The Zinbiel operad was defined by J.-L. Loday [Math. Scand. 77, No. 2, 189–196 (1995; Zbl 0859.17015)] as the Koszul dual of Leibniz operad. A Zinbiel algebra is a vector space \(A\) equipped with a binary operation \(\prec\) satisfying the relation \[ (x\prec y)\prec=x\prec (y\prec z+z\prec y) \] for all \(x,y,z \in A\). The symmetrized operation \[ x\dot y = x\prec y+y\prec x \] is commutative and associative. The commutative algebra associated to the free Zinbiel algebra is the shuffle algebra [J.-L. Loday, Manuscr. Math. 123, No. 1, 79–93 (2007; Zbl 1126.16029)].
In this paper the authors extend results from the theory of Zinbiel algebras to the theory of Zinbiel superalgebras. In particular, it is proved that any finite dimensional Zinbiel superalgebra is nilpotent. Moreover, the authors study null-filiform Zinbiel superalgebras and characterise complex naturally graded filiform Zinbiel superalgebras. Finally, the authors classify complex Zinbiel superalgebras up to dimension three.