𝜋12 ∙ 21)]), [min (α, 𝜖13 ∙ 31), min (α, 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 ∙ 31), max (β, 𝜋13 ∙ 31)])+ [min (α, 𝜖14 ∙ 41), min (α, 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 ∙ 41), max (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max (max (max ([min (α, 𝜖11 ∙ 11), min (α, 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 ∙ 11), max (β, 𝜋11 ∙ 11)], [min (α, 𝜖12 ∙ 21), min (α, 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 ∙ 21), max (β, 𝜋12 ∙ 21)]),[min (α, 𝜖13 ∙ 31), min (α, 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 ∙ 31), max (β, 𝜋13 ∙ 31)]), [min (α, 𝜖14 ∙ 41), min (α, 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 ∙ 41), max (β, 𝜋14 ∙ 41)]). Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. And 𝜔11 = [ max (α, 𝜖11 ∙ 11), max (α, 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 ∙ 11), min (β, 𝜋11 ∙ 11)] + [max (α, 𝜖12 ∙  21 ), max (α, 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 ∙ 21), min (β, 𝜋12 ∙ 21)] +[max (α, 𝜖13 ∙ 31), max (α, 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 ∙ 31), min (β, 𝜋13 ∙ 31)] +[max (α, 𝜖14 ∙ 41), max (α, 𝜃14 ∙ 𝜌41)], [min (β, 𝜎14 ∙  41 ), min (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max ([ max (α, 𝜖11 ∙ 11), max (α, 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 ∙ 11), min (β, 𝜋11 ∙ 11)],[max (α, 𝜖12 ∙ 21), max (α, 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 ∙ 21), min (β, 𝜋12 ∙ 21)]) + [max (α, 𝜖13 ∙ 31), max (α, 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 ∙ 31), min (β, 𝜋13 ∙ 31)] +[max (α, 𝜖14 ∙ 41), max (α, 𝜃14 ∙ 𝜌41)], [min (β, 𝜎14 ∙ 41), min (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max (max ([ max (α, 𝜖11 ∙ 11), max (α, 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 ∙ 11), min (β, 𝜋11 ∙ 11)], [max (α, 𝜖12 ∙ 21), max (α, 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 ∙ 21), min (β, 𝜋12 ∙ 21)]),[max (α, 𝜖13 ∙ 31), max (α, 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 ∙ 31), min (β, 𝜋13 ∙ 31)]) + [max (α, 𝜖14 ∙ 41), max (α, 𝜃14 ∙ 𝜌41)], [min Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 54 https://internationalpubls.com (β, 𝜎14 ∙ 41), min (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max (max (max ([ max (α, 𝜖11 ∙ 11), max (α, 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 ∙ 11), min (β, 𝜋11 ∙  11 )],[max (α, 𝜖12 ∙ 21), max (α, 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 ∙ 21), min (β, 𝜋12 ∙ 21)]), [max (α, 𝜖13 ∙  31 ), max (α, 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 ∙ 31), min (β, 𝜋13 ∙ 31)]), [max (α, 𝜖14 ∙ 41), max (α, 𝜃14 ∙ 𝜌41)], [min (β, 𝜎14 ∙ 41), min (β, 𝜋14 ∙ 41)]). Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. Hence Πα,β(Ạ ×1 Ḅ) and Ωα,β(Ạ ×1 Ḅ) is an interval valued intuitionistic fuzzy matrices. Theorem 4.2. If Ạ ×2 Ḅ are an interval valued intuitionistic fuzzy matrices, then Πα,β(Ạ ×2 Ḅ) and Ωα,β(Ạ ×2 Ḅ) is also an interval valued intuitionistic fuzzy matrices. Proof. Let us consider Ạ have 3×4 matrix and Ḅ have 4×3 matrix both are an interval valued intuitionistic fuzzy matrices. Then Πα,β(Ạ ×2 Ḅ) = [(min [𝛼, 𝜇𝐴 𝐿+ 𝜇𝑩 𝐿 − 𝜇𝐴 𝐿 ∙ 𝜇𝑩 𝐿 ], min [𝛼, 𝜇𝐴 𝑈 + 𝜇𝐵 𝑈 − 𝜇𝐴 𝑈. 𝜇𝐵 𝑈])], [(max [β,𝐴 𝐿 . 𝑩 𝐿 ], max [β, 𝐴 𝑈. 𝐵 𝑈 ])] and Ωα,β(Ạ ×2 Ḅ) = [(max [𝛼, 𝜇𝐴 𝐿 + 𝜇𝑩 𝐿 − 𝜇𝐴 𝐿 ∙ 𝜇𝑩 𝐿 ], max [𝛼, 𝜇𝐴 𝑈 + 𝜇𝐵 𝑈 − 𝜇𝐴 𝑈. 𝜇𝐵 𝑈])], [(min [β, 𝐴 𝐿 . 𝑩 𝐿 ], min [β, 𝐴 𝑈. 