Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 131 https://internationalpubls.com Lukasiewicz Fuzzy BM-Algebra and BM-Ideal T. Gokila1, Dr. M. Mary Jansirani2 1Research Scholar; School of Sciences; Division of Mathematics; SRM- Institute of Science and Technology (Deemed to be University); Irungalur, Trichy – 621105, Tamil Nadu 2Associate Professor; School of Science; Division of Mathematics; SRM-Institute of Science and Technology (Deemed to be University); Irungalur, Trichy – 621105, Tamil Nadu Article History: Received: 24-05-2024 Revised: 13-07-2024 Accepted: 26-07-2024 Abstract: Introduction: ℱ𝑢𝑧𝑧𝑦 Sets is a mathematical framework that expands the traditional concept of sets by enabling elements to have degrees of membership. This enables partial membership based on degree of likeness. In classical set theory, an element can be represented as a crisp set, indicated by 𝑥, which either belongs to or does not belong to the set. In contrast, an ℱ𝑢𝑧𝑧𝑦 Sets allows for various levels of membership. The level of membership has a value somewhere between 0 and 1, with 0 representing non- participation and 1 representing full participation. The shape of the member function varies according to the application and intended behaviour. Jan Lukasiewicz was a logical thinker and philosopher. He contributed to the advancement of proportional logic. Lukasiewicz or Lukasz logic is an uncommon and highly appreciated logic that follows the Lukasz t-norm and t-conorm operations to compute the intersection and union of ℱ𝑢𝑧𝑧𝑦 Sets. This logic enables reasoning with unclear or incomplete knowledge, making it appropriate for a variety of applications including ambiguity and imprecision. Objectives: Incorporation of Lukasz logic theory to ℱ𝑢𝑧𝑧𝑦 set in 𝐵𝑀-algebra for the betterment of algorithms to address a variety of real-world issues, including risk management, decision making, managing public transit, diagnosing medical conditions and more. Methods: Applying 𝐵𝑀-algebra to ℱ𝑢𝑧𝑧𝑦 set theory and incorporating Lukasz logic theory with the inclusion of certain attributes, in order to facilitate the production of Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-algebra and 𝐵𝑀-ideal, wherein the characteristics and attributes of the Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-algebra and 𝐵𝑀-ideal are examined, and the relationships between them are demonstrated by a few examples. Results: Theorem 3.5. Every Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊 iff it satisfies: 𝐿𝑈 𝜀 (�̇� ∗ �̇�) ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇�), 𝐿𝑈 𝜀 (�̇�)}, ∀ �̇�, �̇� ∈ 𝔊. Theorem 3.6. Show that 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 in 𝔊 is an 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀- 𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊, if 𝑈 is a ℱ𝑢𝑧𝑧𝑦 sub algebra of 𝔊. An example has been provided to show that the converse is not true. Theorem 4.3. Every Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 of a ℱ𝑢𝑧𝑧𝑦 set 𝑈 in 𝔊 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊 if and only if it satisfies (i) ∀ �̇� ∈ 𝔊, ∀ 𝑢𝑎 ∈ (0,1], [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 ⇒ [0 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 (ii) ∀ �̇�, �̇� ∈ 𝔊, 𝐿𝑈 𝜀 (�̇�) ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇� ∗ �̇�), 𝐿𝑈 𝜀 (�̇�)} Conclusions: The application of 𝐵𝑀-algebra within Lukasiewicz ℱ𝑢𝑧𝑧𝑦 logic operation can optimize public transportation system by scheduling time and routing Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 132 https://internationalpubls.com based on passengers need. It also improves service reliability using operational constraints taken from the field. This study give rise to the notion of Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-algebra and Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀- ideal along with some of their properties are investigated. In addition to the characterization of both Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-algebra and 𝐵𝑀-ideal, the relations of ℱ𝑢𝑧𝑧𝑦 subalgebra, ℱ𝑢𝑧𝑧𝑦 ideal, Lukasz ℱ𝑢𝑧𝑧𝑦 set, Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-algebra and Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-ideal are discussed. Some examples are provided based on those relations. In the future, we will construct an algorithm for the advancement of transportation, making use of the ideas and results of this study. Keywords: BM-Algebra, ℱ𝑢𝑧𝑧𝑦 subalgebra, Lukasz ℱ𝑢𝑧𝑧𝑦 set, Lukasz ℱ𝑢𝑧𝑧𝑦 BM- Algebra, Lukasz ℱ𝑢𝑧𝑧𝑦 BM-Ideal. 1. Introduction In 1966, 𝐵𝐶𝐾/𝐵𝐶𝐼-algebra are developed by Y. Imai, K. Iseki and S. Tanaka [4]. There were other algebraic structures besides 𝐵𝐶𝐼 and 𝐵𝐶𝐾 algebras. These structures belong to universal algebra that describes fragments of proportional calculus. Such algebraic structures are 𝐵𝐶𝐶/𝐵𝐶𝐻/𝐵/𝐵𝐸- algebras, etc. These algebras can be explored both theoretically and practically in Mathematics and Computer science. In 2006, a specialized 𝐵-algebra, called 𝐵𝑀-algebras was delivered by [2]. The concept of Lukasz ℱ𝑢𝑧𝑧𝑦 subalgebra in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras was built by Jun using the thoughts of Lukasz t-norm [8]. Later, he extended it to Lukasz ℱ𝑢𝑧𝑧𝑦 ideal in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras in 2023 [7]. In 2002, Jun and Ahn designed the concept of Lukasz ℱ𝑢𝑧𝑧𝑦 set in 𝐵𝐸-𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠 to be Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝐸-𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠 and 𝐵𝐸-filters [10] along with the discussion of relationship between Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝐸-algebra and Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝐸-𝑓𝑖𝑙𝑡𝑒𝑟𝑠. Their capacity to handle partial truth values, include fuzzy principles and integrate them into the decision-making process leads to a variety of applications. To build an advanced algorithm for the solution of real-life problems, we can explore various algebraic structures. This study led to explore the concept of Lukasz ℱ𝑢𝑧𝑧𝑦 BM-algebra and BM-ideal using the notion of Lukasz ℱ𝑢𝑧𝑧𝑦 set to the given ℱ𝑢𝑧𝑧𝑦 set in BM-algebra and investigated some of their properties, characterizations and relations with some examples. 