Multiplicative isomorphisms and derivations on axial algebras

In this paper, we show that the multiplicative derivations on $J\left(\alpha \right)$-axial algebras, with $\alpha \ne 1,0,\frac{1}{2}$, are additive under suitable conditions, which nowadays are called Martindale-type conditions. Besides, with proper assumptions, we proceed to study the additivity of multiplicative isomorphisms and derivations in the context of $M(\alpha ,\beta )$-axial algebras, except for multiplicative derivations when $\beta =\frac{1}{2}$. In this case, we mention a research question at the end.