𝐵 𝑈 ])]. Ạ ×2 Ḅ = ( 𝜏11 𝜏12 𝜏13 𝜏21 𝜏22 𝜏23 𝜏31 𝜏32 𝜏33 ) and Ạ ×2 Ḅ = ( 𝜔11 𝜔12 𝜔13 𝜔21 𝜔22 𝜔23 𝜔31 𝜔32 𝜔33 ) Πα,β(Ạ×2Ḅ)= ( [𝜖11 𝜃11 𝜎11 𝜋11] [𝜖12 𝜃12 𝜎12 𝜋12] [𝜖13 𝜃13 𝜎13 𝜋13] [𝜖21 𝜃21 𝜎21 𝜋21] [𝜖22 𝜃22 𝜎22 𝜋22] [𝜖23 𝜃23 𝜎23 𝜋23] [𝜖31 𝜃31 𝜎31 𝜋31] [𝜖32 𝜃32 𝜎32 𝜋32] [𝜖33 𝜃33 𝜎33 𝜋33] [𝜖14 𝜃14 𝜎14 𝜋14] [𝜖24 𝜃24 𝜎24 𝜋24] [𝜖34 𝜃34 𝜎34 𝜋34] ) ×2 ( [11 𝜌11  11  11] [12 𝜌12  12  12] [13 𝜌13  13  13] [21 𝜌21  21  21] [22 𝜌22  22  22] [23 𝜌23  23  23] [31 𝜌31  31  31] [32 𝜌32  32  32] [33 𝜌33  33  33] [41 𝜌41  41  41] [42 𝜌42  42  42] [43 𝜌43  43  43]) 𝜏11 = [min (α, 𝜖11 +  11 − 𝜖11 ∙ 11 ), min (α, 𝜃11 + 𝜌11 − 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 ∙ 11), max (β, 𝜋11 ∙ 11)] +[min (α, 𝜖12 +  21 − 𝜖12 ∙ 21), min (α, 𝜃12 + 𝜌21 − 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 ∙  21 ), max (β, 𝜋12 ∙ 21)] +[min (α, 𝜖13 +  31 − 𝜖13 ∙ 31), min (α, 𝜃13 + 𝜌31 − 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 ∙ 31), max (β, 𝜋13 ∙ 31)] +[min (α, 𝜖14 +  41 − 𝜖14 ∙ 41), min (α, 𝜃14 + 𝜌41 − 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 ∙ 41), max (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max ([min (α, 𝜖11 +  11 − 𝜖11 ∙ 11 ), min (α, 𝜃11 + 𝜌11 − 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 ∙ 11), max (β, 𝜋11 ∙ 11)],[min (α, 𝜖12 +  21 − 𝜖12 ∙ 21), min (α, 𝜃12 + 𝜌21 − 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 ∙ 21), max (β, 𝜋12 ∙ 21)]) +[min (α, 𝜖13 +  31 − 𝜖13 ∙ 31), min (α, 𝜃13 + 𝜌31 − 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 ∙  31 ), max (β, 𝜋13 ∙ 31)] +[min (α, 𝜖14 +  41 − 𝜖14 ∙ 41), min (α, 𝜃14 + 𝜌41 − 𝜃14 ∙ 𝜌41)], [max (β, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 55 https://internationalpubls.com 𝜎14 ∙ 41), max (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max (max ([min (α, 𝜖11 +  11 − 𝜖11 ∙ 11 ), min (α, 𝜃11 + 𝜌11 − 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 ∙ 11), max (β, 𝜋11 ∙ 11)],[min (α, 𝜖12 +  21 − 𝜖12 ∙ 21), min (α, 𝜃12 + 𝜌21 − 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 ∙  21 ), max (β, 𝜋12 ∙ 21)]),[min (α, 𝜖13 +  31 − 𝜖13 ∙ 31), min (α, 𝜃13 + 𝜌31 − 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 ∙ 31), max (β, 𝜋13 ∙ 31)]) +[min (α, 𝜖14 +  41 − 𝜖14 ∙ 41), min (α, 𝜃14 + 𝜌41 − 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 ∙ 41), max (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max (max (max ([min (α, 𝜖11 +  11 − 𝜖11 ∙ 11 ), min (α, 𝜃11 + 𝜌11 − 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 ∙  11 ), max (β, 𝜋11 ∙ 11)], [min (α, 𝜖12 +  21 − 𝜖12 ∙ 21), min (α, 𝜃12 + 𝜌21 − 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 ∙ 21), max (β, 𝜋12 ∙ 21)]),[min (α, 𝜖13 +  31 − 𝜖13 ∙ 31), min (α, 𝜃13 + 𝜌31 − 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 ∙ 31), max (β, 𝜋13 ∙ 31)]),[min (α, 𝜖14 +  41 − 𝜖14 ∙ 41), min (α, 𝜃14 + 𝜌41 − 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 ∙ 41), max (β, 𝜋14 ∙ 41)]). Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. And 𝜔11 = [max (α, 𝜖11 +  11 − 𝜖11 ∙ 11 ), max (α, 𝜃11 + 𝜌11 − 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 ∙ 11), min (β, 𝜋11 ∙ 11)] +[max (α, 𝜖12 +  21 − 𝜖12 ∙ 21), max (α, 𝜃12 + 𝜌21 − 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 ∙ 21), min (β, 𝜋12 ∙ 21)] + [max (α, 𝜖13 +  31 − 𝜖13 ∙ 31), max (α, 𝜃13 + 𝜌31 − 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 ∙ 31), min (β, 𝜋13 ∙ 31)] +[max (α, 𝜖14 +  41 − 𝜖14 ∙ 41), max (α, 𝜃14 + 𝜌41 − 𝜃14 ∙ 𝜌41)], [min (β, 𝜎14 ∙  41 ), min (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max ([max (α, 𝜖11 +  11 − 𝜖11 ∙ 11 ), max (α, 𝜃11 + 𝜌11 − 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 ∙ 11), min (β, 𝜋11 ∙ 11)],[max (α, 𝜖12 +  21 − 𝜖12 ∙ 21), max (α, 𝜃12 + 𝜌21 − 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 ∙ 21), min (β, 𝜋12 ∙ 21)]) +[max (α, 𝜖13 +  31 − 𝜖13 ∙ 31), max (α, 𝜃13 + 𝜌31 − 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 ∙  31 ), min (β, 𝜋13 ∙ 31)] +[max (α, 𝜖14 +  41 − 𝜖14 ∙ 41), max (α, 𝜃14 + 𝜌41 − 𝜃14 ∙ 𝜌41)], [min (β, 𝜎14 ∙ 41), min (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max (max ([max (α, 𝜖11 +  11 − 𝜖11 ∙ 11 ), max (α, 𝜃11 + 𝜌11 − 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 ∙ 11), min (β, 𝜋11 ∙ 11)],[max (α, 𝜖12 +  21 − 𝜖12 ∙ 21), max (α, 𝜃12 + 𝜌21 − 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 ∙  21 ), min (β, 𝜋12 ∙ 21)]),[max (α, 𝜖13 +  31 − 𝜖13 ∙ 31), max (α, 𝜃13 + 𝜌31 − 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 ∙ 31), min (β, 𝜋13 ∙ 31)]) +[max (α, 𝜖14 +  41 − 𝜖14 ∙ 41), max (α, 𝜃14 + 𝜌41 − 𝜃14 ∙ 𝜌41)], [min (β, 𝜎14 ∙ 41), min (β, 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max (max (max ([max (α, 𝜖11 +  11 − 𝜖11 ∙ 11 ), max (α, 𝜃11 + 𝜌11 − 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 ∙ 11), min (β, 𝜋11 ∙ 11)],[max (α, 𝜖12 +  21 − 𝜖12 ∙ 21), max (α, 𝜃12 + 𝜌21 − 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 ∙ 21), min (β, 𝜋12 ∙ 21)]),[max (α, 𝜖13 +  31 − 𝜖13 ∙ 31), max (α, 𝜃13 + 𝜌31 − 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 ∙ 31), min (β, 𝜋13 ∙ 31)]),[max (α, 𝜖14 +  41 − 𝜖14 ∙ 41), max (α, 𝜃14 + 𝜌41 − 𝜃14 ∙ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 56 https://internationalpubls.com 𝜌41)], [min (β, 𝜎14 ∙ 41), min (β, 𝜋14 ∙ 41)]). Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. Hence Πα,β(Ạ ×2 Ḅ) and Ωα,β(Ạ ×2 Ḅ) is an interval valued intuitionistic fuzzy matrices. Theorem 4.3. If Ạ ×3 Ḅ are an interval valued intuitionistic fuzzy matrices, then Πα,β(Ạ ×3 Ḅ) and Ωα,β(Ạ ×3 Ḅ) is also an interval valued intuitionistic fuzzy matrices. Proof. Let us consider Ạ have 3×4 matrix and Ḅ have 4×3 matrix both are an interval valued intuitionistic fuzzy matrices. Then Πα,β(Ạ ×3 Ḅ) = [(min [𝛼, 𝜇𝐴 𝐿 ∙ 𝜇𝑩 𝐿 ], min [𝛼, 𝜇𝐴 𝑈. 𝜇𝐵 𝑈])], [(max [β, 𝐴 𝐿 + 𝐵 𝐿 − 𝐴 𝐿 . 𝐵 𝐿 ], max [β, 𝐴 𝑈 + 𝐵 𝑈 − 𝐴 𝑈. 𝐵 𝑈 ])] and Ωα,β(Ạ ×3 Ḅ) = [(max [𝛼, 𝜇𝐴 𝐿 ∙ 𝜇𝑩 𝐿 ], max [𝛼, 𝜇𝐴 𝑈. 𝜇𝐵 𝑈])], [(min [β, 𝐴 𝐿 + 𝐵 𝐿 − 𝐴 𝐿 . 𝐵 𝐿 ], min [β, 𝐴 𝑈 + 𝐵 𝑈 − 𝐴 𝑈. 𝐵 𝑈 ])]. Ạ ×3 Ḅ = ( 𝜏11 𝜏12 𝜏13 𝜏21 𝜏22 𝜏23 𝜏31 𝜏32 𝜏33 ) and Ạ ×3 Ḅ = ( 𝜔11 𝜔12 𝜔13 𝜔21 𝜔22 𝜔23 𝜔31 𝜔32 𝜔33 ) Πα,β(Ạ×3Ḅ)= ( [𝜖11 𝜃11 𝜎11 𝜋11] [𝜖12 𝜃12 𝜎12 𝜋12] [𝜖13 𝜃13 𝜎13 𝜋13] [𝜖21 𝜃21 𝜎21 𝜋21] [𝜖22 𝜃22 𝜎22 𝜋22] [𝜖23 𝜃23 𝜎23 𝜋23] [𝜖31 𝜃31 𝜎31 𝜋31] [𝜖32 𝜃32 𝜎32 𝜋32] [𝜖33 𝜃33 𝜎33 𝜋33] [𝜖14 𝜃14 𝜎14 𝜋14] [𝜖24 𝜃24 𝜎24 𝜋24] [𝜖34 𝜃34 𝜎34 𝜋34] ) ×3 ( [11 𝜌11  11  11] [12 𝜌12  12  12] [13 𝜌13  13  13] [21 𝜌21  21  21] [22 𝜌22  22  22] [23 𝜌23  23  23] [31 𝜌31  31  31] [32 𝜌32  32  32] [33 𝜌33  33  33] [41 𝜌41  41  41] [42 𝜌42  42  42] [43 𝜌43  43  43]) 𝜏11 = [min (α, 𝜖11 ∙ 11 ), min (α, 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 +  11 − 𝜎11 ∙ 11), max (β, 𝜋11 +  11 − 𝜋11 ∙ 11)] +[min (α, 𝜖12 ∙ 21), min (α, 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 +  21 − 𝜎12 ∙ 21), max (β, 𝜋12 +  21 − 𝜋12 ∙ 21)] +[min (α, 𝜖13 ∙ 31), min (α, 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 +  31 − 𝜎13 ∙ 31), max (β, 𝜋13 +  31 − 𝜋13 ∙ 31)] +[min (α, 𝜖14 ∙ 41), min (α, 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 +  41 − 𝜎14 ∙ 41), max (β, 𝜋14 +  41 − 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max ([min (α, 𝜖11 ∙ 11 ), min (α, 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 +  11 − 𝜎11 ∙ 11), max (β, 𝜋11 +  11 − 𝜋11 ∙ 11)],[min (α, 𝜖12 ∙ 21), min (α, 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 +  21 − 𝜎12 ∙ 21), max (β, 𝜋12 +  21 − 𝜋12 ∙ 21)]) +[min (α, 𝜖13 ∙ 31), min (α, 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 +  31 − 𝜎13 ∙ 31), max (β, 𝜋13 +  31 − 𝜋13 ∙ 31)] +[min (α, 𝜖14 ∙ 41), min (α, 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 +  41 − 𝜎14 ∙ 41), max (β, 𝜋14 +  41 − 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max (max ([min (α, 𝜖11 ∙ 11 ), min (α, 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 +  11 − 𝜎11 ∙ 11), max (β, 𝜋11 +  11 − 𝜋11 ∙ 11)],[min (α, 𝜖12 ∙ 21), min (α, 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 +  21 − 𝜎12 ∙ 21), max (β, 𝜋12 +  21 − 𝜋12 ∙ 21)]),[min (α, 𝜖13 ∙ 31), min (α, 