2. Objectives Definition 2.1 The set 𝔊 be a non-empty set.The 𝑩𝑴-𝒂𝒍𝒈𝒆𝒃𝒓𝒂 satisfies the given axioms: (𝐵𝑀1) �̇� ∗ 0 = �̇� (𝐵𝑀2)(�̇� ∗ �̇�) ∗ (�̇� ∗ �̇�) = �̇� ∗ �̇�, for all �̇�, �̇�, �̇� ∈ 𝔊 under binary operation " ∗ " with a constant element "0" Remark 2.2 Every 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 satisfies (i) (�̇� ∗ 𝓅)̇ = 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 133 https://internationalpubls.com (ii) (0 ∗ (0 ∗ �̇�)) = �̇� (iii) (0 ∗ (�̇� ∗ �̇�)) = �̇� ∗ �̇� (iv) (�̇� ∗ �̇�) ∗ (�̇� ∗ �̇�) = �̇� ∗ �̇� (v) (�̇� ∗ 𝓆)̇ = 0 ⇔ (�̇� ∗ 𝓅)̇ = 0 for all �̇�, �̇�, �̇� ∈ 𝔊. Definition 2.3 A 𝐵𝑀-algebra 𝔊 is called subalgebra of 𝔊 if 𝔖 be a subset of 𝔊 then �̇� ∗ �̇� ∈ 𝔖, ∀�̇�, �̇� ∈ 𝔖 Definition 2.4 A 𝐵𝑀-algebra 𝔊 is called ideal of 𝔊 if 𝔖 be a subset of 𝔊 then 0 ∈ 𝔊 and �̇� ∗ �̇� ∈ 𝔖, �̇� ∈ 𝔖 ⟹ �̇� ∈ 𝔖, ∀�̇�, �̇� ∈ 𝔊 Definition 2.5 Zadeh’s ℱ𝑢𝑧𝑧𝑦 set 𝑈 in 𝔊 takes the form 𝑈(�̇�) = { 𝑢 ∈ (0,1] 𝑖𝑓 �̇� = �̇� 0 𝑖𝑓 �̇� ≠ �̇� is regarded as ℱ𝑢𝑧𝑧𝑦 point with support �̇� and value 𝑢. It is viewed by [�̇� 𝑢⁄ ]. Definition 2.6 A ℱ𝑢𝑧𝑧𝑦 point [�̇� 𝑢⁄ ] in every ℱ𝑢𝑧𝑧𝑦 set 𝑈 in 𝔊 is (i) contained in 𝑈, noted by [�̇� 𝑢⁄ ] ∈ 𝑈 if 𝑈(�̇�) ≥ 𝑢. (ii) quasi-coincident with 𝑈, noted by [�̇� 𝑢⁄ ]𝔮𝑈 if 𝑈(�̇�) + 𝑢 > 1 Definition 2.7 A ℱ𝑢𝑧𝑧𝑦 set 𝑈 is called ℱ𝑢𝑧𝑧𝑦 subalgebra of a 𝐵𝑀-algebra 𝔊 if it satisfies (𝐹𝐴1) 𝑈(�̇� ∗ �̇�) ≥ 𝑚𝑖𝑛{𝑈(�̇�), 𝑈(�̇�)} , ∀ �̇�, �̇� ∈ 𝔊 Definition 2.8 A ℱ𝑢𝑧𝑧𝑦 set 𝑈 is called ℱ𝑢𝑧𝑧𝑦 ideal of a 𝐵𝑀-algebra 𝔊 if it satisfies (𝐹𝐼1) 𝑈(0) ≥ 𝑈(�̇�) (𝐹𝐼2) 𝑈(�̇�) ≥ 𝑚𝑖𝑛{𝑈(�̇� ∗ �̇�), 𝑈(�̇�)}, ∀ �̇�, �̇� ∈ 𝔊 3. Methods Definition 3.1 An 𝜺 − 𝑳𝒖𝒌𝒂𝒔𝒛 𝓕𝐮𝐳𝐳𝐲 𝑺𝒆𝒕 of ℱ𝑢𝑧𝑧𝑦 set 𝑈 in 𝔊 is a function from the BM-algebra 𝔊 to [0,1] and 𝜀 ∈ [0,1]. 𝐿𝑈 𝜀 : 𝔊 → [0,1], �̇� ↦ 𝑚𝑎𝑥 {0, 𝑈(�̇�) + 𝜀 − 1} (3.1) Remark 3.2 If 𝑈 is a ℱ𝑢𝑧𝑧𝑦 set in 𝔊, then its 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 satisfies Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 134 https://internationalpubls.com 𝑈(�̇�) ≥ 𝑈(�̇�) ⇒ 𝐿𝑈 𝜀 (�̇�) ≥ 𝐿𝑈 𝜀 (�̇�), ∀ �̇�, �̇� ∈ 𝔊 (3.2) Proof Suppose 𝑈 be a ℱ𝑢𝑧𝑧𝑦 set in 𝔊 and 𝑈(�̇�) ≥ 𝑈(�̇�) then 𝐿𝑈 𝜀 (�̇�) = 𝑚𝑎𝑥{0, 𝑈(�̇�) + 𝜀 − 1} ≥ 𝑚𝑎𝑥{0, 𝑈(�̇�) + 𝜀 − 1} = 𝐿𝑈 𝜀 (�̇�). Thus 𝐿𝑈 𝜀 (�̇�) ≥ 𝐿𝑈 𝜀 (�̇�). Definition 3.3 An 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 in 𝔊 is called an 𝜺- Lukasz ℱ𝑢𝑧𝑧𝑦 subalgebra of 𝕲 or 𝜺- Lukasz ℱ𝑢𝑧𝑧𝑦 𝑩𝑴 − 𝒂𝒍𝒈𝒆𝒃𝒓𝒂 of 𝕲 if it satisfies (𝐿𝐹𝐴1) [�̇� 𝑢𝑎⁄ ], [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 ⇒ [(�̇� ∗ �̇�) 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}⁄ ] ∈ 𝐿𝑈 𝜀 (3.3) for all �̇�, �̇� ∈ 𝐺,𝜀 ∈ (0,1) and 𝑢𝑎, 𝑢𝑏 ∈ (0,1]. Example 3.4 Let 𝔊 = {0, 𝓅1̇, 𝓅2̇, 𝓅3̇} be a set and Table 3.1 shows the Cayley table of 𝔊 under " ∗ " ∗ 0 𝓅1̇ 𝓅2̇ 𝓅3̇ 0 0 𝓅1̇ 𝓅2̇ 𝓅3̇ 𝓅1̇ 𝓅1̇ 0 0 𝓅2̇ 𝓅2̇ 𝓅2̇ 0 0 𝓅1̇ 𝓅3̇ 𝓅3̇ 𝓅2̇ 𝓅1̇ 0 TABLE 3.1 Cayley table with respect to " ∗ " Then 𝔊 is a 𝐵𝑀 − 𝑎𝑙𝑔𝑒𝑏𝑟𝑎. Defining a ℱ𝑢𝑧𝑧𝑦 set 𝑈 in 𝔊 as follows: 𝑈: 𝔊 → [0,1], �̇� ↦ { 0.88 𝑖𝑓 �̇� = 0 0.69 𝑖𝑓 �̇� = {𝓅1̇, 𝓅2̇} 0.77 𝑖𝑓 �̇� = 𝓅3 ̇ . If it is taken that 𝜀 = 0.61, then the Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 of 𝑈 in 𝔊 is provided as follows: 𝐿𝑈 𝜀 : 𝔊 → [0,1], �̇� ↦ { 0.49 𝑖𝑓 �̇� = 0 0.3 𝑖𝑓 �̇� = {𝓅1̇, 𝓅2̇} 0.36 𝑖𝑓 𝓅 = 𝓅3̇ Typically, it is verified that 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊. Definition 3.5 A Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 in 𝔊 is called Lukasz ℱ𝑢𝑧𝑧𝑦 𝑩𝑴-𝒊𝒅𝒆𝒂𝒍 of 𝔊 if it satisfies (𝐿𝐹𝐼1) 𝐿𝑈 𝜀 (0) is an upper bound of {𝐿𝑈 𝜀 (�̇�)|�̇� ∈ 𝔊} (3.4) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 135 https://internationalpubls.com (𝐿𝐹𝐼2) [(�̇� ∗ �̇�) 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 , [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 ⇒ [�̇� 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}⁄ ] ∈ 𝐿𝑈 𝜀 (3.5) for all �̇�, �̇� ∈ 𝔊 and 𝑢𝑎 , 𝑢𝑏 ∈ (0,1]. Example 3.6 Suppose the set 𝔊 = {0, 𝓅1̇, 𝓅2̇, 𝓅3̇} be a 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 with respect to a binary operation " ∗ "given by Table 3.2 ∗ 0 𝓅1̇ 𝓅2̇ 𝓅3̇ 0 0 𝓅1̇ 𝓅2̇ 𝓅3̇ 𝓅1̇ 𝓅1̇ 0 𝓅3̇ 𝓅2̇ 𝓅2̇ 𝓅2̇ 𝓅3̇ 0 𝓅1̇ 𝓅3̇ 𝓅3̇ 𝓅2̇ 𝓅1̇ 0 TABLE 3.2 Cayley table with respect to " ∗ " Then 𝔊 is a 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎. ℱ𝑢𝑧𝑧𝑦 set 𝑈 in 𝔊 is defined as follows: 𝑈: 𝔊 → [0,1], �̇� ↦ { 0.91 𝑖𝑓 �̇� = 0 0.78 𝑖𝑓 �̇� = {𝓅1̇, 𝓅2̇} 0.83 𝑖𝑓 �̇� = 𝓅3 ̇ . If it is taken that 𝜀 = 0.54, then the Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 of 𝑈 in 𝔊 is provided as below 𝐿𝑈 𝜀 : 𝔊 → [0,1], �̇� ↦ { 0.45 𝑖𝑓 �̇� = 0 0.32 𝑖𝑓 �̇� = {𝓅1̇, 𝓅2̇} 0.37 𝑖𝑓 �̇� = 𝓅3̇ Typically, it is verified that 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊. 4. Results Theorem 4.1 Every Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊 iff it satisfies: 𝐿𝑈 𝜀 (�̇� ∗ �̇�) ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇�), 𝐿𝑈 𝜀 (�̇�)}, ∀ �̇�, �̇� ∈ 𝔊 (4.1) Proof Suppose 𝑈 be a ℱ𝑢𝑧𝑧𝑦 set in 𝔊. For instance, 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊. Let �̇�, �̇� ∈ 𝔊 and it is clear that [�̇� 𝐿𝑈 𝜀 (�̇�)⁄ ] ∈ 𝐿𝑈 𝜀 and [�̇� 𝐿𝑈 𝜀 (�̇�)⁄ ] ∈ 𝐿𝑈 𝜀 . From (3.3), it is evident that [(�̇� ∗ �̇�) 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇�), 𝐿𝑈 𝜀 (�̇�)}⁄ ] ∈ 𝐿𝑈 𝜀 , and hence 𝐿𝑈 𝜀 (�̇� ∗ �̇�) ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇�), 𝐿𝑈 𝜀 (�̇�)} for all �̇�, �̇� ∈ 𝔊. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 136 https://internationalpubls.com Conversely, suppose that 𝐿𝑈 𝜀 satisfies (4.1). Also let �̇�, �̇� ∈ 𝔊 and 𝑢𝑎 , 𝑢𝑏 ∈ (0,1] be such that [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 , [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 . Then 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑎 and 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑏, which imply from (4.1) that 𝐿𝑈 𝜀 (�̇� ∗ �̇�) ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇�), 𝐿𝑈 𝜀 (�̇�)} ≥ 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}. Thus [(�̇� ∗ �̇�) 𝑚𝑖𝑛{𝑢𝑎 , 𝑢𝑏}⁄ ] ∈ 𝐿𝑈 𝜀 . Therefore 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊. Theorem 4.2 Show that 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 in 𝔊 is an 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊, if 𝑈 is a ℱ𝑢𝑧𝑧𝑦 sub algebra of 𝔊. Proof For instance, 𝑈 is a ℱ𝑢𝑧𝑧𝑦 subalgebra of 𝔊. Let �̇�, �̇� ∈ 𝔊 and 𝑢𝑎, 𝑢𝑏 ∈ (0,1] be such that [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 , [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 . Then 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑎 and 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑏. Thus 𝐿𝑈 𝜀 (�̇� ∗ �̇�) = 𝑚𝑎𝑥{0, 𝑈(�̇� ∗ �̇�) + 𝜀 − 1} [∵ (3.1)] ≥ 𝑚𝑎𝑥{0, 𝑚𝑖𝑛{𝑈(�̇�), 𝑈(�̇�)} + 𝜀 − 1} [∵ (4.1)] = 𝑚𝑎𝑥{0, 𝑚𝑖𝑛{𝑈(�̇�) + 𝜀 − 1, 𝑈(�̇�) + 𝜀 − 1}} = 𝑚𝑖𝑛{𝑚𝑎𝑥{0, 𝑈(�̇�) + 𝜀 − 1}, 𝑚𝑎𝑥{0, 𝑈(�̇�) + 𝜀 − 1}} = 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇�), 𝐿𝑈 𝜀 (�̇�)} [∵ (3.1)] ≥ 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}. So, [(�̇� ∗ �̇�) 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}⁄ ] ∈ 𝐿𝑈 𝜀 . Hence 𝐿𝑈 𝜀 is a 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊. The subsequent example demonstrates why the reverse portion of Theorem 4.2 is false. Example 4.3 Suppose the set 𝔊 = {0, 𝓅1̇, 𝓅2̇} be a 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 and Table 4.1 shows binary operation " ∗ " in 𝔊 ∗ 0 𝓅1̇ 𝓅2̇ 0 0 𝓅2̇ 𝓅1̇ 𝓅1̇ 𝓅1̇ 0 𝓅2̇ 𝓅2̇ 𝓅2̇ 𝓅1̇ 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 137 https://internationalpubls.com TABLE 4.1 Cayley table with respect to " ∗ " Defining a ℱ𝑢𝑧𝑧𝑦 set 𝑈 in 𝔊 as follows: 𝑈: 𝔊 → [0,1], �̇� ↦ { 0.72 𝑖𝑓 �̇� = 0 0.51 𝑖𝑓 �̇� = 𝓅1̇ 0.43 𝑖𝑓 �̇� = 𝓅2̇ . Provided that 𝜀 = 0.49, then the 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 of 𝑈 in 𝔊 is formed as follows: 𝐿𝑈 𝜀 : 𝔊 → [0,1], �̇� ↦ { 0.21 𝑖𝑓 �̇� = 0 0 𝑖𝑓 �̇� = 𝓅1̇ 0 𝑖𝑓 �̇� = 𝓅2̇ Typically, it is verified that 𝐿𝑈 𝜀 is an 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊. But 𝑈 is not a ℱ𝑢𝑧𝑧𝑦 subalgebra of 𝔊 because of 𝑈(0 ∗ 𝓅1̇) = 𝑈(𝓅2̇) = 0.43 ≱ 0.51 = 𝑚𝑖𝑛{𝑈(0), 𝑈(𝓅1̇)}. Theorem 4.4 Every Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 of a ℱ𝑢𝑧𝑧𝑦 set 𝑈 in 𝔊 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊 if and only if it satisfies (i) ∀ �̇� ∈ 𝔊, ∀ 𝑢𝑎 ∈ (0,1], [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 ⇒ [0 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 (4.2) (ii) ∀ �̇�, �̇� ∈ 𝔊, 𝐿𝑈 𝜀 (�̇�) ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇� ∗ �̇�), 𝐿𝑈 𝜀 (�̇�)} (4.3) Proof For instance, 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊. Let �̇� ∈ 𝔊 and 𝑢𝑎 ∈ (0,1] be such that [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 . Utilising (3.4), leads to 𝐿𝑈 𝜀 (0) ≥ 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑎, and so [0 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 . Note that [(�̇� ∗ �̇�) 𝐿𝑈 𝜀 (�̇� ∗ �̇�)⁄ ] ∈ 𝐿𝑈 𝜀 , [�̇� 𝐿𝑈 𝜀 (�̇�)⁄ ] ∈ 𝐿𝑈 𝜀 for all �̇�, �̇� ∈ 𝔊. From (3.5), it is evident that [�̇� 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇� ∗ �̇�), 𝐿𝑈 𝜀 (�̇�)}⁄ ] ∈ 𝐿𝑈 𝜀 , and hence 𝐿𝑈 𝜀 (�̇�) ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇� ∗ �̇�), 𝐿𝑈 𝜀 (�̇�)} for all �̇�, �̇� ∈ 𝔊. Conversely, let us consider 𝐿𝑈 𝜀 satisfies (4.2) and (4.3). Since [�̇� 𝐿𝑈 𝜀 (�̇�)⁄ ] ∈ 𝐿𝑈 𝜀 for all �̇� ∈ 𝔊, we have [0 𝐿𝑈 𝜀 (�̇�)⁄ ] ∈ 𝐿𝑈 𝜀 and so 𝐿𝑈 𝜀 (0) ≥ 𝐿𝑈 𝜀 (�̇�) for all �̇� ∈ 𝔊 by (4.2). Hence, 𝐿𝑈 𝜀 (0) is an upper bound of {𝐿𝑈 𝜀 (�̇�)|�̇� ∈ 𝔊}. Also let �̇�, �̇� ∈ 𝔊 and 𝑢𝑎 , 𝑢𝑏 ∈ (0,1] be such that [(�̇� ∗ �̇�) 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 , [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 . Then 𝐿𝑈 𝜀 (�̇� ∗ �̇�) ≥ 𝑢𝑎 and 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑏, which imply from (4.3) that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 138 https://internationalpubls.com 𝐿𝑈 𝜀 (�̇�) ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇� ∗ �̇�), 𝐿𝑈 𝜀 (�̇�)} ≥ 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}. Thus [�̇� 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}⁄ ] ∈ 𝐿𝑈 𝜀 . Therefore 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊. 