Summary: This paper introduces the concept of \(L\)-partially ordered sets on complete residual lattices, and gives the concepts of \({L_k}\)-prime closure operators and \({L_k}\)-prime interior operators and characterizations on them. Then, it gives the generalization. The concepts of \(n\) multiple \({L_k}\)-prime closure operators and \(b\) multiple \({L_k}\)-prime interior operators and characterization on them are obtained.