# representation theorems of \$L-\$subsets and \$L-\$families on complete residuated lattice

Document Type : Research Paper

Authors

1 Department of Mathematics, Ocean University of China, 266100 Qingdao, P.R. China

2 Department of Mathematics, Ocean University of China, 266100 Qing- dao, P.R. China

Abstract

In this paper, our purpose is twofold. Firstly, the tensor and
residuum operations on \$L-\$nested systems are introduced under the
condition of complete residuated lattice. Then we show that
\$L-\$nested systems form a complete residuated lattice, which is
precisely the classical isomorphic object of complete residuated
power set lattice. Thus the new representation theorem of
\$L-\$subsets on complete residuated lattice is obtained. Secondly, we
introduce the concepts of \$L-\$family and the system of \$L-\$subsets,
then with the tool of the system of \$L-\$subsets, we obtain the
representation theorem of intersection-preserving \$L-\$families on
complete residuated lattice.

Keywords

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