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

in this paper, our purpose is twofold. firstly, the tensor andresiduum operations on $l-$nested systems are introduced under thecondition of complete residuated lattice. then we show that$l-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. thus the new representation theorem of$l-$subsets on complete residuated lattice is obtained. secondly, weintroduce the concepts of $l-$family and the system of $l-$subsets,then with the tool of the system of $l-$subsets, we obtain therepresentation theorem of intersection-preserving $l-$families oncomplete residuated lattice.