Symmetric bi-derivations of subtraction algebras

Abstract

In this paper, we introduce the notion of symmetric bi-derivations on subtraction algebra and investigated some related properties. We prove that  a map $D : X\times X\to X$ is a symmetric bi-derivation  on $ X$ if and only if $D$ is  a symmetric map and  it satisfies $D(x-y, z)=D(x, z)-y$ for all $x, y, z\in X.$