IHJPAS. 36(1)2023 367 This work is licensed under a Creative Commons Attribution 4.0 International License Cubic Ideals of TM-algebras Abstract For the generality of fuzzy ideals in TM-algebra, a cubic ideal in this algebra has been studied, such as cubic ideals and cubic T-ideals. Some properties of these ideals are investigated. Also, we show that the cubic T-ideal is a cubic ideal, but the converse is not generally valid. In addition, a cubic sub-algebra is defined, and new relations between the level subset and a cubic sub-algebra are discussed. After that, cubic ideals and cubic T-ideals under homomorphism are studied, and the image (pre-image) of cubic T-ideals is discussed. Finally, the Cartesian product of cubic ideals in Cartesian product TM-algebras is given. We proved that the product of two cubic ideals of the Cartesian product of two TM-algebras is also a cubic ideal. Key words: TM-algebra, cubic T-ideal, cubic ideal, fuzzy ideals. 1.Introduction In 2010 the notion of TM-algebras was introduced by [1] as a generalization of BCK and BCI algebras. After that, many authors studied this structure differently; see [2-6]. The cubic set is an essential concept for generalizing the fuzzy set. So, Jun et al. [7- 8] introduced subalgebras and ideals in BCK/BCI-algebras and discussed the relationship between a cubic subalgebra and a cubic ideal. In [9], Yaqoob et al. introduced the cubic KU-algebra, a generalization of fuzzy KU-ideals of KU-algebras. After that, some authors introduced a cubic set of different structures. See [10-13]. This paper introduces the concept of cubic T-ideals in TM-algebra, and investigate some properties of these ideals. Also, a few relations between a cubic ideal and a cubic T-ideal are discussed. The Cartesian product of cubic T-ideals in Cartesian product TM-algebras is given. doi.org/10.30526/36.1.2920 Article history: Received 26 June 2022, Accepted 23 Augest 2022, Published in January 2023. Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Fatima M. Ghlaim Department of Mathematics, College of Education for Pure Sciences,Ibn Al – Haitham/ University of Baghdad- Iraq. Fa560fa@gmail.com Fatema F.Kareem Department of Mathematics, College of Education for Pure Sciences,Ibn Al – Haitham/ University of Baghdad- Iraq. fatma.f.k@ihcoedu.uobaghdad.edu.iq https://creativecommons.org/licenses/by/4.0/ mailto:Fa560fa@gmail.com mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq IHJPAS. 