University of Sistan and BaluchestanIranian Journal of Fuzzy Systems1735-065410320130601representation theorems of $L-$subsets and $L-$families on complete residuated latticerepresentation theorems of $L-$subsets and $L-$families on complete residuated lattice12513686610.22111/ijfs.2013.866ENHuiHanDepartment of Mathematics, Ocean University of China, 266100 Qingdao,
P.R. ChinaJinmingFangDepartment of Mathematics, Ocean University of China, 266100 Qing-
dao, P.R. ChinaJournal Article20111015In 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.https://ijfs.usb.ac.ir/article_866_34e66093d052f7fc3dc927be495d286b.pdf