Fuzzy Pseudo-ideal in  Pseudo-TM  Alegbra

Background: Fuzzy sets and ideals play a significant role in the study of algebraic structures, particularly in the context of pseudo-TM algebras, which are non-commutative generalizations of MV-algebras. However, the concept of fuzzy pseudo-ideals within these algebras has not been extensively explored. This paper introduces fuzzy pseudo-ideals in pseudo-TM algebras and investigates their key properties, contributing to the broader understanding of fuzzy algebraic structures. Methods We define fuzzy pseudo-ideals in the framework of pseudo-TM algebras and investigate their properties using level sets. The paper employs techniques from algebraic logic and set theory to characterize fuzzy pseudo-ideals and their interactions with homomorphism’s and Cartesian products. Several theorems are developed to establish the closure of fuzzy pseudo-ideals under intersection, and their relationships with homomorphism’s and Cartesian products are explored. Results The main findings include a comprehensive characterization of fuzzy pseudo-ideals in pseudo-TM algebras, both in terms of their algebraic structure and their behavior under intersections. We also show how these fuzzy pseudo-ideals interact with homomorphism’s and Cartesian products. Concrete examples are provided to illustrate the theoretical results, demonstrating the applicability of the concepts to real-world algebraic problems. Conclusions This research enhances the theoretical understanding of fuzzy sets and ideals within pseudo-TM algebras, offering new insights into their properties and interrelationships. The results pave the way for future work, particularly in extending the concepts to fuzzy pseudo-strong ideals and fuzzy pseudo-TM ideals. These extensions could further advance the study of fuzzy algebraic structures, contributing to the broader field of algebraic logic and fuzzy set theory.