Lukasiewicz Fuzzy BM-Algebra and BM-Ideal

Abstract

Introduction: ℱ???????????????? 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 ℱ???????????????? 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 ℱ???????????????? 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 ℱ???????????????? 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 ℱ???????????????? set theory and incorporating Lukasz logic theory with the inclusion of certain attributes, in order to facilitate the production of Lukasz ℱ???????????????? BM-algebra and BM-ideal, wherein the characteristics and attributes of the Lukasz ℱ???????????????? BM-algebra and BM-ideal are examined, and the relationships between them are demonstrated by a few examples.
Results: 
Theorem 3.5. Every Lukasz ℱ???????????????? set L_U^ε is a Lukasz ℱ???????????????? BM-algebra of G iff it satisfies: L_U^ε (p ̇*q ̇)≥min{L_U^ε (p ̇ ),L_U^ε (q ̇ )}, ∀ p ̇,q ̇∈G. 
Theorem 3.6. Show that ε-Lukasz ℱ???????????????? set L_U^ε in G is an ε-Lukasz ℱ???????????????? BM-algebra of G, if U is a ℱ???????????????? sub algebra of G. An example has been provided to show that the converse is not true.
Theorem 4.3. Every Lukasz ℱ???????????????? set L_U^ε of a ℱ???????????????? set U in G is a Lukasz ℱ???????????????? BM-ideal of G if and only if it satisfies 
         (i) ∀ p ̇∈G, ∀ u_a∈(0,1], [p ̇⁄u_a ]∈L_U^ε  ⇒[0⁄u_a ]∈L_U^ε
          (ii) ∀ p ̇,q ̇∈G, L_U^ε (p ̇)≥min{L_U^ε (p ̇*q ̇ ),L_U^ε (q ̇ )}
Conclusions: The application of BM-algebra within Lukasiewicz ℱ???????????????? logic operation can optimize public transportation system by scheduling time and routing 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 ℱ???????????????? BM-algebra and Lukasz ℱ???????????????? BM-ideal along with some of their properties are investigated. In addition to the characterization of both Lukasz ℱ???????????????? BM-algebra and BM-ideal, the relations of ℱ???????????????? subalgebra, ℱ???????????????? ideal, Lukasz ℱ???????????????? set, Lukasz ℱ???????????????? BM-algebra and Lukasz ℱ???????????????? 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.