  Let L be a finite-dimensional n-Lie algebra with free presentation F/R. Then
the concept of c-nilpotent multiplier of L, denoted by M(c)(L), is defined as
follows: M(c)(L) =(gamma c+1(F) R)/gamma c+1(R, F, . . . , F). In this paper,
we obtain some inequalities and certain bounds for the dimension of M(c)(L) by
using the basic commutators. Also, we discuss the relationship between the
dimension of the c-nilpotent multiplier of L and the c-nilpotent multiplier of
some factor of L. We further obtain an inequality between dimensions of
c-nilpotent multiplier of n-Lie algebra and non-abelian tensor (exterior)
product of a central ideal by its abelianized factor n-Lie algebra. Finally, we
also determine the dimension and structure of c-nilpotent multipliers
Heisenberg n-Lie algebras, which can be a useful tool for determining the
dimension of the multiplier of nilpotent n-Lie algebras of class 2.