36(1)2023 368 2. Basic concepts We will recall some concepts related to TM algebra and cubic sets. Definition (1)[1]. A TM-algebra is a nonempty subset with a constant “0” and a binary operation “*” satisfying the following: (tm1) 𝜌 ∗ 0 = 𝜌, (tm2)(𝜌 ∗ 𝜏) ∗ (𝜌 ∗ ) = ∗ 𝜏,∀ 𝜌, 𝜏, ∈ ℵ . For ℵ we can define a binary operation ≤ by 𝜌 ≤ 𝜏 if and only if 𝜌 ∗ 𝜏 = 0. For any TM-algebra(ℵ,∗ ,0),the following axioms hold.∀ 𝜌, 𝜏 , ∈ ℵ a) 𝜌 ∗ 𝜌 = 0, b) (𝜌 ∗ 𝜏) ∗ 𝜌 = 0 ∗ 𝜏, c) 𝜌 ∗ (𝜌 ∗ 𝜏) = 𝜏, d) (𝜌 ∗ ) ∗ (𝜏 ∗ ) ≤ 𝜌 ∗ 𝜏, e) (𝜌 ∗ 𝜏) ∗ = (𝜌 ∗ ) ∗ 𝜏, f) 𝜌 ∗ 0 = 0 ⇒ 𝜌 = 0, g) 𝜌 ≤ 𝜏 ⇒ 𝜌 ∗ ≤ 𝜏 ∗ 𝑎𝑛𝑑 ∗ 𝜏 ≤ ∗ 𝜌, h) 𝜌 ∗ (𝜌 ∗ (𝜌 ∗ 𝜏)) = 𝜌 ∗ 𝜏, i) 0 ∗ (𝜌 ∗ 𝜏) = 𝜏 ∗ 𝜌 = (0 ∗ 𝜌) ∗ (0 ∗ 𝜏), j) (𝜌 ∗ (𝜌 ∗ 𝜏)) ∗ 𝜏 = 0, k) If 𝜌 ∗ 𝜏 = 0 and 𝜏 ∗ 𝜌 = 0 imply 𝜌 = 𝜏. Example (2) [1]. Letℵ= {0,1,2,3}be a set with the following table. * 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 Then,(ℵ,∗ ,0) is a TM-algebra. Definition (3) [2]. A non-empty subset 𝑆 of a TM-algebra (ℵ,∗ ,0) is called a TM-subalgebra ofℵif𝜌 ∗ 𝜏 ∈ 𝑆whenever 𝜌, 𝜏 ∈ 𝑆. Definition (4) [2]. A non-empty subset 𝜓 ofanTM-algebra (ℵ,∗ ,0) is said to be an ideal of ℵif it satisfies, for any 𝜌, 𝜏 ∈ 𝜓 i) 0 ∈ 𝜓, IHJPAS. 36(1)2023 369 ii) 𝜌 ∗ 𝜏 ∈ 𝜓 and 𝜏 ∈ 𝜓 implies that 𝜌 ∈ 𝜓. Example (5) [2]. Letℵ = {0, 1 , 2 , 3} be a set with a binary operation∗defined inthe following Table: * 0 1 2 3 0 0 0 3 2 1 1 0 3 2 2 2 2 0 3 3 3 3 b 0 Then (ℵ,∗ ,0)is a TM-algebra and 𝜓 = {0,1} is an ideal of ℵ. Definition (6) [1]. A non-empty subset 𝐸 of a TM-algebra ℵ is a T-ideal, if i) 0 ∈ 𝐸 ii) ∀ 𝜌, 𝜏, ∈ ℵ, (𝜌 ∗ 𝜏) ∗ ∈ 𝐸 and 𝜏 ∈ 𝐸 imply (𝜌 ∗ ) ∈ 𝐸. Definition (7) [5]. Let (ℵ ,∗ ,0) 𝑎𝑛𝑑 (ℵ′,∗′, 0′)be a TM-algebras. A homomorphism is a map 𝑓: ℵ → ℵ′ satisfying 𝑓(𝜌 ∗ 𝜏) = 𝑓(𝜌) ∗′ 𝑓(𝜏), for all 𝜌, 𝜏 ∈ ℵ. Now, we review an interval-valued fuzzy set concepts. Definition (8) [5]. Let �̃� = [𝑎𝐿 , 𝑎𝑈 ] be an interval number, where 0 ≤ 𝑎𝐿 ≤ 𝑎𝑈 ≤ 1 and let 𝐷[0,1]be denoted the family of all closed subinterval of [0,1], that is , 𝐷[0,1] = {�̃� = [𝑎𝐿 , 𝑎𝑈 ] ∶ 𝑎𝐿 ≤ 𝑎𝑈 , 𝑓𝑜𝑟 𝑎𝐿 ≤ 𝑎𝑈 ∈ [0,1]} . The operations≥ , ≤ , = , 𝑟𝑚𝑖𝑛, 𝑎𝑛𝑑 𝑟𝑚𝑎𝑥 of two elements in 𝐷[0,1] is defined as follows: let �̃� = [𝑎𝐿 , 𝑎𝑈 ], �̃� = [𝑏𝐿 , 𝑏𝑈 ] in 𝐷[0,1], then (1) �̃� ≥ �̃� 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑎𝐿 ≥ 𝑏𝐿 𝑎𝑛𝑑 𝑎𝑈 ≥ 𝑏𝑈 , (2) �̃� ≤ �̃� 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑎𝐿 ≤ 𝑏𝐿 𝑎𝑛𝑑 𝑎𝑈 ≤ 𝑏𝑈 , (3) �̃� = �̃� 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑎𝐿 = 𝑏𝐿 𝑎𝑛𝑑 𝑎𝑈 = 𝑎𝑈, (4) 𝑟𝑚𝑖𝑛{�̃�, �̃�} = [𝑚𝑖𝑛{𝑎𝐿 , 𝑏𝐿 }, 𝑚𝑖𝑛{𝑎𝑈 , 𝑏𝑈 }], (5) 𝑟𝑚𝑎𝑥{�̃�, �̃�} = [𝑚𝑎𝑥{𝑎𝐿 , 𝑏𝐿 }, 𝑚𝑎𝑥{𝑎𝑈 , 𝑏𝑈 }], And if �̃�𝑖 ∈ 𝐷[0,1] where 𝑖 ∈ ⋀. We define 𝑟 inf 𝑖∈⋀ �̃�𝑖 = [inf 𝑖∈⋀ 𝑎𝑖 𝐿 , inf 𝑖∈⋀ 𝑎𝑖 𝑈 ] , 𝑟 sup 𝑖∈⋀ �̃�𝑖 = [sup 𝑖∈⋀ 𝑎𝑖 𝐿 , sup 𝑖∈⋀ 𝑎𝑖 𝑈 ] . An interval-valued fuzzy set 𝑉 =< 𝜌, �̃�(𝜌) > on ℵ is defined as �̃�(𝜌) = {〈𝜌, [𝜗𝐿 (𝜌), 𝜗𝐿 (𝜌)]〉 ∶ 𝜌 ∈ ℵ}, where 𝜗𝐿 (𝜌) ≤ 𝜗𝑈 (𝜌), for all 𝜌 ∈ ℵ. Then,𝜗𝐿 (𝜌) ∶ ℵ → [0,1] 𝑎𝑛𝑑 𝜗𝑈 ∶ ℵ → [0,1] are called a lower fuzzy set and an upper fuzzy set of 𝜗,̃ respectively. Definition (9) [5]. Let (ℵ,∗ ,0) be a TM-algebra and �̃�: ℵ → 𝐷[0,1]. Then, 𝑉 =< 𝜌, �̃�(𝜌) > is called an interval valued fuzzy sub TM-algebra ℵ, if IHJPAS. 