ON THE SYSTEM OF LEVEL-ELEMENTS INDUCED BY AN L-SUBSET

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.