5. Discussion Remark 5.1 Prove that 𝜀 -Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 satisfies 𝐿𝑈 𝜀 (0) ≥ 𝐿𝑈 𝜀 (�̇�), ∀ �̇� ∈ 𝔊, if 𝑈 is a ℱ𝑢𝑧𝑧𝑦 sub algebra of 𝔊, then it Proof Let 𝑈 be a ℱ𝑢𝑧𝑧𝑦 subalgebra of 𝔊. Then, 𝑈(0) = 𝑈(�̇� ∗ �̇�) ≥ 𝑚𝑖𝑛{𝑈(�̇�), 𝑈(�̇�)} = 𝑈(�̇�). Therefore 𝑈(0) ≥ 𝑈(�̇�) for all �̇� ∈ 𝔊. From (3.2), it is evident that 𝐿𝑈 𝜀 (0) ≥ 𝐿𝑈 𝜀 (�̇�) for all �̇� ∈ 𝔊. Remark 5.2 Every ℱ𝑢𝑧𝑧𝑦 subalgebra 𝑈 of 𝔊 is said to be 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 if [∀ �̇�, �̇� ∈ 𝔊] [𝐿𝑈 𝜀 (�̇�) = 𝐿𝑈 𝜀 (0) ⇔ 𝐿𝑈 𝜀 (�̇� ∗ �̇�) ≥ 𝐿𝑈 𝜀 (�̇�)]. Proof Let 𝑈 be a ℱ𝑢𝑧𝑧𝑦 subalgebra of 𝔊. For instance, 𝐿𝑈 𝜀 (�̇�) = 𝐿𝑈 𝜀 (0) for all �̇� ∈ 𝔊. Then 𝐿𝑈 𝜀 (�̇� ∗ �̇�) ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇�), 𝐿𝑈 𝜀 (�̇�)} = 𝑚𝑖𝑛{𝐿𝑈 𝜀 (0), 𝐿𝑈 𝜀 (�̇�)} = 𝐿𝑈 𝜀 (�̇�). Combining the results of Theorem 4.2 and Remark 3.2 leads to 𝐿𝑈 𝜀 (�̇� ∗ �̇�) ≥ 𝐿𝑈 𝜀 (�̇�) for all �̇�, �̇� ∈ 𝔊. Conversely, suppose that 𝐿𝑈 𝜀 (�̇� ∗ �̇�) ≥ 𝐿𝑈 𝜀 (�̇�) for all �̇�, �̇� ∈ 𝔊. Utilising (𝐵𝑀1) leads to 𝐿𝑈 𝜀 (�̇�) = 𝐿𝑈 𝜀 (�̇� ∗ 0) ≥ 𝐿𝑈 𝜀 (0). Combining the results of above inequality and Remark 5.1 leads to 𝐿𝑈 𝜀 (�̇�) = 𝐿𝑈 𝜀 (0) for all �̇� ∈ 𝔊. Remark 5.3 Every ℱ𝑢𝑧𝑧𝑦 subalgebra 𝑈 of 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 𝔊 is said to be an 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 if Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 139 https://internationalpubls.com [∀ �̇� ∈ 𝔊] [𝐿𝑈 𝜀 (0 ∗ �̇�) ≥ 𝐿𝑈 𝜀 (�̇�)]. Proof Let 𝑈 be a ℱ𝑢𝑧𝑧𝑦 subalgebra of 𝐵𝑀-algebra 𝔊. Then, 𝑈(0 ∗ �̇�) ≥ 𝑚𝑖𝑛{𝑈(0), 𝑈(�̇�)} = 𝑈(�̇�) for all �̇� ∈ 𝔊. From (3.2), it is evident that 𝐿𝑈 𝜀 (0 ∗ �̇�) ≥ 𝐿𝑈 𝜀 (�̇�) for all �̇� ∈ 𝔊. Remark 5.4 If 𝑈 is a ℱ𝑢𝑧𝑧𝑦 sub algebra of 𝔊, then prove that 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 satisfies [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 , [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 ⇒ [(0 ∗ (�̇� ∗ �̇�))/𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}] ∈ 𝐿𝑈 𝜀 , ∀ �̇�, �̇� ∈ 𝔊 and 𝑢𝑎 , 𝑢𝑏 ∈ (0,1]. Proof Let 𝑈 be a ℱ𝑢𝑧𝑧𝑦 subalgebra of 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 𝔊. It is given that �̇�, �̇� ∈ 𝔊 and 𝑢𝑎 , 𝑢𝑏 ∈ (0,1] which implies [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 , [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 . Then 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑎 and 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑏. Thus 𝐿𝑈 𝜀 (0 ∗ (�̇� ∗ �̇�)) = 𝑚𝑎𝑥{0, 𝑈(0 ∗ (�̇� ∗ �̇�)) + 𝜀 − 1} = 𝑚𝑎𝑥{0, 𝑈(�̇� ∗ �̇�) + 𝜀 − 1} ≥ 𝑚𝑎𝑥{0, 𝑚𝑖𝑛{𝑈(�̇�), 𝑈(�̇�)} + 𝜀 − 1} ≥ 𝑚𝑎𝑥{0, 𝑚𝑖𝑛{𝑈(�̇�) + 𝜀 − 1, 𝑈(�̇�) + 𝜀 − 1}} ≥ 𝑚𝑖𝑛{𝑚𝑎𝑥{0, 𝑈(�̇�) + 𝜀 − 1}, 𝑚𝑎𝑥{0, 𝑈(�̇�) + 𝜀 − 1}} ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇�), 𝐿𝑈 𝜀 (�̇�)} ≥ 𝑚𝑖𝑛{𝑢𝑏 , 𝑢𝑎}. So [(0 ∗ (�̇� ∗ �̇�))/𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}] ∈ 𝐿𝑈 𝜀 . Lemma 5.5 Every Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 𝐿𝑈 𝜀 of 𝔊 