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 +  31 − 𝜎13 ∙ 31), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 57 https://internationalpubls.com max (β, 𝜋13 +  31 − 𝜋13 ∙ 31)]) +[min (α, 𝜖14 ∙ 41), min (α, 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 +  41 − 𝜎14 ∙  41 ), max (β, 𝜋14 +  41 − 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max (max (max ([min (α, 𝜖11 ∙ 11 ), min (α, 𝜃11 ∙ 𝜌11)], [max (β, 𝜎11 +  11 − 𝜎11 ∙ 11), max (β, 𝜋11 +  11 − 𝜋11 ∙ 11)],[min (α, 𝜖12 ∙ 21), min (α, 𝜃12 ∙ 𝜌21)], [max (β, 𝜎12 +  21 − 𝜎12 ∙ 21), max (β, 𝜋12 +  21 − 𝜋12 ∙ 21)]),[min (α, 𝜖13 ∙ 31), min (α, 𝜃13 ∙ 𝜌31)], [max (β, 𝜎13 +  31 − 𝜎13 ∙  31 ), max (β, 𝜋13 +  31 − 𝜋13 ∙ 31)]),[min (α, 𝜖14 ∙ 41), min (α, 𝜃14 ∙ 𝜌41)], [max (β, 𝜎14 +  41 − 𝜎14 ∙ 41), max (β, 𝜋14 +  41 − 𝜋14 ∙ 41)]). Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. And 𝜔11 = [max (α, 𝜖11 ∙ 11 ), max (α, 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 +  11 − 𝜎11 ∙ 11), min (β, 𝜋11 +  11 − 𝜋11 ∙ 11)] +[max (α, 𝜖12 ∙ 21), max (α, 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 +  21 − 𝜎12 ∙ 21), min (β, 𝜋12 +  21 − 𝜋12 ∙ 21)] +[max (α, 𝜖13 ∙ 31), max (α, 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 +  31 − 𝜎13 ∙ 31), min (β, 𝜋13 +  31 − 𝜋13 ∙ 31)] +[max (α, 𝜖14 ∙ 41), max (α, 𝜃14 ∙ 𝜌41)], [min (β, 𝜎14 +  41 − 𝜎14 ∙ 41), min (β, 𝜋14 +  41 − 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max ([max (α, 𝜖11 ∙ 11 ), max (α, 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 +  11 − 𝜎11 ∙ 11), min (β, 𝜋11 +  11 − 𝜋11 ∙ 11)],[max (α, 𝜖12 ∙ 21), max (α, 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 +  21 − 𝜎12 ∙ 21), min (β, 𝜋12 +  21 − 𝜋12 ∙ 21)]) +[max (α, 𝜖13 ∙ 31), max (α, 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 +  31 − 𝜎13 ∙ 31), min (β, 𝜋13 +  31 − 𝜋13 ∙ 31)] +[max (α, 𝜖14 ∙ 41), max (α, 𝜃14 ∙ 𝜌41)], [min (β, 𝜎14 +  41 − 𝜎14 ∙ 41), min (β, 𝜋14 +  41 − 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max (max ([max (α, 𝜖11 ∙ 11 ), max (α, 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 +  11 − 𝜎11 ∙ 11), min (β, 𝜋11 +  11 − 𝜋11 ∙ 11)],[max (α, 𝜖12 ∙ 21), max (α, 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 +  21 − 𝜎12 ∙ 21), min (β, 𝜋12 +  21 − 𝜋12 ∙ 21)]),[max (α, 𝜖13 ∙ 31), max (α, 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 +  31 − 𝜎13 ∙ 31), min (β, 𝜋13 +  31 − 𝜋13 ∙ 31)]) +[max (α, 𝜖14 ∙ 41), max (α, 𝜃14 ∙ 𝜌41)], [min (β, 𝜎14 +  41 − 𝜎14 ∙  41 ), min (β, 𝜋14 +  41 − 𝜋14 ∙ 41)]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max (max (max ([max (α, 𝜖11 ∙ 11 ), max (α, 𝜃11 ∙ 𝜌11)], [min (β, 𝜎11 +  11 − 𝜎11 ∙ 11), min (β, 𝜋11 +  11 − 𝜋11 ∙ 11)],[max (α, 𝜖12 ∙ 21), max (α, 𝜃12 ∙ 𝜌21)], [min (β, 𝜎12 +  21 − 𝜎12 ∙ 21), min (β, 𝜋12 +  21 − 𝜋12 ∙ 21)]),[max (α, 𝜖13 ∙ 31), max (α, 𝜃13 ∙ 𝜌31)], [min (β, 𝜎13 +  31 − 𝜎13 ∙  31 ), min (β, 𝜋13 +  31 − 𝜋13 ∙ 31)]),[max (α, 𝜖14 ∙ 41), max (α, 𝜃14 ∙ 𝜌41)], [min (β, 𝜎14 +  41 − 𝜎14 ∙ 41), min (β, 𝜋14 +  41 − 𝜋14 ∙ 41)]). Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. Hence Πα,β(Ạ ×3 Ḅ) and Ωα,β(Ạ ×3 Ḅ) is an interval valued intuitionistic fuzzy matrices. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 58 https://internationalpubls.com Theorem 4.4. If Ạ ×4 Ḅ are an interval valued intuitionistic fuzzy matrices, then Πα,β(Ạ ×4 Ḅ) and Ωα,β(Ạ ×4 Ḅ) is also an interval valued intuitionistic fuzzy matrices. Proof. Let us consider Ạ have 3×4 matrix and Ḅ have 4×3 matrix both are an interval valued intuitionistic fuzzy matrices. Then Πα,β(Ạ ×4 Ḅ) = [(min [min (𝛼, 𝜇𝐴 𝐿 , 𝜇𝑩 𝐿 )], min [min (𝛼, 𝜇𝐴 𝑈, 𝜇𝐵 𝑈)])], [(max [max (β,𝐴 𝐿 , 𝐵 𝐿 )], max [max (β, 𝐴 𝑈 , 𝐵 𝑈)])] And Ωα,β(Ạ ×4 Ḅ) = [(max [max (𝛼, 𝜇𝐴 𝐿 , 𝜇𝑩 𝐿 )], max [max (𝛼, 𝜇𝐴 𝑈, 𝜇𝐵 𝑈)])], [(min [min (β,𝐴 𝐿 , 𝐵 𝐿 )], min [min (β, 𝐴 𝑈, 𝐵 𝑈)])]. Ạ ×4 Ḅ = ( 𝜏11 𝜏12 𝜏13 𝜏21 𝜏22 𝜏23 𝜏31 