454 This work is licensed under a Creative Commons Attribution 4.0 International License IHJPAS. 37 (1) 2024 Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq PISSN: 1609-4042, EISSN: 2521-3407 1 Samy M. Mostafa 2Fatema. F. Kareem* 3Omniat. A. Hasan 1 Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt. 2,3 Department of Mathematics, College of Education for Pure Sciences, Ibn Al – Haitham, University of Baghdad, Baghdad, Iraq. *Corresponding Author. fatma.f.k@ihcoedu.uobaghdad.edu.iq Abstract In this paper, we define a cubic positive implicative-ideal, a cubic implicative-ideal and a cubic commutative-ideal of a semigroup in KU-algebra as a generalization of a fuzzy (positive implicative-ideal, an implicative-ideal and a commutative-ideal) of a semigroup in KU-algebra. Some relations between these types of cubic ideals are discussed. Also, some important properties of these ideals are studied. Finally, some important theories are discussed. It is proved that every cubic commutative-ideal, cubic positive implicative-ideal, and cubic implicative-ideal are a cubic ideal, but not conversely. Also, we show that if Θ is a cubic positive implicative-ideal and a cubic commutative-ideal then Θ is a cubic implicative-ideal. Some examples of the opposite direction of the previous theories are obtained. Keywords A KU-semigroup, cubic ideal, cubic k-ideal, cubic positive implicative ideal, cubic implicative ideal, cubic commutative ideal. 1. Introduction The structure of KU-algebra was studied by Prabpayak and Leerawat [1,2]. They gave a homomorphism of KU-algebras and proved some important theories. The idea of a fuzzy set was initialed in 1965 by the author Zadeh [3]. Since then this concept has been applied in many different branches of mathematic. The study of fuzzy algebraic structures was started with the introduction of the concept of fuzzy groups by Rosenfeld [4]. Fuzzy p-ideals and fuzzy H-ideals in BCI-algebras are introduced in [5]. Jun et al, in [6] studied of the Fuzzy Implicative ideals of BCK- algebras. Also, Mostafa et al [7] introduced the notion of