Some Generalizations of Derivations on Jacobi-Jordan Algebras

This paper explores several generalizations of derivations on Jacobi-Jordan algebras, namely, anti-biderivations, almost inner (anti-)derivations, and local (anti-)derivations. We begin by investigating anti-biderivations, concentrating on their applications to diverse algebraic systems as well as their structural properties. Regarding the remaining generalizations of derivations, we focus on Jacobi-Jordan algebras of dimension less than 7. We prove that all almost inner derivations are inner, while almost inner anti-derivations are inner with a few exceptions. Finally, we demonstrate the existence of proper local (anti-)derivations and prove that they constitute a Lie algebra. This finding provides affirmative support for the conjecture concerning the Lie structure of local derivations.