𝜏32 𝜏33 ) and Ạ ×4 Ḅ = ( 𝜔11 𝜔12 𝜔13 𝜔21 𝜔22 𝜔23 𝜔31 𝜔32 𝜔33 ) Πα,β(Ạ×4Ḅ)= ( [𝜖11 𝜃11 𝜎11 𝜋11] [𝜖12 𝜃12 𝜎12 𝜋12] [𝜖13 𝜃13 𝜎13 𝜋13] [𝜖21 𝜃21 𝜎21 𝜋21] [𝜖22 𝜃22 𝜎22 𝜋22] [𝜖23 𝜃23 𝜎23 𝜋23] [𝜖31 𝜃31 𝜎31 𝜋31] [𝜖32 𝜃32 𝜎32 𝜋32] [𝜖33 𝜃33 𝜎33 𝜋33] [𝜖14 𝜃14 𝜎14 𝜋14] [𝜖24 𝜃24 𝜎24 𝜋24] [𝜖34 𝜃34 𝜎34 𝜋34] ) ×4 ( [11 𝜌11  11  11] [12 𝜌12  12  12] [13 𝜌13  13  13] [21 𝜌21  21  21] [22 𝜌22  22  22] [23 𝜌23  23  23] [31 𝜌31  31  31] [32 𝜌32  32  32] [33 𝜌33  33  33] [41 𝜌41  41  41] [42 𝜌42  42  42] [43 𝜌43  43  43]) 𝜏11 = [min (min (α, 𝜖11,11 )), min (min (α, 𝜃11, 𝜌11))], [max (max (β, 𝜎11, 11)), max (max (β, 𝜋11, 11))] +[min (min (α, 𝜖12,21)), min (min (α, 𝜃12, 𝜌21))], [max (max (β, 𝜎12, 21)), max (max (β, 𝜋12, 21))] +[min (min (α, 𝜖13,31)), min (min (α, 𝜃13, 𝜌31))], [max (max (β, 𝜎13, 31)), max (max (β, 𝜋13, 31))] +[min (min (α, 𝜖14,41)), min (min (α, 𝜃14, 𝜌41))], [max (max (β, 𝜎14, 41)), max (max (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max ([min (min (α, 𝜖11,11 )), min (min (α, 𝜃11, 𝜌11))], [max (max (β, 𝜎11, 11)), max (max (β, 𝜋11, 11))],[min (min (α, 𝜖12,21)), min (min (α, 𝜃12, 𝜌21))], [max (max (β, 𝜎12, 21)), max (max (β, 𝜋12, 21))]) +[min (min (α, 𝜖13,31)), min (min (α, 𝜃13, 𝜌31))], [max (max (β, 𝜎13, 31)), max (max (β, 𝜋13, 31))] +[min (min (α, 𝜖14,41)), min (min (α, 𝜃14, 𝜌41))], [max (max (β, 𝜎14, 41)), max (max (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max (max ([min (min (α, 𝜖11,11 )), min (min (α, 𝜃11, 𝜌11))], [max (max (β, 𝜎11, 11)), max (max (β, 𝜋11, 11))],[min (min (α, 𝜖12,21)), min (min (α, 𝜃12, 𝜌21))], [max (max (β, 𝜎12, 21)), max (max (β, 𝜋12, 21))]),[min (min (α, 𝜖13,31)), min (min (α, 𝜃13, 𝜌31))], [max (max (β, 𝜎13, 31)), max (max (β, 𝜋13, 31))]) +[min (min (α, 𝜖14,41)), min (min (α, 𝜃14, 𝜌41))], [max (max (β, 𝜎14, 41)), max (max (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max (max (max ([min (min (α, 𝜖11,11 )), min (min (α, 𝜃11, 𝜌11))], [max (max (β, 𝜎11, 11)), max (max (β, 𝜋11, 11))],[min (min (α, 𝜖12,21)), min (min (α, 𝜃12, 𝜌21))], [max (max (β, 𝜎12, 21)), max (max (β, 𝜋12, 21))]),[min (min (α, 𝜖13,31)), min (min (α, 𝜃13, 𝜌31))], [max (max (β, 𝜎13, 31)), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 59 https://internationalpubls.com max (max (β, 𝜋13, 31))]),[min (min (α, 𝜖14,41)), min (min (α, 𝜃14, 𝜌41))], [max (max (β, 𝜎14, 41)), max (max (β, 𝜋14, 41))]). Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. And 𝜔11 = [max (max (α, 𝜖11,11 )), max (max (α, 𝜃11, 𝜌11))], [min (min (β, 𝜎11, 11)), min (min (β, 𝜋11, 11))] +[max (max (α, 𝜖12,21)), max (max (α, 𝜃12, 𝜌21))], [min (min (β, 𝜎12, 21)), min (min (β, 𝜋12, 21))] +[max (max (α, 𝜖13,31)), max (max (α, 𝜃13, 𝜌31))], [min (min (β, 𝜎13, 31)), min (min (β, 𝜋13, 31))] +[max (max (α, 𝜖14,41)), max (max (α, 𝜃14, 𝜌41))], [min (min (β, 𝜎14, 41)), min (min (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max ([max (max (α, 𝜖11,11 )), max (max (α, 𝜃11, 𝜌11))], [min (min (β, 𝜎11, 11)), min (min (β, 𝜋11, 11))],[max (max (α, 𝜖12,21)), max (max (α, 𝜃12, 𝜌21))], [min (min (β, 𝜎12, 21)), min (min (β, 𝜋12, 21))]) +[max (max (α, 𝜖13,31)), max (max (α, 𝜃13, 𝜌31))], [min (min (β, 𝜎13, 31)), min (min (β, 𝜋13, 31))] +[max (max (α, 𝜖14,41)), max (max (α, 𝜃14, 𝜌41))], [min (min (β, 𝜎14, 41)), min (min (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max (max ([max (max (α, 𝜖11,11 )), max (max (α, 𝜃11, 𝜌11))], [min (min (β, 𝜎11, 11)), min (min (β, 𝜋11, 11))],[max (max (α, 𝜖12,21)), max (max (α, 𝜃12, 𝜌21))], [min (min (β, 𝜎12, 21)), min (min (β, 𝜋12, 21))]),[max (max (α, 𝜖13,31)), max (max (α, 𝜃13, 𝜌31))], [min (min (β, 𝜎13, 31)), min (min (β, 𝜋13, 31))]) +[max (max (α, 𝜖14,41)), max (max (α, 𝜃14, 𝜌41))], [min (min (β, 𝜎14, 41)), min (min (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max (max (max ([max (max (α, 𝜖11,11 )), max (max (α, 𝜃11, 𝜌11))], [min (min (β, 𝜎11, 11)), min (min (β, 𝜋11, 11))],[max (max (α, 𝜖12,21)), max (max (α, 𝜃12, 𝜌21))], [min (min (β, 𝜎12, 21)), min (min (β, 𝜋12, 21))]),[max (max (α, 𝜖13,31)), max (max (α, 𝜃13, 𝜌31))], [min (min (β, 𝜎13, 31)), min (min (β, 𝜋13, 31))]),[max (max (α, 𝜖14,41)), max (max (α, 𝜃14, 𝜌41))], [min (min (β, 𝜎14, 41)), min (min (β, 𝜋14, 41))]). Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. Hence Πα,β(Ạ ×4 Ḅ) and Ωα,β(Ạ ×4 Ḅ) is an interval valued intuitionistic fuzzy matrices. Theorem 4.5. If Ạ ×5 Ḅ are an interval valued intuitionistic fuzzy matrices, then Πα,β(Ạ ×5 Ḅ) and Ωα,β(Ạ ×5 Ḅ) is also an interval valued intuitionistic fuzzy matrices. Proof. Let us consider Ạ have 3×4 matrix and Ḅ have 4×3 matrix both are an interval valued intuitionistic fuzzy matrices. Then Πα,β(Ạ ×5 Ḅ) = [(min [max (𝛼, 𝜇𝐴 𝐿 , 𝜇𝑩 𝐿 )], min [max (𝛼, 𝜇𝐴 𝑈, 𝜇𝐵 𝑈)])], [(max [min (β, 𝐴 𝐿 , 𝐵 𝐿 )], max [min (β, 𝐴 𝑈 , 𝐵 𝑈)])] and Ωα,β(Ạ ×5 Ḅ) = [(max [min (𝛼, 𝜇𝐴 𝐿 , 𝜇𝑩 𝐿 )], max [min (𝛼, 𝜇𝐴 𝑈, 𝜇𝐵 𝑈)])], [(min [max (β,𝐴 𝐿 , 𝐵 𝐿 )], min [max (β, 𝐴 𝑈, 𝐵 𝑈)])]. Ạ ×5 Ḅ = ( 𝜏11 𝜏12 𝜏13 𝜏21 𝜏22 𝜏23 𝜏31 𝜏32 𝜏33 ) and Ạ ×5 Ḅ = ( 𝜔11 𝜔12 𝜔13 𝜔21 𝜔22 𝜔23 𝜔31 𝜔32 𝜔33 ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 60 https://internationalpubls.com Πα,β(Ạ×5Ḅ)= ( [𝜖11 𝜃11 𝜎11 𝜋11] [𝜖12 𝜃12 𝜎12 𝜋12] [𝜖13 𝜃13 𝜎13 𝜋13] [𝜖21 𝜃21 𝜎21 𝜋21] [𝜖22 𝜃22 𝜎22 𝜋22] [𝜖23 𝜃23 𝜎23 𝜋23] [𝜖31 𝜃31 𝜎31 𝜋31] [𝜖32 𝜃32 𝜎32 𝜋32] [𝜖33 𝜃33 𝜎33 𝜋33] [𝜖14 𝜃14 𝜎14 𝜋14] [𝜖24 𝜃24 𝜎24 𝜋24] [𝜖34 𝜃34 𝜎34 𝜋34] ) ×5 ( [11 𝜌11  11  11] [12 𝜌12  12  12] [13 𝜌13  13  13] [21 𝜌21  21  21] [22 𝜌22  22  22] [23 𝜌23  23  23] [31 𝜌31  31  31] [32 𝜌32  32  32] [33 𝜌33  33  33] [41 𝜌41  41  41] [42 𝜌42  42  42] [43 𝜌43  43  43]) 𝜏11 = [min (max (α, 𝜖11,11 )), min (max (α, 𝜃11, 𝜌11))], [max (min (β, 𝜎11, 11)), max (min (β, 𝜋11, 11))] +[min (max (α, 𝜖12,21)), min (max (α, 𝜃12, 𝜌21))], [max (min (β, 𝜎12, 21)), max (min (β, 𝜋12, 21))] +[min (max (α, 𝜖13,31)), min (max (α, 𝜃13, 𝜌31))], [max (min (β, 𝜎13, 31)), max (min (β, 𝜋13, 31))] +[min (max (α, 𝜖14,41)), min (max (α, 𝜃14, 𝜌41))], [max (min (β, 𝜎14, 41)), max (min (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max ([min (max (α, 𝜖11,11 )), min (max (α, 𝜃11, 𝜌11))], [max (min (β, 𝜎11, 11)), max (min (β, 𝜋11, 11))],[min (max (α, 𝜖12,21)), min (max (α, 𝜃12, 𝜌21))], [max (min (β, 𝜎12, 21)), max (min (β, 𝜋12, 21))]) +[min (max (α, 𝜖13,31)), min (max (α, 𝜃13, 𝜌31))], [max (min (β, 𝜎13, 31)), max (min (β, 𝜋13, 31))] +[min (max (α, 𝜖14,41)), min (max (α, 𝜃14, 𝜌41))], [max (min (β, 𝜎14, 41)), max (min (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max (max ([min (max (α, 𝜖11,11 )), min (max (α, 𝜃11, 𝜌11))], [max (min (β, 𝜎11, 11)), max (min (β, 𝜋11, 11))],[min (max (α, 𝜖12,21)), min (max (α, 𝜃12, 𝜌21))], [max (min (β, 𝜎12, 21)), max (min (β, 𝜋12, 21))]),[min (max (α, 𝜖13,31)), min (max (α, 𝜃13, 𝜌31))], [max (min (β, 𝜎13, 31)), max (min (β, 𝜋13, 31))]) +[min (max (α, 𝜖14,41)), min (max (α, 𝜃14, 𝜌41))], [max (min (β, 𝜎14, 41)), max (min (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. 𝜏11 = max (max (max ([min (max (α, 𝜖11,11 )), min (max (α, 𝜃11, 𝜌11))], [max (min (β, 𝜎11, 11)), max (min (β, 𝜋11, 11))],[min (max (α, 𝜖12,21)), min (max (α, 𝜃12, 𝜌21))], [max (min (β, 𝜎12, 21)), max (min (β, 𝜋12, 21))]),[min (max (α, 𝜖13,31)), min (max (α, 𝜃13, 𝜌31))], [max (min (β, 𝜎13, 31)), max (min (β, 𝜋13, 31))]),[min (max (α, 𝜖14,41)), min (max (α, 𝜃14, 𝜌41))], [max (min (β, 𝜎14, 41)), max (min (β, 𝜋14, 41))]). Similarly, we can expand the expression of 𝜏12,𝜏13,𝜏21,𝜏22,𝜏23,𝜏31,𝜏32,𝜏33. And 𝜔11 = [max (min (α, 𝜖11,11 )), max (min (α, 𝜃11, 𝜌11))], [min (max (β, 𝜎11, 11)), min (max (β, 𝜋11, 11))] +[max (min (α, 𝜖12,21)), max (min (α, 𝜃12, 𝜌21))], [min (max (β, 𝜎12, 21)), min (max (β, 𝜋12, 21))] +[max (min (α, 