Lifts of $L$-valued powerset mapping systems

Motivated by Zadeh's extension of mappings, this paper introduces a concept of lifts of powerset mapping systems. In this framework, $L$-subsets are lifts of ordinary subsets, $L$-Zadeh functions are lifts of ordinary mappings, $L$-fuzzy rough set operators are lifts of classical rough set operators, $L$-interior/closure operators of $L$-topology are lifts of interior/closure operators of level topologies. Consequently, lifts of powerset mapping systems can be considered as a useful fuzzy structures for future study.