fuzzy KU-ideals of KU-algebras and they investigated several basic properties which are associated to fuzzy KU-ideals. Many Mathematicians have studied “a fuzzy” for some algebraic structures, seeˑ[8-12]. There are several kinds of fuzzy set extensions in the fuzzy set theory, for example, interval-valued fuzzy sets and bipolar-valued fuzzy sets, which were introduced in [13-17]. The concept of cubic subalgebras /ideals in BCK/BCI-algebras was introduced by Jun et al. [18, 19]. They discussed the relationship between a cubic subalgebra and a cubic ideal. And then, Yaqoob et al [20] introduced the notion of cubic KU-algebra which is a generalization of the concept of fuzzy KU-ideals of KU-algebras. After that, Kareem and Hasan[21] introduced the Received 25 November 2022, Received 18 May 2023, Accepted 22 May 2023, Published 20 January 2024 Cubic of Positive Implicative Ideals in KU-Semigroup doi.org/10.30526/37.1.3112 https://creativecommons.org/licenses/by/4.0/ https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042 https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407 mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-5069-2434 mailto:%20%20%20%20%20%20%20samymostafa@yahoo.com https://orcid.org/0000-0002-6141-8721 mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0005-7019-2655 mailto:umniyatadnan@gmail.com IHJPAS. 37 (1) 2024 455 notion of a KU-algebra with semigroup which is called a KU-semigroup and defined some types of ideals in this concept. Also, they studied the fuzzy ideals of a KU-semigroup. After that, Kareem and Hasan [22] introduced the Cubic ideals of a semigroup in KU-algebra and defined some types of ideals in this concept. Senapati et al[23,24] introduced the two concepts which are cubic ideal and implicative ideal. Some authors introduced a cubic set of different structures. See [25-30]. In this work, the notion of cubic (positive implicative, implicative and commutative)-ideal are discussed and the relationship among these types are studied. 