satisfies the condition if �̇� ≤ �̇� and �̇� ∗ �̇� = 0, then [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 ⇒ [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 , ∀ �̇�, �̇� ∈ 𝔊, ∀ 𝑢𝑎 ∈ (0,1] (5.1) Proof Let �̇�, �̇� ∈ 𝔊 and 𝑢𝑎 ∈ (0,1] be such that �̇� ≤ �̇� , �̇� ∗ �̇� = 0 and [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 then (�̇� ∗ �̇�) = 0 and 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑎 so, 𝐿𝑈 𝜀 (�̇�) ≥ min{𝐿𝑈 𝜀 (�̇� ∗ �̇�), 𝐿𝑈 𝜀 (�̇�)} = 𝑚𝑖𝑛{𝐿𝑈 𝜀 (0), 𝐿𝑈 𝜀 (�̇�)} = 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑎. Hence [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 140 https://internationalpubls.com Lemma 5.6 Every Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 𝐿𝑈 𝜀 of 𝔊 fulfils the condition if �̇� ∗ �̇� ≤ �̇� and �̇� ∗ (�̇� ∗ �̇�) = 0, then [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 , [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 ⇒ [�̇� 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}⁄ ] ∈ 𝐿𝑈 𝜀 (5.2) for all �̇�, �̇�, �̇� ∈ 𝔊, and 𝑢𝑎 , 𝑢𝑏 ∈ (0,1] Proof Let �̇�, �̇�, �̇� ∈ 𝔊 and 𝑢𝑎 , 𝑢𝑏 ∈ (0,1] be such that �̇� ∗ �̇� ≤ �̇�, �̇� ∗ (�̇� ∗ �̇�) = 0, [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 and [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 . Then, (�̇� ∗ �̇�) ∗ �̇� = 0, 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑎 and 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑏. Hence, 𝐿𝑈 𝜀 (�̇�) ≥ min{𝐿𝑈 𝜀 (�̇� ∗ �̇�), 𝐿𝑈 𝜀 (�̇�)} ≥ 𝑚𝑖𝑛{𝑚𝑖𝑛{𝐿𝑈 𝜀 ((�̇� ∗ �̇�) ∗ �̇�), 𝐿𝑈 𝜀 (�̇�)}, 𝐿𝑈 𝜀 (�̇�)} = 𝑚𝑖𝑛{𝑚𝑖𝑛{𝐿𝑈 𝜀 (0), 𝐿𝑈 𝜀 (�̇�)}, 𝐿𝑈 𝜀 (�̇�)} = min{𝐿𝑈 𝜀 (�̇�), 𝐿𝑈 𝜀 (�̇�)} ≥ 𝑚𝑖𝑛{𝑢𝑏 , 𝑢𝑎} and so [�̇� 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}⁄ ] ∈ 𝐿𝑈 𝜀 . Remark 5.7 If 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 ideal of 𝔊, then it satisfies the following inequalities (i) [�̇� ≤ �̇� 𝑎𝑛𝑑 �̇� ∗ �̇� = 0 ⇒ 𝐿𝑈 𝜀 (�̇�) ≥ 𝐿𝑈 𝜀 (�̇�)] (5.3) (ii) [�̇� ∗ �̇� ≤ �̇� 𝑎𝑛𝑑 �̇� ∗ (�̇� ∗ �̇�) = 0 ⇒ 𝐿𝑈 𝜀 (�̇�) ≥ 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇�), 𝐿𝑈 𝜀 (�̇�)}] (5.4) for all �̇�, �̇�, �̇� ∈ 𝔊 Theorem 5.8 Every Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 in 𝔊 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊 if 𝑈 is a ℱ𝑢𝑧𝑧𝑦 ideal of 𝐵𝑀-algebra 𝔊. Proof For instance, 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 set of a ℱ𝑢𝑧𝑧𝑦 ideal 𝑈 in 𝔊. Let �̇�, �̇� ∈ 𝔊 and 𝑢𝑎, 𝑢𝑏 ∈ (0,1] be such that [(�̇� ∗ �̇�) 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 , [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 . Then 𝐿𝑈 𝜀 (�̇� ∗ �̇�) ≥ 𝑢𝑎 and 𝐿𝑈 𝜀 (�̇�) ≥ 𝑢𝑏. Thus 𝐿𝑈 𝜀 (�̇�) = 𝑚𝑎𝑥{0, 𝑈(�̇�) + 𝜀 − 1} [∵ (3.1)] ≥ 𝑚𝑎𝑥{0, 𝑚𝑖𝑛{𝑈(�̇� ∗ �̇�), 𝑈(�̇�)} + 𝜀 − 1} [∵ (4.3)] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 141 https://internationalpubls.com = 𝑚𝑎𝑥{0, 𝑚𝑖𝑛{𝑈(�̇� ∗ �̇�) + 𝜀 − 1, 𝑈(�̇�) + 𝜀 − 1}} = 𝑚𝑖𝑛{𝑚𝑎𝑥{0, 𝑈(�̇� ∗ �̇�) + 𝜀 − 1}, 𝑚𝑎𝑥{0, 𝑈(�̇�) + 𝜀 − 1}} = 𝑚𝑖𝑛{𝐿𝑈 𝜀 (�̇� ∗ �̇�), 𝐿𝑈 𝜀 (�̇�)} [∵ (3.1)] ≥ 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}. So, [�̇� 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}⁄ ] ∈ 𝐿𝑈 𝜀 . Hence 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊. The subsequent example demonstrates why the reverse portion of Theorem 5.8 is