Jinming Fang - Department of Mathematics, Ocean University of China, Qing Dao ۲۶۶۰۷۱, PR China
Youyan Li - Department of Mathematics, Ocean University of China, Qing Dao ۲۶۶۰۷۱, PR China
Wenyi Chen - Department of Mathematics, Ocean University of China, Qing Dao ۲۶۶۰۷۱, PR China

This paper focuses on the relationship between an L-subset and the system of level-elements induced by it, where the underlying lattice L is a complete residuated lattice and the domain set of L-subset is an L-partially ordered set (X,P). Firstly, we obtain the sufficient and necessary condition that an L-subset is represented by its system of level-elements. Then, a new representation theorem of intersection-preserving L-subsets is shown by using union-preserving system of elements. At last, another representation theorem of compatible intersection-preserving L-subsets is obtained by means of compatible union-preserving system of elements.

Complete residuated lattice, L-partially ordered set, L-subset, System of level-elements, Union-preserving system of elements, Compatible union-preserving system of elements, Representation theorem