2. Basic Concepts Definition (1) [1]. An algebra(Ν,∗ ,0) is named a KU-algebra if, for all ς , ω, κ ∈ Ν, (ku1) (ς ∗ ω) ∗ [(ω ∗ κ) ∗ (ς ∗ κ)] = 0, (ku2) ς ∗0 = 0 , (ku3) 0∗ ς = ς , (ku4) ς ∗ ω = 0 and ω ∗ ς = 0 implies ς = ω, (ku5) ς ∗ ς = 0. The binary relation ≤ on Ν is define by ς ≤ ω ⟺ ω ∗ ς = 0. Theorem(2)[2].ˑLet(Ν,∗ ,0) be a KU-algebra. Then for all ς,ω, κ ∈ Ν: (1) If ς ≤ ω implyω ∗ κ ≤ ς ∗ κ. (2) ς ∗ (ω ∗ κ) = ω ∗ (ς ∗ κ). (3) (ω ∗ ς) ∗ ς ≤ ω. (4)((ω ∗ ς) ∗ ς) ∗ ς)) = ω ∗ ς. Definition(3) [21]. A nonempty set Ν with~∗,∘ and 0 is called a KU-semigroup if~~ (I)The set Ν with ∗ and 0 is ˑa KU-algebra. (II) The set Νwith ∘ and 0 is aˑsemigroup. (III) ς ∘ (ω ∗ κ) = (ς ∘ ω) ∗ (ς ∘ κ) and(ς ∗ ω) ∘ κ = (ς ∘ κ) ∗ (ω ∘ κ), for all ς,ω, κ ∈ Ν. Example (4) [21]. LetˑΝ = {0,1,2,3} with two operations ∗ and ∘ defined by Table 1: Table 1. a KU-semigroup It follows that ˑ(Ν,∗,∘ ,0) is aˑKU-semigroup. * 0 1 2 3 0 0 1 3 2 1 0 0 0 2 2 2 0 0 1 3 0 0 0 0 ∘ 0 1 2 3 0 0 0 0 0 1 0 1 0 1 2 0 0 2 2 3 0 1 2 3 IHJPAS. 37 (1) 2024 456 We recall, an interval valued fuzzy set μ̃ in Ν is defined as μ̃ = {〈𝜍, [𝜇𝐿(𝜍), 𝜇𝑈(𝜍)], 𝜍 ∈ 𝑁〉} , where the ordinary fuzzy sets 𝜇𝐿: 𝑁 → [0,1] and 𝜇 𝑈: 𝑁 → [0,1] are called a lower fuzzy set and an upper fuzzy set of μ̃ respectively. Also, we recall some definitions of a cubic subset of aˑKU-semigroup from [22]. Definition(5) [18]. The cubic set Θ of a non-empty set Ν is Θ = {〈ς, 𝜇Θ(ς), 𝜆Θ(ς)〉: ς ∈ Ν}, which is briefly indicated by Θ = 〈𝜇Θ, 𝜆Θ〉 where 𝜇Θ(ς) = [𝜇Θ 𝐿(ς), 𝜇Θ 𝑈(ς)] is an interval valued fuzzy set in Ν and 𝜆Θ(ς) is a fuzzy set in Ν.The set {ς ∈ Ν ∶ 𝜇Θ(ς) ≥ �̃�, 𝜆Θ(ς) ≤ 𝛼} is called a cubic level set of Θ = 〈𝜇Θ, 𝜆Θ〉, where [0,0]≤ �̃� ≤ [1,1] and 𝛼 ∈ [0,1]. Definition(6) [22]. The cubic set Θ of Ν is named a cubic sub KU-semigroup if for all ς, ω ∈ Ν, 1. 𝜇Θ(ς ∗ ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ς)ˑ, 𝜇Θ(ω)},𝜆Θ(ς ∗ ω) ≤ 𝑚𝑎𝑥{𝜆Θ(ς)ˑ, 𝜆Θ(ω)}. 2. 