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

Abstract

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