On Derivations of Tensor Products of Perm Algebras and Associative Algebras

The study of derivations and their generalizations on non-associative algebras has proven to be fundamental in understanding the internal symmetries and algebraic dynamics of such structures. In this paper, we investigate derivations and diderivations of tensor product dialgebras arising from the combination of a perm algebra and a unital associative algebra. We provide decomposition theorems that characterize these operators in terms of derivations of the individual factors and suitable multiplication maps. Explicit coordinate formulas are also derived, allowing concrete descriptions of the action of derivations and diderivations with respect to natural bases. These results extend classical decomposition theorems for tensor products beyond the associative setting, highlighting the interplay between perm algebras and non-associative algebraic frameworks.