  In this paper we construct a non-skewsymmetric version of a Poisson bracket
on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to
such Poisson-like brackets as Loday-Poisson brackets. We examine the relations
between the Hamiltonian vector fields with respect to both the odd Jacobi
structure and the Loday-Poisson structure. Furthermore, we show that the
Loday-Poisson bracket satisfies the Leibniz rule over the noncommutative
product derived from the homological vector field.