𝜇Θ(ς ∘ ω) ≥ 𝑟𝑚𝑖 𝑛{𝜇𝛩(𝜍)ˑ, ˑ𝜇𝛩(𝜔)}, 𝜆Θ(ς ∘ ω) ≤ 𝑚𝑎𝑥 {𝜆Θ(ς)ˑ, ˑ𝜆Θ(ω)}. Definition(7) [22]. The cubic set Θ of Ν is named a cubic ideal, if for all ς, ω ∈ Ν. (CI1) 𝜇Θ(0) ≥ 𝜇Θ(ς)and 𝜆Θ(0) ≤ 𝜆Θ(ς). (CI2) 𝜇Θ(ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ς ∗ ω), 𝜇Θ(ς)}, 𝜆Θ(ω) ≤ 𝑚𝑎𝑥 {𝜆Θ(ς ∗ ω)ˑ, 𝜆Θ(ς)}. (CI3) 𝜇Θ(ς ∘ ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ς), ˑ𝜇Θ(ω)}, 𝜆Θ(ς ∘ ω) ≤ 𝑚𝑎𝑥 {𝜆Θ(ς)ˑ, ˑ𝜆Θ(ω)}. Definition(8) [22]. The cubic set Θ of Ν is named a cubic k-ideal if for all ς, ω, κ ∈ Ν (𝑪𝒌𝟏 ) �̃�Θ(0)) ≥ 𝜇Θ(ς) , and 𝜆Θ(0) ≤ 𝜆Θ(ς). (𝑪𝒌𝟐) �̃�Θ(ς ∗ κ) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ς ∗ (ω ∗ κ)), 𝜇Θ(ω)}. 𝜆Θ(ς ∗ κ) ≤ 𝑚𝑎𝑥 {𝜆Θ(ς ∗ (ω ∗ κ)), 𝜆Θ(ω)}. (𝑪𝒌𝟑) �̃�Θ(ς ∘ ω) ≥ 𝑟𝑚𝑖𝑛 {�̃�Θ(ς)ˑ, 𝜇Θ(ω)} ,𝜆Θ(ς ∘ ω) ≤ 𝑚𝑎𝑥 {𝜆Θ(ς)ˑ, ˑ𝜆Θ(ω)}. Theorem(9) [22]. Let (Ν,∗,∘ ,0) be a KU-semigroup. A non-empty subset Θ is a cubic k-ideal if and only if it is a cubic ideal of Ν . 3. Cubic Positive Implicative-Ideals of 𝚴 Definition(10). The cubic set Θ of Ν is named a cubic positive implicative-ideal if for all ς, ω, κ ∈ Ν (𝑪𝒑𝟏 )�̃�Θ(0)) ≥ 𝜇Θ(ς) , and 𝜆Θ(0) ≤ 𝜆Θ(ς). (𝑪𝒑𝟐)�̃�Θ(κ ∗ ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ (ς ∗ ω)), 𝜇Θ(κ ∗ ς)}, 𝜆Θ(κ ∗ ω) ≤ 𝑚𝑎𝑥 {𝜆Θ(κ ∗ (ς ∗ ω)), 𝜆Θ(κ ∗ ς)}. (𝑪𝒑𝟑)�̃�Θ(ς ∘ ω) ≥ 𝑟𝑚𝑖𝑛 {𝜇Θ(ς)ˑ, 𝜇Θ(ω)} ,𝜆Θ(ς ∘ ω) ≤ 𝑚𝑎𝑥 {𝜆Θ(ς)ˑ, ˑ𝜆Θ(ω)}. Definition(11). The cubic set Θ of Ν is named a cubic implicative-ideal if for all ς, ω, κ ∈ Ν (𝑪𝑽𝟏 )�̃�Θ(0)) ≥ 𝜇Θ(ς) , and 𝜆Θ(0) ≤ 𝜆Θ(ς). (𝑪𝑽𝟐)𝜇Θ((ς ∗ ω) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ ((ς ∗ ω) ∗ ς)), 𝜇Θ(κ)}, IHJPAS. 37 (1) 2024 457 𝜆Θ((ς ∗ ω) ∗ ς) ≤ 𝑚𝑎𝑥 {𝜆Θ(κ ∗ ((ς ∗ ω) ∗ ς) , 𝜆Θ(κ)}. (𝑪𝑽𝟑)𝜇Θ(ς ∘ ω) ≥ 𝑟𝑚𝑖𝑛 {𝜇Θ(ς)ˑ, 𝜇Θ(ω)}, 𝜆Θ(ς ∘ ω) ≤ 𝑚𝑎 𝑥{𝜆Θ(ς)ˑ, ˑ𝜆Θ(ω)}. Definition(12). The cubic set Θ of Ν is named a cubic commutative-ideal if for all ς, ω, κ ∈ Ν (𝑪𝑪𝟏 ) �̃�Θ(0)) ≥ 𝜇Θ(ς) , and 𝜆Θ(0) ≤ 𝜆Θ(ς). (𝑪𝑪𝟐) 𝜇Θ((ς ∗ ω) ∗ ω) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ω ∗ (κ ∗ ς)), �̃�Θ(κ)}, 𝜆Θ((ς ∗ ω) ∗ ω) ∗ ς) ≤ 𝑚𝑎𝑥 {𝜆Θ(ω ∗ (κ ∗ ς)) , 𝜆Θ(κ)}. (𝑪𝑪𝟑) 𝜇Θ(ς ∘ ω) ≥ 𝑟𝑚𝑖𝑛 {𝜇Θ(ς)ˑ, 𝜇Θ(ω)}, 𝜆Θ(ς ∘ ω) ≤ 𝑚𝑎𝑥 {𝜆Θ(ς)ˑ, ˑ𝜆Θ(ω)}. Lemma(13). In a cubic positive implicative-ideal Θ of Ν, if ς ≤ ω, then 𝜇Θ(ς) ≥ 𝜇Θ(ω) and 𝜆Θ(ς) ≤ 𝜆Θ(ω), for all ς, ω ∈ Ν. Proof. Since ς ≤ ω ⇒ ω ∗ ς = 0, since Θ is a cubic positive implicative-ideal, then 𝜇Θ(κ ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ (ω ∗ ς)), 𝜇Θ(κ ∗ ω)}, put κ=0 𝜇Θ(0 ∗ ς) ≥ 𝑟 𝑚𝑖𝑛{�̃�Θ(0 ∗ (ω ∗ ς)), 𝜇Θ(0 ∗ ω)}, 𝜇Θ(ς) ≥ 𝑟 𝑚𝑖𝑛{�̃�Θ(0 ∗ 0), 𝜇Θ(ω)} = 𝜇Θ(ω), and 𝜆Θ(κ ∗ ς) ≤ 𝑚𝑎𝑥 {𝜆Θ(κ ∗ (ω ∗ ς)), 𝜆Θ(κ ∗ ω)}, put κ=0 𝜆Θ(0 ∗ ς) ≤ 𝑚𝑎𝑥{ 𝜆Θ(0 ∗ (ω ∗ ς) , 𝜆Θ(0 ∗ ω)}, 𝜆Θ(ς) ≤ 𝑚𝑎𝑥{ 𝜆Θ(0 ∗ 0) , 𝜆Θ(ω)} = 𝜆Θ(ω). ■ Theorem(14). In a cubic positive implicative-ideal Θ of Ν, if ς ∗ ω ≤ κ, then 𝜇Θ(ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ) , 𝜇Θ(ς)}, and 𝜆Θ(ω) ≤ max {𝜆Θ(κ) , 𝜆Θ(ς)}, for all ς, ω, κ ∈ Ν. Proof. Suppose ς ∗ ω ≤ κholds, then by Lemma(13) we get 𝜇Θ(ς ∗ ω) ≥ 𝜇Θ(κ), and 𝜆Θ(ς ∗ ω) ≤ 𝜆Θ(κ). Since Θ is a cubic positive implicative-ideal, that is 𝜇Θ(κ ∗ ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ (ς ∗ ω)), 𝜇Θ(κ ∗ ς)}, put κ = 0 𝜇Θ(0 ∗ ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(0 ∗ (ς ∗ ω)), 𝜇Θ(0 ∗ ς)}, 𝜇Θ(ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ς ∗ ω), 𝜇Θ(ς)}, but 𝜇Θ(ς ∗ ω) ≥ 𝜇Θ(κ) , then 𝜇Θ(ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ), 𝜇Θ(ς)} , and 𝜆Θ(κ ∗ ω) ≤ 𝑚𝑎𝑥{𝜆Θ(κ ∗ (ς ∗ ω)), 𝜆Θ(κ ∗ ς)}, put κ = 0, we obtain 𝜆Θ(ω) ≤ 𝑚𝑎𝑥{𝜆Θ((ς ∗ ω)), 𝜆Θ(ς)}, but 𝜆Θ(ς ∗ ω) ≤ 𝜆Θ(κ) , then 𝜆Θ(ω) ≤ 𝑚𝑎𝑥{𝜆Θ(κ), 𝜆Θ(ς)}, which is the required.■ Theorem(15). Every cubic positive implicative-ideal Θ of Ν is a cubic ideal. Proof. Let Θ be a cubic positive implicative-ideal, then 𝜇Θ(κ ∗ ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ (ς ∗ ω)) , 𝜇Θ(κ ∗ ς)}, 𝜆Θ(κ ∗ ω) ≤ 𝑚𝑎𝑥 {𝜆Θ(κ ∗ (ς ∗ ω)), 𝜆Θ(κ ∗ ς)}, put κ = 0 we get 𝜇Θ(0 ∗ ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(0 ∗ (ς ∗ ω)), 𝜇Θ(0 ∗ ς)}, IHJPAS. 37 (1) 2024 458 𝜇Θ(ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ς ∗ ω), 𝜇Θ(ς)} … … . . (1) , and 𝜆Θ(0 ∗ ω) ≤ 𝑚𝑎𝑥 {𝜆Θ(0 ∗ (ς ∗ ω)), 𝜆Θ(0 ∗ ς)}, 𝜆Θ(ω) ≤ 𝑚𝑎𝑥 {𝜆Θ(ς ∗ ω), 𝜆Θ(ς)} … … … (2) From (1) and (2) , Θ is a cubic ideal. The following example shows the converse of this theorem is not true, in general. Example(16). Let Ν = {0, 𝑎, 𝑏} with two operations ∗ and ∘ defined by Table 2: Table 2. a cubic ideal It follows that (Ν,∗,∘ ,0) is a KU-semigroup and define Θ by: 𝜇Θ(ς) = { [0.5,0.8] , 𝑖𝑓 ς = 0 [0.1,0.2] , 𝑖𝑓 ς = 𝑎 [0.1,0.3] , 𝑖𝑓 ς = 𝑏 and λΘ(ς) = { 0.1, 𝑖𝑓 ς = 0 0.5, 𝑖𝑓 ς = 𝑎 0.3, 𝑖𝑓 ς = 𝑏 And then we can