The two applications of similarity on residuated lattices

In this paper, we study the properties of C. C. Chang’s functions on residuated lattices. We prove that Chang’s function is a distance iff the residuated lattice is involutory. Since the number of involutory residuated lattices is very fewer, we use similarity, instead of Chang’s function, to give two applications on residuated lattices, including topology and Chinese Remainder Theorem. Upon using similarity and lattice filter, we construct a topology on a residuated lattice. Furthermore, a sufficient and necessary condition is presented that the topology makes all operations on residuated lattices be continuous. In addition, by similarity and filter, we construct a concrete solution of congruence systems of Chinese Remainder Theorem on some residuated lattices. Finally, an effective algorithm is designed to compute the concrete solution of Chinese Remainder Theorem.