The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the derivations' arguments. Let us explore the implications of this principle: in effect, we develop the calculus of variations on the infinite jet spaces for maps from sheaves of free associative algebras over commutative manifolds to the generators and then, quotients of free associative algebras over the linear relation of equivalence under cyclic shifts. In the frames of such variational noncommutative symplectic (super)geometry we prove the main properties of the Batalin-Vilkovisky Laplacian and variational Schouten bracket.
We show as by-product that the structures which arise in the variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption.