prove that {0,a,b} is a cubic ideal but not a cubic positive implicative-ideal, since 𝜇Θ(0 ∗ 𝑎) = [0.1 , 0.2} ≤ 𝑟𝑚𝑖𝑛{𝜇Θ(0 ∗ (𝑏 ∗ 𝑎)), 𝜇Θ(0 ∗ 𝑏)} = [0.1,0.3]. Corollary(17). Every cubic positive implicative-ideal in a KU-semigroup is a cubic k-ideal. Proof. By referring to Theorem (15), we get Θ is a cubic ideal and by referring to Theorem (9) and upon it Θ is achieved the required. Lemma(18). If Θ is a cubic ideal of Ν, then Θ is a cubic positive implicative-ideal if and only if the following conditions hold (a) μ̃Θ((κ ∗ ς) ∗ (κ ∗ ω)) ≥ μ̃Θ(κ ∗ (ς ∗ ω)) and λΘ((κ ∗ ς) ∗ (κ ∗ ω)) ≤ λΘ(κ ∗ (ς ∗ ω)). Proof. Since Θ is a cubic ideal of Ν, then 𝜇Θ(κ ∗ ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ((κ ∗ ς) ∗ (κ ∗ ω)) , 𝜇Θ(κ ∗ ς)} and by above condition, we have 𝜇Θ(κ ∗ ω) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ (ς ∗ ω)) , 𝜇Θ(κ ∗ ς)} , also 𝜆Θ(κ ∗ ω) ≤ 𝑚𝑎𝑥{𝜆Θ((κ ∗ ς) ∗ (κ ∗ ω)) , 𝜆Θ(κ ∗ ς)} and by above condition, we have 𝜆Θ(κ ∗ ω) ≤ 𝑚𝑎𝑥{𝜆Θ(κ ∗ (ς ∗ ω)) , 𝜆Θ(κ ∗ ς)}, which is 𝑪𝒑𝟐, the conditions 𝑪𝒑𝟏 and 𝑪𝒑𝟑 are verify from Definition(10). On the contrary, by Theorem (15),Θ is a cubic ideal. * 0 a b 0 0 0 0 a 0 a 0 b 0 0 b ∘ 0 A b 0 0 A b a 0 0 a b 0 0 0 IHJPAS. 37 (1) 2024 459 Let α = ς ∗ ω 𝑎𝑛𝑑 β = (κ ∗ ς) ∗ ω, then 𝜇Θ(κ ∗ (α ∗ β)) = 𝜇Θ(κ ∗ ((ς ∗ ω) ∗ ((κ ∗ ς) ∗ ω)) ≥ 𝜇Θ(κ ∗ ((κ ∗ ς) ∗ ς)) by ku1 =𝜇Θ(0) by Theorem(2) . So 𝜇Θ(κ ∗ (α ∗ β)) = 𝜇Θ(0) and 𝜆Θ(κ ∗ (α ∗ β)) = 𝜆Θ(κ ∗ ((ς ∗ ω) ∗ (κ ∗ ς) ∗ ω)) ≥ 𝜆Θ(κ ∗ ((κ ∗ ς) ∗ ς)) by ku1 = 𝜆Θ(0) by Theorem (2). So 𝜆Θ(κ ∗ (α ∗ β)) = 𝜆Θ(0). Now by using Theorem (2) and the condition 𝑪𝒑𝟐 from definition of a cubic positive implicative-ideal we obtain: 𝜇Θ((κ ∗ ς) ∗ (κ ∗ ω)) = 𝜇Θ(κ ∗ (κ ∗ ς) ∗ ω) = 𝜇Θ(κ ∗ β) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ (α ∗ β)), 𝜇Θ(κ ∗ α)} = 𝑟𝑚𝑖𝑛{𝜇Θ(0), 𝜇Θ(κ ∗ α)} = 𝜇Θ(κ ∗ α) =𝜇Θ(κ ∗ (ς ∗ ω)) , and 𝜆Θ((κ ∗ ς) ∗ (κ ∗ ω)) = 𝜆Θ(κ ∗ (κ ∗ ς) ∗ ω) = 𝜆Θ(κ ∗ β) ≤ 𝑚𝑎𝑥{𝜆Θ(κ ∗ (α ∗ β)), 𝜆Θ(κ ∗ α)} = 𝑚𝑎𝑥{𝜆Θ(0), 𝜆Θ(κ ∗ α)} = 𝜆Θ(κ ∗ α) =𝜆Θ(κ ∗ (ς ∗ ω)) , which is the condition (a).