  Higher nilpotent analogues of the $A-\infty$-structure are explicitly defined
on arbitrary simplicial complexes, generalizing explicit construction of
/hep-th/0704.2609. These structures are associated with the higher nilpotent
differential $d_n$, satisfying $d_n^n =0$, which is naturally defined on
triangulated manifolds (tetrahedral lattices). The deformation $D_n = (I +
\epsilon_n) d_n (I + \epsilon_n)^{-1}$ is defined with the help of the
$n$-versions of discrete exterior product $\wedge_n$ and the $K_n$-operator.
