representation theorems of $L-$subsets and $L-$families on complete residuated lattice

In this paper, our purpose is twofold. Firstly, the tensor and residuum operations on $L-$nested systems are introduced under the condition of complete residuated lattice. Then we show that $L-$nested systems form a complete residuated lattice, which is precisely the classical isomorphic object of complete residuated power set lattice. Thus the new representation theorem of $L-$subsets on complete residuated lattice is obtained. Secondly, we introduce the concepts of $L-$family and the system of $L-$subsets, then with the tool of the system of $L-$subsets, we obtain the representation theorem of intersection-preserving $L-$families on complete residuated lattice.