\(f_q\)-derivations of \(G\)-algebra (Q1751497)

Summary: We introduce the notion of \(f_q\)-derivation as a new derivation of \(G\)-algebra. For an endomorphism map \(f\) of any \(G\)-algebra \(X\), we show that at least one \(f_q\)-derivation of \(X\) exists. Moreover, for such a map, we show that a self-map \(d_q^f\) of \(X\) is \(f_q\)-derivation of \(X\) if \(X\) is an associative medial \(G\)-algebra. For a medial \(G\)-algebra \(X\), \(d_q^f\) is \(f_q\)-derivation of \(X\) if \(d_q^f\) is an outside \(f_q\)-derivation of \(X\). Finally, we show that if \(f\) is the identity endomorphism of \(X\) then the composition of two \(f_q\)-derivations of \(X\) is a \(f_q\)-derivation. Moreover, we give a condition to get a commutative composition.