36(1)2023 370 �̃�(𝜌 ∗ 𝛾) ≥ 𝑟𝑚𝑖𝑛{�̃�(𝜌), �̃�(𝛾)}, ∀𝜌, 𝛾 ∈ ℵ. Definition (10) [5]. Let(ℵ,∗ ,0) be a TM-algebra and �̃�: ℵ → 𝐷[0,1]. Then 𝑉 =< 𝜌, �̃�(𝜌) > is said to be an interval valued fuzzy ideal if (i1) �̃�(0) ≥ �̃�(𝜌), ∀𝜌 ∈ ℵ, (i2) For all 𝜌, 𝛾 ∈ ℵ, �̃�(𝜌) ≥ 𝑟𝑚𝑖𝑛{�̃�(𝜌 ∗ 𝛾), �̃�(𝛾)}. Definition (11) [5]. Let(ℵ,∗ ,0) be a TM-algebra and �̃�: ℵ → 𝐷[0,1]. Then 𝑉 =< 𝜌, �̃�(𝜌) >is said to bean interval valued fuzzy T-ideal if (i1) �̃�(0) ≥ �̃�(𝜌), ∀𝜌 ∈ ℵ, (i2) For all 𝜌, 𝛾, ∈ ℵ, �̃�(𝜌 ∗ ) ≥ 𝑟𝑚𝑖𝑛{�̃�((𝜌 ∗ 𝛾) ∗ ), �̃�(𝛾)}. 3. Cubic T-ideals of TM-Algebra We recall that a cubic set 𝛿 in a set ℵ is the structure 𝛿 = {〈𝜌, �̃�𝛿 (𝜌), 𝛼𝛿 (𝜌)〉 ∶ 𝜌 ∈ ℵ}, where �̃�𝛿 ∶ ℵ → 𝐷[0,1] such that �̃�𝛿 (𝜌) = [𝜗𝛿 𝐿 (𝜌), 𝜗𝛿 𝑈 (𝜌)] is an interval valued fuzzy set in ℵ and 𝛼𝛿 is a fuzzy set in ℵ.We write a cubic set by as follows. 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 and we can define the level subset of 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 which is denoted by 𝑈(𝛿, �̃�, 𝑠) as follows𝑈(𝛿, �̃�, 𝑠) = {𝜌 ∈ ℵ: �̃�𝛿 (𝜌) ≥ �̃�, 𝛼𝛿 ≤ 𝑠}, for every [0,0] ≤ �̃� ≤ [1,1] and 𝑠 ∈ [0,1]. Definition (12).Let ℵ be a TM-algebra. A cubic set 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 in ℵ is called a cubic sub-algebra if (1) �̃�δ(𝜌 ∗ 𝜏) ≥ 𝑟𝑚𝑖𝑛 {�̃�(𝜌), �̃�(𝜏)}. (2) 𝛼δ(𝜌 ∗ 𝜏) ≤ 𝑚𝑎𝑥{𝛼δ(𝜌), 𝛼δ(𝜏)},∀ 𝜌, 𝜏 ∈ ℵ. Example (13). Let ℵ = {0, a, b, c} be a set with the following Table: * 0 a b c 0 0 a b c a a 0 c b b b c 0 a c c b a 0 Then (ℵ,∗ ,0) is a TM-algebra. Define�̃�δ(𝜌)ˑand 𝛼δ (𝜌) by �̃�δ(𝜌) = { [0.2,0.9] 𝑖𝑓 𝜌 = {0, 𝑎, 𝑏} [0.1,0.3] 𝑖𝑓 𝜌 = 𝑐 , 𝛼δ(𝜌) = { 0.2 𝑖𝑓 𝜌 = {0, 𝑎, 𝑏} 0.4 𝑖𝑓 𝜌 = 𝑐 , By apply definition(12), we can prove that𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉is a cubic sub-algebra ofℵˑ. IHJPAS. 36(1)2023 371 Proposition (14). If 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉is a cubic sub-algebra ofℵˑ, then �̃�δ(0) ≥ �̃�δ(𝜌) and 𝛼δ(0) ≤ 𝛼δ(𝜌), ∀𝜌 ∈ ℵ Proof.Since 𝜌 ∗ 𝜌 = 0, then �̃�δ(0) = �̃�δ(𝜌 ∗ 𝜌) ≥ 𝑟𝑚𝑖 𝑛{�̃�δ(𝜌)ˑ, �̃�δ(𝜌)} = �̃�δ(𝜌)and 𝛼δ(0) = 𝛼δ(𝜌 ∗ 𝜌) ≤ 𝑚𝑎 𝑥{𝛼δ(𝜌)ˑ, ˑ𝛼δ(𝜌)} = 𝛼δ(𝜌). Theorem (15). Let 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 be a cubic set in ℵ, then 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 is a cubic sub-algebra of ℵ if and only if for all �̃� ∈ 𝐷[0,1] and 𝑠 ∈ [0,1], the set 𝑈(𝛿; �̃�, 𝑠) is either empty or a sub-algebra of ℵ . Proof. Assumethat 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 is a cubic sub-algebra of ℵ , let �̃� ∈ 𝐷[0,1] and 𝑠 ∈ [0,1], be such that 𝑈(𝛿; �̃�, 𝑠) ≠ ∅. Then, for any 𝜌, 𝜏 ∈ 𝑈(𝛿, �̃�, 𝑠) we have �̃�𝛿 (𝜌) ≥ �̃�, �̃�𝛿 (𝜏) ≥ �̃� and 𝛼𝛿 (𝜌) ≤ 𝑠,𝛼𝛿 (𝜏) ≤ 𝑠 and since 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 is a cubic sub-algebra, we have �̃�δ(𝜌 ∗ 𝜏) ≥ 𝑟𝑚𝑖𝑛{�̃�(𝜌), �̃�(𝜏)} = �̃�. 