On the Structural Behavior of Multiplicative (Generalized)‐Derivations via d ‐Algebra Structures

In the context of a d ‐algebra structure (℧, ∗, 0), this paper aims to introduce the concept of a multiplicative (generalized)‐derivation associated with a self‐map Ξ (not necessarily a derivation). Based on this concept, the operations ∧ and composition ° will be defined, and several interesting related properties will be investigated, such as regularity, d ‐ideal, kernel, d ‐subalgebra, fixed set, and ‐invariant. Moreover, we will demonstrate that by strengthening the condition of a d ‐algebra ℧ to include the property ( ℘ ∗( ℘ ∗ ϖ )) = ϖ , the collection of all multiplicative (generalized)‐derivations forms a semigroup on ℧. Furthermore, several relevant consequences and examples will be explored.