  We introduce a strong homotopy notion of a cyclic symmetric inner product of
an A-infinity algebra and prove a characterization theorem in the formalism of
the infinity inner products by Tradler. We also show that it is equivalent to
the notion of a non-constant symplectic structure on the corresponding formal
non-commutative supermanifold. We show that (open Gromov-Witten type) potential
for a cyclic filtered A-infinity algebra is invariant under the cyclic filtered
A-infinity homomorphism up to reparametrization, cyclization and a constant
addition, generalizing the work of Kajiura.