■ Theorem(19). Every cubic commutative-ideal Θ of Ν is a cubic ideal. Proof. By definition of a cubic commutative-ideal, we have CI1 and CI3 are fulfilled (𝐂𝑪𝟐) 𝜇Θ((ς ∗ ω) ∗ ω) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ω ∗ (κ ∗ ς)), �̃�Θ(κ)}, put ω = 0 , we get 𝜇Θ((ς ∗ 0) ∗ 0) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(0 ∗ (κ ∗ ς)), �̃�Θ(κ)}, 𝜇Θ(ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ ς), �̃�Θ(κ)}, and 𝜆Θ((ς ∗ ω) ∗ ω) ∗ ς) ≤ 𝑚𝑎𝑥 {𝜆Θ(ω ∗ (κ ∗ ς)) , 𝜆Θ(κ)}, put ω = 0 𝜆Θ((ς ∗ 0) ∗ 0) ∗ ς) ≤ 𝑚𝑎𝑥 {𝜆Θ(0 ∗ (κ ∗ ς)) , 𝜆Θ(κ)}, 𝜆Θ(ς) ≤ 𝑚𝑎𝑥 {𝜆Θ(κ ∗ ς) , 𝜆Θ(κ)}, which is a CI2 condition, therefore Θ is a cubic ideal.■ Example (20). Let Ν = {0, 𝑎, 𝑏} with two operations ∗ and ∘ defined by the tables in Example (16). Define 𝜇Θ(ν) and 𝜆Θ(ν) as follows: 𝜇Θ(ς) = { [0.5 ,0.9] 𝑖𝑓 ς = 0, 𝑎 [0.2, 0.4] 𝑖𝑓 ς = 𝑏 𝑎𝑛𝑑 𝜆Θ(ς) = { 0.2 𝑖𝑓 ς = 0, 𝑎 0.5 𝑖𝑓 ς = 𝑏 Then we can prove that Θ is a cubic ideal but not a cubic commutative-ideal, since 𝜇Θ(((𝑏 ∗ 0) ∗ 0) ∗ 𝑏)) ≤ 𝑟𝑚𝑖𝑛{𝜇Θ(0 ∗ (𝑎 ∗ 𝑏)), 𝜇Θ(𝑎)}, 𝜇Θ(𝑏) = [0.2 , 0.4] ≤ 𝜇Θ(𝑎) = [0.5 , 0.9] and 𝜆Θ(((𝑏 ∗ 0) ∗ 0) ∗ 𝑏)) ≥ 𝑚𝑎𝑥{𝜆Θ(0 ∗ (𝑎 ∗ 𝑏)), 𝜆Θ(𝑎)} , So 𝜆Θ(𝑏) = 0.5 ≥ 𝜆Θ(𝑎) = 0.2, this is also a wrong phrase. So we conclude that the converse of the previous theorem is not true in general, as we will clarify in the following theorem IHJPAS. 37 (1) 2024 460 Theorem(21). A cubic ideal Θ of Ν is a cubic commutative-ideal if and only if it is satisfies the following inequalities: (a) μ̃Θ(ω ∗ ς) ≤ 𝜇Θ((ς ∗ ω) ∗ ω) ∗ ς). (b) λΘ(ω ∗ ς) ≥ 𝜆Θ((ς ∗ ω) ∗ ω) ∗ ς), for all ς, ω ∈ Ν. Proof. Let Θ is a cubic ideal which satisfies (a) and (b), then 𝜇Θ(ω ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ (ω ∗ ς)), 𝜇Θ(κ)}, by substituting (a) and using Theorem(2), we obtain 𝜇Θ((ς ∗ ω) ∗ ω) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ω ∗ (κ ∗ ς)), �̃�Θ(κ)} and 𝜆Θ(ω ∗ ς) ≤ 𝑚𝑎𝑥 {𝜆Θ(κ ∗ (ω ∗ ς)) , 𝜆Θ(κ)}, substituting (b) and using Theorem (2) we get 𝜆Θ((ς ∗ ω) ∗ ω) ∗ ς) ≤ 𝑚𝑎𝑥 {𝜆Θ(ω ∗ (κ ∗ ς)) , 𝜆Θ(κ)}, then Θ is a cubic commutative-ideal. On the contrary, suppose Θ = 〈𝜇Θ, 𝜆Θ〉 is a cubic commutative ideal, so 𝜇Θ((ς ∗ ω) ∗ ω) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ω ∗ (κ ∗ ς)), 𝜇Θ(κ)}, put κ = 0 𝜇Θ((ς ∗ ω) ∗ ω) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ω ∗ (0 ∗ ς)), 𝜇Θ(0)}, 𝜇Θ((ς ∗ ω) ∗ ω) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(ω ∗ ς), 𝜇Θ(0)}, 𝜇Θ((ς ∗ ω) ∗ ω) ∗ ς) ≥ 𝜇Θ(ω ∗ ς), which is (a) likewise put κ = 0 in 𝜆Θ((ς ∗ ω) ∗ ω) ∗ ς) ≤ 𝑚𝑎 𝑥{𝜆Θ(ω ∗ (κ ∗ ς)) , 𝜆Θ(κ)}, so we get 𝜆Θ((ς ∗ ω) ∗ ω) ∗ ς) ≥ 𝜆Θ(ω ∗ ς).