On the $c$-nilpotent multiplier of a pair of Lie algebras

Let $1\rightarrow R\rightarrow F\rightarrow L\rightarrow1$ be a free presentation of a Lie algebra $L$, and $N$ be an ideal in $L$. In [14], the first two authors of the present paper defined and studied the notion of the $c$-nilpotent multiplier of a pair $(N,L)$ of Lie algebras as $\mathcal{M}^{(c)}(N,L)=(R\cap[S,_cF])/[R,_cF]$, where $S$ is an ideal in $F$ such that $S/R\cong N$. In this paper, we give some results on $c$-covers of a pair of Lie algebras as well as several inequalities for the dimension of the $c$-nilpotent multiplier of a pair of nilpotent Lie algebras.