In this paper we define Akivis superalgebra and study enveloping superalgebras for this class of algebras, proving an analogous of the PBW Theorem.Lie and Malcev superalgebras are examples of Akivis superalgebras. For these particular superalgebras, we describe the connection between the classical enveloping superalgebras and the corresponding generalized concept defined in this work. . Financial Support by CMUC-FCT gratefully acknowledged 2 Examples of Akivis superalgebras Lie superalgebras and more generally Malcev superalgebras are Akivis superalgebras. For the first class, we consider the trilinear map A to be the zero map and, for the second class, we take A(x, y, z) = 1/6 SJ(x, y, z). Here SJ(x, y, z) denotes the superjacobian SJ(x, y, z) = [[x, y], z] + (−1) α(β+γ) [[y, z], x] + (−1) γ(β+α) [[z, x], y], of the homogeneous elements x ∈ M α , y ∈ M β , z ∈ M γ , (α, β, γ ∈ Z 2 ). Next, we give two examples of Akivis superalgebras which are not included in these classes. Consider the algebra of octonions O as the algebra obtained by Cayley-Dickson Process from the quaternions Q, with the Z 2 gradation O 0 = Q =< 1, e 1 , e 2 , e 3 > and O 1 = e 4 Q =< e 4 , e 5 , e 6 , e 7 > .
