Symmetric bi-derivations in 𝑑-algebras

In this thesis, 𝐵𝐶𝐾-algebras, one of the fundamental examples of algebraic structures, are introduced, and their natural generalization, 𝑑-algebras, is examined with a focus on their fundamental properties. The structural implications of the derivation concept on 𝑑-algebras are elaborated; in this context, symmetric bi-derivations are defined and their algebraic properties are thoroughly analyzed. The relationships between this type of derivation and certain subsets of 𝑑-algebras are demonstrated with supporting examples. Furthermore, generalized symmetric bi-derivations defined on edge 𝑑- algebras, along with the invariance conditions of 𝑑-ideals under these derivations, are evaluated, leading to a variety of structural results.