■ Theorem(22). Every cubic implicative-ideal of Ν is a cubic ideal. Proof. By Definition(11), we have CI1 and CI3 are hold, then by (𝐂𝐕𝟐)𝜇Θ((ς ∗ ω) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ ((ς ∗ ω) ∗ ς)), 𝜇Θ(κ)}, take ω=0 𝜇Θ((ς ∗ 0) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ ((ς ∗ 0) ∗ ς)), 𝜇Θ(κ)}, 𝜇Θ(ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(κ ∗ ς), 𝜇Θ(κ)}, also take ω=0 in 𝜆Θ((ς ∗ ω) ∗ ς) ≤ 𝑚𝑎𝑥 {𝜆Θ(κ ∗ ((ς ∗ ω) ∗ ς) , 𝜆Θ(κ)}, we get 𝜆Θ(ς) ≤ 𝑚𝑎𝑥 {𝜆Θ(κ ∗ ς) , 𝜆Θ(κ)}.■ Theorem(23). A cubic ideal of Ν is a cubic implicative if it satisfies the following: μ̃Θ(ς) ≥ 𝜇Θ((ς ∗ ω) ∗ ς) and 𝜆Θ(ς) ≤ 𝜆Θ((ς ∗ ω) ∗ ς) , 𝑓𝑜𝑟 𝑎𝑙𝑙 ς, ω ∈ Ν Proof. Suppose Θ is a cubic ideal satisfies the above two inequalities, hence 𝜇Θ(ς) ≥ 𝜇Θ((ς ∗ ω) ∗ ς) ≥ 𝑟𝑚𝑖𝑛{�̃�Θ(κ ∗ ((ς ∗ ω) ∗ ς)), 𝜇Θ(κ)} and 𝜆Θ(ς) ≤ 𝜆Θ((ς ∗ ω) ∗ ς) ≤ 𝑚𝑎𝑥{𝜆𝛩(κ ∗ ((ς ∗ ω) ∗ ς)), 𝜆𝛩(κ)}.■ Theorem(24). If Θ of Ν is a cubic positive implicative-ideal and a cubic commutative-ideal, then Θ is a cubic implicative-ideal. IHJPAS. 37 (1) 2024 461 Proof. By Theorem (21) and Lemma (18), we have: 𝜇Θ((ς ∗ ω) ∗ ς) ∗ ς) ≥ 𝜇Θ((ς ∗ (ς ∗ ω)) ∗ (ς ∗ ω)) ≥ 𝜇Θ(ς ∗ ((ς ∗ ω) ∗ ω)) = 𝜇Θ((ς ∗ ω) ∗ (ς ∗ ω)) and by Theorem(2) = 𝜇Θ(0) . Also, 𝜆Θ((ς ∗ ω) ∗ ς) ∗ ς) ≤ 𝜆Θ((ς ∗ (ς ∗ ω)) ∗ (ς ∗ ω)) ≤ 𝜆Θ(ς ∗ (ω ∗ (ς ∗ ω))) = 𝜆Θ((ς ∗ ω) ∗ (ς ∗ ω)) and by Theorem(2) = 𝜆Θ(0) By two Theorems (15) and (19) and Θ is a cubic-ideal, then 𝜇Θ(ς) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(((ς ∗ ω) ∗ ς) ∗ ς) , 𝜇Θ((ς ∗ ω) ∗ ς)} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ(0) , 𝜇Θ((ς ∗ ω) ∗ ς)} = 𝜇Θ((ς ∗ ω) ∗ ς), and 𝜆Θ(ς) ≤ 𝑚𝑎𝑥{𝜆Θ((ς ∗ ω) ∗ ς) ∗ ς), 𝜆Θ((ς ∗ ω) ∗ ς)} ≤ 𝑚𝑎𝑥{𝜆Θ(0) , 𝜆Θ((ς ∗ ω) ∗ ς)} = 𝜆Θ((ς ∗ ω) ∗ ς) so from Theorem(23), we get the required.■ 4. 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