𝛼δ(𝜌 ∗ 𝜏) ≤ 𝑚𝑎𝑥{𝛼δ(𝜌), 𝛼δ(𝜏)} = 𝑠, So that 𝜌 ∗ 𝜏 ∈ 𝑈(𝛿; �̃�, 𝑠). Hence, 𝑈(𝛿; �̃�, 𝑠) is a sub-algebraof ℵ. Conversely, suppose that 𝑈(𝛿; �̃�, 𝑠) is a sub-algebraof ℵand let 𝜌, 𝜏 ∈ ℵ. Take �̃� = 𝑟𝑚𝑖𝑛{�̃�(𝜌), �̃�(𝜏)} and 𝑠 = 𝑚𝑎𝑥{𝛼δ(𝜌), 𝛼δ(𝜏)} By assumption 𝑈(𝛿; �̃�, 𝑠) is sub algebra of ℵ implies: 𝜌 ∗ 𝜏 ∈ 𝑈(𝛿; �̃�, 𝑠), therefore �̃�δ(𝜌 ∗ 𝜏) ≥ �̃� = 𝑟𝑚𝑖𝑛{�̃�(𝜌), �̃�(𝜏)} and 𝛼δ(𝜌 ∗ 𝜏) ≤ 𝑠 = 𝑚𝑎𝑥{𝛼δ(𝜌), 𝛼δ(𝜏)}. Hence 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 is a cubic sub-algebra of ℵ . Definition(16). Let ℵ be a TM-algebra. A cubic set𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 in ℵ is said to be a cubic ideal if: (H1)�̃�δ(0) ≥ �̃�δ(𝜌) and 𝛼δ(0) ≤ 𝛼δ(𝜌). (H2)�̃�δ(𝜌) ≥ 𝑟𝑚𝑖𝑛{�̃�δ(𝜌 ∗ 𝜏), �̃�δ(𝜏)} and 𝛼δ(𝜌) ≤ 𝑚𝑎𝑥{𝛼δ(𝜌 ∗ 𝜏), 𝛼𝛿 (𝜏)}, for all 𝜌, 𝜏 ∈ ℵ. Definition (17). Let ℵ be a TM-algebra. A cubic set 𝛿 = 〈�̌�𝛿 , 𝛼𝛿 〉 in ℵ is said to be a cubic T- ideal if: (𝐵1)�̃�𝛿 (0) ≥ �̃�𝛿 (𝜌) 𝑎𝑛𝑑 𝛼𝛿 (0) ≤ 𝛼𝛿 (𝜌), (𝐵2)�̃�𝛿 (𝜌 ∗ ) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝜌 ∗ 𝜏) ∗ ), �̃�𝛿 (𝜏)} 𝑎𝑛𝑑 𝛼𝛿 (𝜌 ∗ ) ≤ 𝑚𝑎𝑥{𝛼𝛿 ((𝜌 ∗ 𝜏) ∗ ), 𝛼𝛿 (𝜏)}. Example (18). Let ℵ = {0, 𝑎, 𝑏, 𝑐} in example(13). Define a cubic set 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 in ℵ as follows: IHJPAS. 36(1)2023 372 �̃�𝛿 (𝜌) = { [0.1,0.7], if 𝜌 = 0, [0.4,0.5], if 𝜌 ∈ {𝑎, 𝑏} [0.1,0.3], if 𝜌 ∈ 𝑐 , 𝛼𝛿 (𝜌) = { 0.1, if 𝜌 = 0, 0.3, if 𝜌 ∈ {𝑎, 𝑏}, 0.6, if 𝜌 ∈ 𝑐 Then, we can easy show that a cubic set 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 is a cubic T-ideal of ℵ . Proposition (19). If 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 is a cubic T-ideal of TM-algebra ℵ, then �̃�𝛿 (𝜌 ∗ (𝜌 ∗ 𝜏)) ≥ �̃�𝛿 (𝜏), 𝛼𝛿 (𝜌 ∗ (𝜌 ∗ 𝜏)) ≤ 𝛼𝛿 (𝜏). Proof. Taking = 𝜌 ∗ 𝜏 in Definition 3.6 We get �̃�𝛿 (𝜌 ∗ ) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝜌 ∗ 𝜏) ∗ ), �̃�𝛿 (𝜏)} �̃�𝛿 (𝜌 ∗ (𝜌 ∗ 𝜏) ) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝜌 ∗ 𝜏) ∗ (𝜌 ∗ 𝜏 )), �̃�𝛿 (𝜏)} = 𝑟𝑚𝑖𝑛{�̃�𝛿 (0), �̃�𝛿 (𝜏)} = �̃�𝛿 (𝜏) and 𝛼𝛿 (𝜌 ∗ ) ≤ 𝑚𝑎𝑥{𝛼𝛿 ((𝜌 ∗ 𝜏) ∗ ), 𝛼𝛿 (𝜏)} 𝛼𝛿 (𝜌 ∗ (𝜌 ∗ 𝜏) ) ≤ 𝑚𝑎𝑥{𝛼𝛿 ((𝜌 ∗ 𝜏) ∗ (𝜌 ∗ 𝜏)), 𝛼𝛿 (𝜏)} = 𝑚𝑎𝑥{𝛼𝛿 (0), 𝛼𝛿 (𝜏)} = 𝛼𝛿 (𝜏). Proposition (20). Let𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 be a cubic T-ideal of TM-algebraℵ . If the inequality 𝜌 ∗ 𝜏 ≤ holds in ℵ , then�̃�𝛿 (𝜌) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ( ), �̃�𝛿 (𝜏)} and 𝛼𝛿 (𝜌) ≤ 𝑚𝑎𝑥{𝛼𝛿 ( ), 𝛼𝛿 (𝜏)}. Proof. Assume that the inequality 𝜌 ∗ 𝜏 ≤ holds inℵ, then (𝜌 ∗ 𝜏) ∗ = 0 and by �̃�𝛿 (𝜌 ∗ ) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝜌 ∗ 𝜏) ∗ ), �̃�𝛿 (𝜏)}, if we put = 0 Then,�̃�𝛿 (𝜌 ∗ 0) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝜌 ∗ 𝜏) ∗ 0), �̃�𝛿 (𝜏)} �̃�𝛿 (𝜌) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 (𝜌 ∗ 𝜏), �̃�𝛿 (𝜏)} ………(𝑖) But�̃�𝛿 (𝜌 ∗ 𝜏) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝜌 ∗ ) ∗ 𝜏), �̃�𝛿 ( )} �̃�𝛿 (𝜌 ∗ 𝜏) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝜌 ∗ 𝜏) ∗ ), �̃�𝛿 ( )} = 𝑟𝑚𝑖𝑛{�̃�𝛿 (0), �̃�𝛿 ( )} = �̃�𝛿 ( ) ……(𝑖𝑖) From (𝑖) and (𝑖𝑖), we get �̃�𝛿 (𝜌) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ( ), �̃�𝛿 (𝜏)}. Similarly, we can show that 𝛼𝛿 (𝜌) ≤ 𝑚𝑎𝑥{𝛼𝛿 ( ), 𝛼𝛿 (𝜏)}. Proposition (21). If 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 is a cubic T-ideal of TM-algebra ℵand 𝜌 ≤ 𝜏 then �̃�𝛿 (𝜌) ≥ �̃�𝛿 (𝜏) 𝑎𝑛𝑑 𝛼𝛿 (𝜌) ≤ 𝛼𝛿 (𝜏). Proof. If 𝜌 ≤ 𝜏then 𝜌 ∗ 𝜏 = 0. This is together with 𝜌 ∗ 0 = 𝜌 and �̃�𝛿 (0) ≥ �̃�𝛿 (𝜏) also 𝛼𝛿 (0) ≤ 𝛼𝛿 (𝜏), we get IHJPAS. 36(1)2023 373 �̃�𝛿 (𝜌 ∗ 0) = �̃�𝛿 (𝜌) ≥ 𝑟𝑚𝑖𝑛{((𝜌 ∗ 𝜏) ∗ 0), �̃�𝛿 (𝜏)} = 𝑟𝑚𝑖𝑛{�̃�𝛿 (0 ∗ 0), �̃�𝛿 (𝜏)} = 𝑟𝑚𝑖𝑛{�̃�𝛿 (0), �̃�𝛿 (𝜏)} = �̃�𝛿 (𝜏), also 𝛼𝛿 (𝜌 ∗ 0) = 𝛼𝛿 (𝜌) ≤ 𝑚𝑎𝑥{𝛼𝛿 ((𝜌 ∗ 𝜏) ∗ 0), 𝛼𝛿 (𝜏)} = 𝑚𝑎𝑥{𝛼𝛿 (0 ∗ 0), 𝛼𝛿 (𝜏)} = 𝑚𝑎𝑥{𝛼𝛿 (0), 𝛼𝛿 (𝜏)} = 𝛼𝛿 (𝜏). Theorem (22). Let ℵ be a TM-algebra, a cubic set𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉of ℵ is a cubic T-ideal if 𝛿 is a cubic ideal of ℵ. Proof. If we put = 0 in (𝐵2), then �̃�𝛿 (𝜌) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝜌 ∗ 𝜏)), �̃�𝛿 (𝜏)} 𝑎𝑛𝑑𝛼𝛿 (𝜌) ≤ 𝑚𝑎𝑥{𝛼𝛿 ((𝜌 ∗ 𝜏)), 𝛼𝛿 (𝜏)}. Hence 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉is a cubic ideal of ℵ. Remark (23). The converse of Theorem (22) is not true. The following example shows the reverse direction of Theorem (22). Example (24). Let ℵ = {0, a, b, c} be a set with the following Table: * 0 a b c 0 0 a b c a a 0 c b b b c 0 a c c b a 0 Then, (ℵ,∗ ,0) is a TM-algebra. Define�̃�δ(𝜌)ˑand 𝛼δ (𝜌) by �̃�δ(𝜌) = { [0.1,0.8] 𝑖𝑓 𝜌 = {0, 𝑎, 𝑏} [0.1,0.3] 𝑖𝑓 𝜌 = 𝑐 , 𝛼δ(𝜌) = { 0.1 𝑖𝑓 𝜌 = {0, 𝑎, 𝑏} 0.8 𝑖𝑓 𝜌 = 𝑐 , Then, it is easy to show that 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 is a cubic ideal of ℵ. But not a cubic T-ideal since �̃�𝛿 (𝑎 ∗ 𝑏) ≤ 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝑎 ∗ 𝑐) ∗ 𝑏), �̃�𝛿 (𝑐)} 𝑎𝑛𝑑 𝛼𝛿 (𝑎 ∗ 𝑏) ≥ 𝑚𝑎𝑥{𝛼𝛿 ((𝑎 ∗ 𝑐) ∗ 𝑏), 𝛼𝛿 (𝑐)}. 4. Image and Pre-image of cubic T-ideals Definition (25). Let 𝑓: ℵ → 𝑌 be a mapping. If 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 is a cubic set of ℵ, then the cubic set 𝜔 = 〈�̃�𝜔 , 𝛼𝜔 〉 of 𝑌 is define by IHJPAS. 