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 thatL-nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. Thus the new representation theorem ofL-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.

Keywords:

Complete residuated lattices , L-subsets , L-nested systems , L-families , Level L-subsets , Representation theorems