On Derivations of Pseudo BN-algebras - Mahiroh; Sri Gemawati; Syamsudhuha (Universitas Islam Negeri Raden Intan Lampung, 2026)
Derivation theory plays a fundamental role in implication-based algebra by providing operator frameworks that reveal the structural behavior of algebraic systems. Although numerous derivation concepts have been established for BN-algebras and related implication algebras, an appropriate derivation framework for pseudo BN-algebras has remained unavailable because their dual-operation structure cannot be accommodated by existing derivation formulations. This study aims to establish a unified derivation framework for pseudo BN-algebras through the introduction of algebraically compatible operator mappings. Using a deductive mathematical approach, two auxiliary binary operations, ⊛ and , are constructed to define Type 1 and Type 2 (l,r)-derivations, (r,l)-derivations, and left derivations. Their fundamental properties are then investigated through formal definitions, propositions, and rigorous mathematical proofs. The obtained results show that both derivation systems satisfy regularity conditions, preserve essential identities involving the distinguished zero element, and maintain structural consistency with the defining axioms of pseudo BN-algebras. More importantly, the proposed framework demonstrates that derivation theory for pseudo BN-algebras cannot be obtained by directly extending existing derivation concepts but instead requires new algebraic constructions that are intrinsically determined by the interaction of their two binary operations. Consequently, this study establishes the first derivation framework for pseudo BN-algebras, broadens the scope of derivation theory within implication-based algebra, and provides a rigorous theoretical foundation for future investigations of generalized derivations, derivation-induced ideals, homomorphisms, congruence relations, fuzzy derivations, and other operator structures on generalized implication algebras.