36(1)2023 374 𝑓(�̃�𝛿 )(𝜏) = �̃�𝜔 (𝜏) = { 𝑟𝑠𝑢𝑝 𝜌∈𝑓−1(𝜏) �̃�𝛿 (𝜌), 𝑖𝑓 𝑓 −1(𝜏) = {𝜌 ∈ ℵ, 𝑓(𝜌) = 𝜏} ≠ ∅ 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓(𝛼𝛿 )(𝜏) = 𝛼𝜔 (𝜏) = { inf 𝜌∈𝑓−1(𝜏) 𝛼𝛿 (𝜌) 𝑖𝑓 𝑓 −1(𝜏) = {𝜌 ∈ ℵ, 𝑓(𝜌) = 𝜏} ≠ ∅ 1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 It is called the image of 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 under 𝑓. Similarly, if 𝜔 = 〈�̃�𝜔 , 𝛼𝜔 〉 is a cubic subset of 𝑌, then the cubic subset defined by �̃�𝛿 (𝜌) = �̃�𝜔(𝑓(𝜌)) and 𝛼𝛿 (𝜌) = 𝛼𝜔 (𝑓(𝜌)),for any 𝜌 ∈ ℵis said to be the pre-image of 𝜔under 𝑓. Theorem (26). An epimorphism pre-image of a cubic T-ideal is also a cubic T-ideal. Proof. Let 𝑓 ∶ ℵ → ℵ′ be an epimorphism mapping of TM-algebra, 𝜔 = 〈�̃�𝜔 , 𝛼𝜔 〉be a cubic T- ideal of ℵ′ and 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉be the pre-image of 𝜔under 𝑓, then �̃�𝛿 (𝜌) = �̃�𝜔 (𝑓(𝜌)) and 𝛼𝛼 (𝜌) = 𝛼𝜔 (𝑓(𝜌)) for any 𝜌 ∈ ℵ, then �̃�𝛿 (0) = �̃�𝜔(𝑓(0)) ≥ �̃�𝜔 (𝑓(𝜌)) = �̃�𝛿 (𝜌), 𝛼𝛿 (0) = 𝛼𝜔 (𝑓(0)) ≤ 𝛼𝜔 (𝑓(𝜌)) = 𝛼𝛿 (𝜌). Now, let 𝜌, 𝜏, ∈ ℵ , then �̃�𝛿 (𝜌 ∗ ) = �̃�𝜔 (𝑓(𝜌 ∗ )) = �̃�𝜔 (𝑓(𝜌) ∗ ′ 𝑓( )) ≥ 𝑟𝑚𝑖𝑛 {�̃�𝜔 ((𝑓(𝜌) ∗ ′ 𝑓(𝜏)) ∗′ 𝑓( )) , �̃�𝜔 (𝑓(𝜏))} = 𝑟𝑚𝑖𝑛 {�̃�𝜔 (𝑓((𝜌 ∗ 𝜏) ∗ )) , �̃�𝜔 (𝑓(𝜏))} = 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝜌 ∗ 𝜏) ∗ ), �̃�𝛿 (𝜏)}, 𝛼𝛿 (𝜌 ∗ ) = 𝛼𝜔 (𝑓(𝜌 ∗ )) = 𝛼𝜔 (𝑓(𝜌) ∗ ′ 𝑓( )) ≤ 𝑚𝑎𝑥 {𝛼𝜔 ((𝑓(𝜌) ∗ ′ 𝑓(𝜏)) ∗′ 𝑓( )) , 𝛼𝜔 (𝑓(𝜏))} = 𝑚𝑎𝑥 {𝛼𝜔 (𝑓((𝜌 ∗ 𝜏) ∗ )) , 𝛼𝜔 (𝑓(𝜏))} = 𝑚𝑎𝑥{𝛼𝛿 ((𝜌 ∗ 𝜏) ∗ ), 𝛼𝛿 (𝜏)}. Definition (27). A cubic subset 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 of ℵhas sup and inf properties if for any subset 𝑇 𝑜𝑓 ℵ , there exist 𝑡, 𝑠 ∈ 𝑇 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 �̃�𝛿 (𝑡) = 𝑟𝑠𝑢𝑝𝑡∈𝑇 �̃�𝛿 (𝑡) 𝑎𝑛𝑑 𝛼𝛿 (𝑠) = 𝑖𝑛𝑓𝑡∈𝑇 𝛼𝛿 (𝑠). Theorem (28). Let 𝑓: ℵ → 𝑌 be anepimorphism between TM-algebra ℵ IHJPAS. 36(1)2023 375 and 𝑌. For every cubic T-ideal 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 𝑖𝑛 ℵ, 𝑡ℎ𝑒𝑛 𝑓(𝛿)is cubic T-ideal of 𝑌. Proof. By definition25�̃�𝜔 (𝜏 ′) = 𝑓(�̃�𝛿 )(𝜏 ′) = 𝑟𝑠𝑢𝑝𝜌∈𝑓−1(𝜏′)�̃�𝛿 (𝜌) and 𝛼𝜔 (𝜏 ′) = 𝑓(𝛼𝛿 )(𝜏 ′) = 𝑖𝑛𝑓𝜌∈𝑓−1(𝜏′)𝛼𝛿 (𝜌) for any 𝜏 ′ ∈ 𝑌 and 𝑟𝑠𝑢𝑝 ∅ = [0,0] = 0. We must prove that �̃�𝜔 (𝜌 ′ ∗ ′) ≥ 𝑟𝑚𝑖𝑛{�̃�𝜔((𝜌 ′ ∗ 𝜏′) ∗ ′), �̃�𝜔 (𝜏 ′)} and 𝛼𝜔(𝜌 ′ ∗ ′) ≤ 𝑚𝑎𝑥{𝛼𝜔 ((𝜌 ′ ∗ 𝜏′) ∗ ′), 𝛼𝜔 (𝜏 ′)} 𝑓𝑜𝑟 𝑎𝑛𝑦 𝜌′, 𝜏′, ′ ∈ 𝑌. Let 𝑓: ℵ → 𝑌 be an epimorphism mappingof ℵ , 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉be a cubic T-ideal of ℵwith sup and inf properties and ω = 〈�̃�𝜔, 𝛼𝜔 〉be the image of 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 under 𝑓. Since 𝛿 = 〈�̃�𝛿 , 𝛼𝛿 〉 is a cubicT-ideal of ℵ , we have �̃�𝛿 (0) ≥ �̃�𝛿 (𝜌), 𝛼𝛿 (0) ≤ 𝛼𝛿 (𝜌) ∀𝜌 ∈ ℵ . Note that 0 ∈ 𝑓 −1(0′) where 0, 0′ are the zero of ℵ 𝑎𝑛𝑑 𝑌, respectively. Thus, �̃�𝛿 (0 ′) = rsup 𝑡∈𝑓−1(0) �̃�𝛿 (𝑡) = �̃�𝛿 (0) ≥ �̃�𝛿 (𝜌) ∀𝜌 ∈ ℵ , 𝛼𝜔 (0 ′) ≤ inf 𝑡∈𝑓−1(0′) 𝛼𝜔 (𝑡) = 𝛼𝜔 (0) ≤ 𝛼𝜔 (𝜌) ∀𝜌 ∈ ℵ , Which implies that �̃�𝜔 (0 ′) ≥ rsup 𝑡∈𝑓−1(𝜌′) �̃�𝜔 (𝜌 ′) and 𝛼𝜔 (0 ′) ≤ inf 𝑡∈𝑓−1(𝜌′) 𝛼𝛿 (𝑡) = 𝛼𝜔 (𝜌 ′)for any𝜌′ ∈ 𝑌. For any 𝜌′, 𝜏′, ′ ∈ 𝑌, let 𝜌0 ∈ 𝑓 −1(𝜌′), 𝜏0 ∈ 𝑓 −1(𝜏′), and 0 ∈ 𝑓 −1( ′) be such that �̃�𝛿 (𝜌0 ∗ 0) = rsup 𝑡∈𝑓−1(𝜌′∗ ′) �̃�𝛿 (𝑡), �̃�𝛿 (𝜏0) = rsup 𝑡∈𝑓−1(𝜏′) �̃�𝛿 (𝑡), �̃�𝛿 ((𝜌0 ∗ 𝜏0) ∗ 0) = �̃�𝜔 {𝑓((𝜌0 ∗ 𝜏0) ∗ 0)} = �̃�𝜔((𝜌 ′ ∗ 𝜏′) ∗ ′) = rsup ((𝜌0∗𝜏0)∗ 0)∈𝑓 −1((𝜌′∗𝜏′)∗ ′) �̃�𝛿 ((𝜌0 ∗ 𝜏0) ∗ 0) = rsup 𝑡∈𝑓−1((𝜌′∗𝜏′)∗ ′) �̃�𝛿 (𝑡) .Also 𝛼𝛿 (𝜌0 ∗ 0) = inf 𝑡∈𝑓−1(𝜌′∗ ′) 𝛼𝛿 (𝑡). 𝛼𝛿 (𝜏0) = inf 𝑡∈𝑓−1(𝜏′) 𝛼𝛿 (𝑡), 𝛼𝛿 ((𝜌0 ∗ 𝜏0) ∗ 0) = 𝛼𝜔 {𝑓((𝜌0 ∗ 𝜏0) ∗ 0)} IHJPAS. 36(1)2023 376 = 𝛼𝜔 ((𝜌 ′ ∗ 𝜏′) ∗ ′) = inf ((𝜌0∗𝜏0)∗ 0)∈𝑓 −1((𝜌′∗𝜏′)∗ ′) 𝛼𝛿 ((𝜌0 ∗ 𝜏0) ∗ 0) = inf 𝑡∈𝑓−1((𝜌′∗𝜏′)∗ ′) 𝛼𝛿 (𝑡). Then �̃�𝜔 (𝜌 ′ ∗ 𝜏′) = rsup 𝑡∈𝑓−1(𝜌′∗𝜏′) �̃�𝛿 (𝑡) = �̃�𝛿 (𝜌0 ∗ 𝜏0) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿 ((𝜌0 ∗ 𝜏0) ∗ 0), �̃�𝛿 (𝜏0)} = 𝑟𝑚𝑖𝑛 { rsup 𝑡∈𝑓−1((𝜌′∗𝜏′)∗ ′) �̃�𝛿 (𝑡) , rsup 𝑡∈𝑓−1(𝜏′) �̃�𝛿 (𝑡)} = 𝑟𝑚𝑖𝑛{�̃�𝜔 ((𝜌 ′ ∗ 𝜏′) ∗ ′), �̃�𝜔(𝜏 ′)} , 𝛼𝜔 (𝜌 ′ ∗ ′) = inf 𝑡∈𝑓−1(𝜌′∗ ′) 𝛼𝛿 (𝑡) = 𝛼𝛿 (𝜌0 ∗ 𝜏0) ≤ 𝑚𝑎𝑥{𝛼𝛿 ((𝜌0 ∗ 𝜏0) ∗ 0), 𝛼𝛿 (𝜏0)} = 𝑚𝑎𝑥 { inf 𝑡∈𝑓−1((𝜌′∗𝜏′)∗ ′) 𝛼𝛿 (𝑡), inf 𝑡∈𝑓−1(𝜏′) 𝛼𝛿 (𝑡)} = 𝑚𝑎𝑥{𝛼𝜔 ((𝜌 ′ ∗ 𝜏′) ∗ ′), 𝛼𝜔 (𝜏 ′)}. Hence, 𝜔is a cubic T-ideal of 𝑌. 5. Cartesian Product of Cubic T-Ideals In this section, we provide some definitions of the Cartesian product of cubic T-ideals in TM- algebras. Definition (29). Let𝛿1 = 〈�̃�𝛿1 , 𝛼𝛿1 〉 𝑎𝑛𝑑 𝛿2 = 〈�̃�𝛿2 , 𝛼𝛿2 〉 be two cubic subsets of TM-algebras ℵ1 𝑎𝑛𝑑 ℵ2, respectively. We definethe Cartesian product of two cubic sets 𝛿1 𝑎𝑛𝑑 𝛿2 by 𝛿1 × 𝛿2 = 〈�̃�𝛿1×𝛿2 , 𝛼𝛿1×𝛿2 〉 and �̃�𝛿1×𝛿2 (𝜌, 𝜏) = 𝑟𝑚𝑖𝑛{�̃�𝛿1 (𝜌), �̃�𝛿2 (𝜏)}, 𝛼𝛿1×𝛿2 (𝜌, 𝜏) = 𝑚𝑎𝑥{𝛼𝛿1 (𝜌), 𝛼𝛿2 (𝜏)}, for any (𝜌, 𝜏) ∈ ℵ1 × ℵ2 . Remark (30). Let ℵ and 𝑌 be TM-algebras. We define ∗ 𝑜𝑛 ℵ × 𝑌 by (𝜌, 𝜏) ∗ (𝑢, 𝑣) = (𝜌 ∗ 𝑢, 𝜏 ∗ 𝑣) for every (𝜌, 𝜏), (𝑢, 𝑣) belong to ℵ × 𝑌, Then, clearly (ℵ × 𝑌,∗, (0,0)) is a TM-algebra. Definition (31). A cubic subset 𝛿1 × 𝛿2 = 〈�̃�𝛿1×𝛿2 , 𝛼𝛿1×𝛿2 〉 of ℵ1 × ℵ2 is called a cubic ideal of ℵ1 × ℵ2 if (𝐂𝐏𝟏)�̃�𝛿1×𝛿2 (0,0) ≥ �̃�𝛿1×𝛿2 (𝜌, 𝜏) and 𝛼𝛿1×𝛿2 (0,0) ≤ 𝛼𝛿1×𝛿2 (𝜌, 𝜏), IHJPAS. 36(1)2023 377 (𝐂𝐏𝟐)�̃�𝛿1×𝛿2 (𝜌1, 𝜏1) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿1×𝛿2 ((𝜌1, 𝜏1) ∗ (𝜌2, 𝜏2)), �̃�𝛿1×𝛿1 (𝜌2, 𝜏2)} (𝐂𝐏𝟑)𝛼𝛿1×𝛿2 (𝜌1, 𝜏1) ≤ 𝑚𝑎𝑥{𝛼𝛿1×𝛿2 ((𝜌1, 𝜏1) ∗ (𝜌2, 𝜏2)), 𝛼𝛿1×𝛿2 (𝜌2, 𝜏2)}, For any (𝜌1, 𝜏1), (𝜌2, 𝜏2) ∈ ℵ1 × ℵ2. Definition (32). A cubic subset 𝛿1 × 𝛿2 = 〈�̃�𝛿1×𝛿2 , 𝛼𝛿1×𝛿2 〉 of ℵ1 × ℵ2 is called a cubic T-ideal of ℵ1 × ℵ2 if (𝐂𝐏𝟏)�̃�𝛿1×𝛿2 (0,0) ≥ �̃�𝛿1×𝛿2 (𝜌, 𝜏) 𝑎𝑛𝑑 𝛼𝛿1×𝛿2 (0,0) ≤ 𝛼𝛿1×𝛿2 (𝜌, 𝜏), (𝐂𝐏𝟐)�̃�𝛿1×𝛿2 ((𝜌1, 𝜏1) ∗ (𝜌3, 𝜏3)) ≥ 𝑟𝑚𝑖𝑛 {�̃�𝛿1×𝛿2 (((𝜌1, 𝜏1) ∗ (𝜌2, 𝜏2)) ∗ (𝜌3, 𝜏3)) , �̃�𝛿1×𝛿2 (𝜌2, 𝜏2)} (𝐂𝐏𝟑)𝛼𝛿1×𝛿2 ((𝜌1, 𝜏1) ∗ (𝜌3, 𝜏3)) ≤ 𝑚𝑎𝑥 {𝛼𝛿1×𝛿2 (((𝜌1, 𝜏1) ∗ (𝜌2, 𝜏2)) ∗ (𝜌3, 𝜏3)) , 𝛼𝛿1×𝛿2 (𝜌2, 𝜏2)} , For any (𝜌1, 𝜏1), (𝜌2, 𝜏2), (𝜌3, 𝜏3) ∈ ℵ1 × ℵ2. Proposition (33). If 𝛿1 × 𝛿2 = 〈�̃�𝛿1×𝛿2 , 𝛼𝛿1×𝛿2 〉 is a cubic T-ideal of TM-algebra ℵ1 × ℵ2 and if (𝜌1, 𝜏1) ≤ (𝜌2, 𝜏2), we have �̃�𝛿1×𝛿2 (𝜌2, 𝜏2) ≤ �̃�𝛿1×𝛿2 (𝜌1, 𝜏1) and 𝛼𝛿1×𝛿2 (𝜌2, 𝜏2) ≥ 𝛼𝛿1×𝛿2 (𝜌1, 𝜏1). For all (𝜌1, 𝜏1), (𝜌2, 𝜏2) ∈ ℵ1 × ℵ2. Proof. Let (𝜌1, 𝜏1), (𝜌2, 𝜏2) ∈ ℵ1 × ℵ2, such that (𝜌1, 𝜏1) ≤ (𝜌2, 𝜏2) ⇒ (𝜌2, 𝜏2) ∗ (𝜌1, 𝜏1) = (0,0).This together with (0,0) ∗ (𝜌1, 𝜏1) = (𝜌1, 𝜏1)and �̃�𝛿1×𝛿2 (𝜌2, 𝜏2) ≤ �̃�𝛿1×𝛿2 (0,0) Also 𝛼𝛿1×𝛿2 (𝜌2, 𝜏2) ≥ 𝛼𝛿1×𝛿2 (0,0). Consider 𝜗𝛿1×𝛿2 ((0,0) ∗ (𝜌1, 𝜏1)) = �̃�𝛿1×𝛿2 (𝜌1, 𝜏1) ≥ 𝑟𝑚𝑖𝑛 {�̃�𝛿1×𝛿2 (((0,0) ∗ (𝜌2, 𝜏2)) ∗ (𝜌1, 𝜏1)) , �̃�𝛿1×𝛿2 (𝜌2, 𝜏2)} = 𝑟𝑚𝑖𝑛{�̃�𝛿1×𝛿2 ((0,0) ∗ (0,0)), �̃�𝛿1×𝛿2 (𝜌2, 𝜏2)} = 𝑟𝑚𝑖𝑛{�̃�𝛿1×𝛿2 (0,0), �̃�𝛿1×𝛿2 (𝜌2, 𝜏2)} = �̃�𝛿1×𝛿2 (𝜌2, 𝜏2), 𝛼𝛿1×𝛿2 ((0,0) ∗ (𝜌1, 𝜏1)) = 𝛼𝛿1×𝛿2 (𝜌1, 𝜏1) ≤ 𝑚𝑎𝑥 {𝛼𝛿1×𝛿2 (((0,0) ∗ (𝜌2, 𝜏2)) ∗ (𝜌1, 𝜏1)) , 𝛼𝛿1×𝛿2 (𝜌2, 𝜏2)} = 𝑚𝑎𝑥{𝛼𝛿1×𝛿2 ((0,0) ∗ (0,0)), 𝛼𝛿1×𝛿2 (𝜌2, 𝜏2)} = 𝑚𝑎𝑥{𝛼𝛿1×𝛿2 (0,0), 𝛼𝛿1×𝛿2 (𝜌2, 𝜏2)} IHJPAS. 36(1)2023 378 = 𝛼𝛿1×𝛿2 (𝜌2, 𝜏2) This shows that �̃�𝛿1×𝛿2 (𝜌2, 𝜏2) ≤ �̃�𝛿1×𝛿2 (𝜌1, 𝜏1) and 𝛼𝛿1×𝛿2 (𝜌2, 𝜏2) ≥ 𝛼𝛿1×𝛿2 (𝜌1, 𝜏2), for all (𝜌1, 𝜏1), (𝜌2, 𝜏2) ∈ ℵ1 × ℵ2. Theorem (34)..Let 𝛿1 = 〈�̃�𝛿1 , 𝛼𝛿1 〉and 𝛿2 = 〈�̃�𝛿2 , 𝛼𝛿2 〉 be two cubic ideal of TM-algebra ℵ1 𝑎𝑛𝑑 ℵ2, respectively. Then 𝛿1 × 𝛿2 = 〈�̃�𝛿1×𝛿2 , 𝛼𝛿1×𝛿2 〉 is a cubic ideal of ℵ1 × ℵ2. Proof. For any (𝜌, 𝜏) ∈ ℵ1 × ℵ2, �̃�𝜹𝟏×𝜹𝟏 (0,0) = 𝑟𝑚𝑖𝑛{�̃�𝛿1 (0), �̃�𝛿2 (0)} ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿1 (𝜌), �̃�𝛿2 (𝜏)} = �̃�𝛿1×𝛿2 (𝜌, 𝜏), 𝛼𝛿1×𝛿2 (0,0) = 𝑚𝑎𝑥{𝛼𝛿1 (0), 𝛼𝛿2 (0)} ≤ 𝑚𝑎𝑥{𝛼𝛿1 (𝜌), 𝛼𝛿2 (𝜏)} = 𝛼𝛿1×𝛿𝟐 (𝜌, 𝜏). For any (𝜌1, 𝜏1), (𝜌2, 𝜏2) ∈ ℵ1 × ℵ2. Then �̃�𝛿1×𝛿2 (𝜌1, 𝜏1) = 𝑟𝑚𝑖𝑛{�̃�𝛿1 (𝜌1), �̃�𝛿2 (𝜏1)} ≥ 𝑟𝑚𝑖𝑛 {𝑟𝑚𝑖𝑛{�̃�𝛿1 (𝜌1 ∗ 𝜌2), �̃�𝛿1 (𝜌2)}, 𝑟𝑚𝑖𝑛{�̃�𝛿2 (𝜏1 ∗ 𝜏2), �̃�𝛿2 (𝜏2)}} = 𝑟𝑚𝑖𝑛 {𝑟𝑚𝑖𝑛{�̃�𝛿1 (𝜌1 ∗ 𝜌2), �̃�𝛿2 (𝜏1 ∗ 𝜏2)}, 𝑟𝑚𝑖𝑛{�̃�𝛿1 (𝜌2), �̃�𝛿2 (𝜏2)}} = 𝑟𝑚𝑖𝑛{�̃�𝛿1×𝛿2 (𝜌1 ∗ 𝜌2, 𝜏1 ∗ 𝜏2), �̃�𝛿1×𝛿2 (𝜌2, 𝜏2)} ≥ 𝑟𝑚𝑖𝑛{�̃�𝛿1×𝛿2 ((𝜌1 ∗ 𝜏1)(𝜌2 ∗ 𝜏2)), �̃�𝛿1×𝛿2 (𝜌2, 𝜏2)}, 𝛼𝛿1×𝛿2 (𝜌1, 𝜏1) = 𝑚𝑎𝑥{𝛼𝛿1 (𝜌1), 𝛼𝛿2 (𝜏1)} ≤ 𝑚𝑎𝑥 {𝑚𝑎𝑥{𝛼𝛿1 (𝜌1 ∗ 𝜌2), 𝛼𝛿2 (𝜏2)}, 𝑚𝑎𝑥{𝛼𝛿2 (𝜏1 ∗ 𝜏2), 𝛼𝛿 2(𝜏2)}} = 𝑚𝑎𝑥 {𝑚𝑎𝑥{𝛼𝛿1 (𝜌1 ∗ 𝜌2), 𝛼𝛿2 (𝜏1 ∗ 𝜏2)}, 𝑚𝑎𝑥{𝛼𝛿2 (𝜌2), 𝛼𝛿2 (𝜏2)}} = 𝑚𝑎𝑥{𝛼𝛿1×𝛿2 (𝜌1 ∗ 𝜌2, 𝜏1 ∗ 𝜏2), 𝛼𝛿1×𝛿2 (𝜌2, 𝜏2)} Hence, for all (𝜌1, 𝜏1), (𝜌2, 𝜏2) ∈ ℵ1 × ℵ1, 𝛿1 × 𝛿2 = 〈�̃�𝛿1×𝛿2 , 𝛼𝛿1×𝛿2 〉 is a cubic idealof TM- algebra ℵ1 × ℵ2. 6.Conclusion The goal of this paper is to introduce the definition of a cubic ideal and a cubic T-ideal. The homomorphism of these ideals is defined, and the Cartesian product of cubic ideals in Cartesian product TM-algebras is given. 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