𝜖13,31)), max (min (α, 𝜃13, 𝜌31))], [min (max (β, 𝜎13, 31)), min (max (β, 𝜋13, 31))] +[max (min (α, 𝜖14,41)), max (min (α, 𝜃14, 𝜌41))], [min (max (β, 𝜎14, 41)), min (max (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max ([max (min (α, 𝜖11,11 )), max (min (α, 𝜃11, 𝜌11))], [min (max (β, 𝜎11, 11)), min (max (β, 𝜋11, 11))],[max (min (α, 𝜖12,21)), max (min (α, 𝜃12, 𝜌21))], [min (max (β, 𝜎12, 21)), min (max (β, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 61 https://internationalpubls.com 𝜋12, 21))]) +[max (min (α, 𝜖13,31)), max (min (α, 𝜃13, 𝜌31))], [min (max (β, 𝜎13, 31)), min (max (β, 𝜋13, 31))] +[max (min (α, 𝜖14,41)), max (min (α, 𝜃14, 𝜌41))], [min (max (β, 𝜎14, 41)), min (max (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max (max ([max (min (α, 𝜖11,11 )), max (min (α, 𝜃11, 𝜌11))], [min (max (β, 𝜎11, 11)), min (max (β, 𝜋11, 11))],[max (min (α, 𝜖12,21)), max (min (α, 𝜃12, 𝜌21))], [min (max (β, 𝜎12, 21)), min (max (β, 𝜋12, 21))]),[max (min (α, 𝜖13,31)), max (min (α, 𝜃13, 𝜌31))], [min (max (β, 𝜎13, 31)), min (max (β, 𝜋13, 31))]) +[max (min (α, 𝜖14,41)), max (min (α, 𝜃14, 𝜌41))], [min (max (β, 𝜎14, 41)), min (max (β, 𝜋14, 41))]. Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. 𝜔11 = max (max (max ([max (min (α, 𝜖11,11 )), max (min (α, 𝜃11, 𝜌11))], [min (max (β, 𝜎11, 11)), min (max (β, 𝜋11, 11))],[max (min (α, 𝜖12,21)), max (min (α, 𝜃12, 𝜌21))], [min (max (β, 𝜎12, 21)), min (max (β, 𝜋12, 21))]),[max (min (α, 𝜖13,31)), max (min (α, 𝜃13, 𝜌31))], [min (max (β, 𝜎13, 31)), min (max (β, 𝜋13, 31))]),[max (min (α, 𝜖14,41)), max (min (α, 𝜃14, 𝜌41))], [min (max (β, 𝜎14, 41)), min (max (β, 𝜋14, 41))]). Similarly, we can expand the expression of 𝜔12,𝜔13,𝜔21,𝜔22,𝜔23,𝜔31,𝜔32,𝜔33. Hence Πα,β(Ạ ×5 Ḅ) and Ωα,β(Ạ ×5 Ḅ) is an interval valued intuitionistic fuzzy matrices. Conclusion: In this article some Cartesian product of intuitionistic fuzzy set and the versions of Cartesian product over interval valued intuitionistic fuzzy matrices is presented in this paper. And some properties, applications are discussed. Also, introduce different types of interval valued intuitionistic fuzzy sets, whether a membership function or a non-membership function, they must be a specific number or value, because they are specific in intuitionistic fuzzy sets. Also, we derive the operations cartesian product of interval valued intuitionistic fuzzy matrices. However, they become an interval or range, because of the uncertainty in interval valued intuitionistic fuzzy matrices. We have proved some type of Cartesian product of interval valued intuitionistic fuzzy matrices, which makes it match to some Cartesian product of interval valued intuitionistic fuzzy sets. We should try prove other Cartesian products of interval valued intuitionistic fuzzy sets shall be study in the future. Acknowledgement: Authors would like to thank to the referees for their valuable comments to improve the presentation of the paper. References [1] K. Atanassov. Intuitionistic fuzzy sets: theory and applications (studies in fuzziness and soft computing). Berlin Heidelberg: Springer-Verlag Telos, 1999. [2] K. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 1986, 20(1): 87 – 96. [3] K. Atanassov, G. Gargov. Interval valued intuitionistic fuzzy sets. Fuzzy sets and Systems, 1989, 31(3): 343 – 349. [4] D. S. Zeng, S. L. Hu. On index of interval valued intuitionistic fuzzy sets. Journal of Sichuan Normal University (Natural Science), 2000, 23(5): 476 – 478. [5] T. Muthuraji, S. Sriram, P. Murugadas. Decomposition of Intuitionistic Fuzzy Matrices. Fuzzy Information and Engineering (2016) 8: 345-354 [6] X. H. Yuan, H. X. Li. Cut sets on interval-valued intuitionistic fuzzy sets. Proc. of the 6th IEEE International Conference on Fuzzy Systems and Knowledge Discovery, 2009, 6: 167 – 171. [7] Annie Varghese1. More on Cartesian Products over Intuitionistic Fuzzy Sets. International Mathematical Forum, Vol. 7, 2012, no. 23, 1129 – 1133. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 62 https://internationalpubls.com [8] R. R. Yager. Some aspects of intuitionistic fuzzy sets. Fuzzy Optimization and Decision Making, 2009, 8(1): 67 – 90. [9] B. Chetia and P. K. Das. Some Results of Intuitionistic Fuzzy Soft Matrix Theory. Advances in Applied Science Research, 2012, 3 (1):412-423. [10] Susanta Kumar Khan and Madhumangal Pal. Interval-Valued Intuitionistic Fuzzy Matrices. arXiv:1404.6949v1[cs.DM] 28 Apr 2014. [11] Q. S. Zhang, H. X. Yao, Z. H. Zhang. Some similarity measures of interval-valued intuitionistic fuzzy sets and application to pattern recognition. International Journal of Applied Mechanics and Materials, 2010, 44 – 47(5): 495 – 500. [12] Jian-qiang Wang, Rong-rong Nie, Hong-yu Zhang, Xiao-hong Chen. Intuitionistic fuzzy multi-criteria decision- making method based on evidential reasoning. Applied soft computing 13(2013) 1823-1831. [13] G. W. Wei, X. F. Zhao. Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making. Expert Systems with Applications, 2012, 39(2): 2026 – 2034. [14] Z. S. Xu, R. R. Yager. Intuitionistic and interval-valued intuitionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group. Fuzzy Optimization and Decision Making, 2009, 8(2): 123 – 139. [15] D. F. Li. Topsis-based nonlinear-programming methodology for multiattribute decision making with Interval-valued intuitionistic fuzzy sets. IEEE Trans. on Fuzzy Systems, 2010, 18(2): 299 – 311. [16] D. F. Li. Mathematical-programming approach to matrix games with payoffs represented by Atanassov’s interval- valued intuitionistic fuzzy sets. IEEE Trans. on Fuzzy Systems, 2010, 18(6):1112 – 1128. [17] H. Bustince, E. Barrenechea, M. Pagola, J. Fernandez. Interval-valued fuzzy sets constructed from matrices: Application to edge detection. Fuzzy Sets and Systems 160 (2009) 1819–1840. [18] Zhang Zhenhua a, b Yang Jingyu a Ye Youpei a Zhang QianSheng b. A Generalized Interval Valued Intuitionistic Fuzzy Sets theory. Advanced in Control Engineering and Information Science, (2011) 2037 – 2041. [19] X. Y. Shen, Y. J. Lei, J. X. Hua, et al. Description and reasoning method of uncertain temporal knowledge based on IFTPN. Control and Decision, 2010, 25(10): 1457 – 1462. [20] S. Senthilkumar, Eswari Prem and C. Ragavan. Intuitionistic fuzzy translation of anti-intuitionistic fuzzy T-idelas of subtraction BCK/BCI-algebras. Malaya Journal of Matematik, Vol. 6, No. 3, 701-710, 2018. [21] F. Y. Meng, X. H. Chen. Entropy and similarity measure for Atannasovs interval-valued intuitionistic fuzzy sets and their application. Fuzzy Optimization and Decision Making, 2016, 15(1): 75 – 101. [22] D. Pandey and 1Kamesh Kumar. Interval Valued Intuitionistic Fuzzy Sets in Medical Diagnosis. Journal of International Academy of Physical Sciences pp. 137-147, Vol. 14 No.2 (2010). [23] A. Khalid, M. Abbas. Distance measures and operations in intuitionistic and interval-valued intuitionistic fuzzy soft set theory. International Journal of Fuzzy Systems, 2015, 17(3): 490 – 497. [24] V. Andonov. On some properties of one Cartesian product over intuitionistic fuzzy sets. Notes on Intuitionistic Fuzzy Sets, 2008, 14(1): 12 – 19. [25] A. Varghese, S. Kuriakose. Cartesian product over intuitionistic fuzzy sets. International Journal of Fuzzy Sets, 2012, 2(1): 21 – 27. [26] A. Varghese, S. Kuriakose. More on Cartesian products over intuitionistic fuzzy sets. International Mathematical Forum, 2012, 7(23): 1129 – 1133. [27] H. Bustince, P. Burillo. Vague sets are intuitionistic fuzzy sets. Fuzzy Sets and Systems 79 (1996) 403-405.