false. Example 5.9 Consider the 𝐵𝑀-algebra set 𝔊 in Example 3.4 and ℱ𝑢𝑧𝑧𝑦 set 𝑈 in 𝔊 defined by 𝑈: 𝔊 → [0,1], �̇� ↦ { 0.81 𝑖𝑓 �̇� = 0 0.42 𝑖𝑓 �̇� = 𝓅1̇ 0.57 𝑖𝑓 �̇� = 𝓅2̇ 0.31 𝑖𝑓 �̇� = 𝓅3̇ . Then 𝑈 is not a ℱ𝑢𝑧𝑧𝑦 ideal of 𝔊. Since 𝑈(𝓅1̇) = 0.42 ≱ 0.57 = 𝑚𝑖𝑛{𝑈(𝓅1̇ ∗ 𝓅2̇), 𝑈(𝓅2̇)} Given that 𝜀 = 0.55, then the Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 of 𝑈 in 𝔊 is provided as below: 𝐿𝑈 𝜀 : 𝔊 → [0,1], �̇� ↦ { 0.36 𝑖𝑓 �̇� = 0 0 𝑖𝑓 �̇� = 𝓅1̇ 0.12 𝑖𝑓 �̇� = 𝓅2̇ 0 𝑖𝑓 �̇� = 𝓅3̇ . and it is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊. Theorem 5.10 Every Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊. Proof For instance, 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊. Let �̇�, �̇� ∈ 𝔊 and 𝑢𝑎, 𝑢𝑏 ∈ (0,1] be such that [�̇� 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 , [�̇� 𝑢𝑏⁄ ] ∈ 𝐿𝑈 𝜀 . Since �̇� ∗ �̇� ≤ �̇� and �̇� ∗ (�̇� ∗ �̇�) = 0 we have [(�̇� ∗ �̇�) 𝑢𝑎⁄ ] ∈ 𝐿𝑈 𝜀 by (5.1). Hence [�̇� 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}⁄ ] ∈ 𝐿𝑈 𝜀 by (3.5), and so [(�̇� ∗ �̇�) 𝑚𝑖𝑛{𝑢𝑎, 𝑢𝑏}⁄ ] ∈ 𝐿𝑈 𝜀 by (5.1). Therefore 𝐿𝑈 𝜀 is a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊. The subsequent example demonstrates why the reverse portion of Theorem 5.10 is false. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 142 https://internationalpubls.com Example 5.11 A set in 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 𝔊 = {0, 𝓅1̇, 𝓅2̇} be considered and the Table 5.1 is built under the " ∗ " operation ∗ 0 𝓅1̇ 𝓅2̇ 0 0 𝓅1̇ 𝓅2̇ 𝓅1̇ 𝓅1̇ 0 𝓅1̇ 𝓅2̇ 𝓅2̇ 𝓅1̇ 0 TABLE 5.1 Cayley table with respect to " ∗ " Defining a ℱ𝑢𝑧𝑧𝑦 set 𝑈 in 𝔊 as follows 𝑈: 𝔊 → [0,1], �̇� ↦ { 0.84 𝑖𝑓 �̇� = 0 0.72 𝑖𝑓 �̇� = 𝓅1̇ 0.51 𝑖𝑓 �̇� = 𝓅2̇ . Given that 𝜀 = 0.58, the 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 set 𝐿𝑈 𝜀 of 𝑈 in 𝔊 is provided as below 𝐿𝑈 𝜀 : 𝔊 → [0,1], �̇� ↦ { 0.42 𝑖𝑓 �̇� = 0 0.3 𝑖𝑓 �̇� = 𝓅1̇ 0.09 𝑖𝑓 �̇� = 𝓅2̇ Typically, it is verified that 𝐿𝑈 𝜀 is an 𝜀-Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎 of 𝔊. But 𝐿𝑈 𝜀 is not a Lukasz ℱ𝑢𝑧𝑧𝑦 𝐵𝑀-𝑖𝑑𝑒𝑎𝑙 of 𝔊 because of 𝑈(𝓅2̇) = 0.09 ≱ 0.3 = 𝑚𝑖𝑛{𝑈(𝓅2̇ ∗ 𝓅1̇), 𝑈(𝓅1̇)}. References [1] A. Paad and A. Jafari, n-fold obstinate and n-fold fantastic (pre)filters of EQ-algebras, J. Algebra Relat. Topics, (1) 9 (2021), 31–50. [2] C. B. Kim, H. S. Kim, On BM-algebra, Sci. Math, Japan, 63 (2006), 421-427. [3] H. S. Kim and Y. H. Kim, On BE-algebras, Sci. Math. Japan, 66 (2007), 113-116. [4] K. Iseki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978), 1–26. [5] K. Iseki, On BCI-algebras, Math. Seminar Notes, 8 (1980), 125–130. [6] L. A. Zadeh, F𝑢𝑧𝑧𝑦 sets, Information and Control, (3) 8 (1965), 338–353. [7] Y.B. Jun, Lukasiewicz F𝑢𝑧𝑧𝑦 Ideals in BCK-Algebras and BCI-Algebras, Journal of Algebra and Related Topics, Vol. 11, No 1, (2023), pp 1-14. [8] Y. B. Jun, Lukasiewicz F𝑢𝑧𝑧𝑦 sub algebras in BCK-algebras and BCI-algebras, Ann. F𝑢𝑧𝑧𝑦 Math. Inform. (2) 23 (2022), 213–223. [9] Y. B. Jun, S. M. Hong, S. J. Kim and S. Z. Song, F𝑢𝑧𝑧𝑦 ideals and F𝑢𝑧𝑧𝑦 sub algebras of BCK-algebras, J. F𝑢𝑧𝑧𝑦 Math. 7 (1999), 411–418. [10] Y. B. Jun, S. S. Ahn, Lukasiewicz F𝑢𝑧𝑧𝑦 BE-algebras and BE-filters, European Journal of Pure and Applied Mathematics,15(2002), 924-937.