To model and analyze systems with multi-valued information, in this paper, we present an extension of Kripke structures in the framework of complete residuted lattices, which we will refer to as lattice-valued Kripke structures (LKSs). We then show how the traditional trace containment and equivalence relations, can be lifted to the lattice-valued setting, and we introduce two families of lattice-valued versions of the relations. Further, we explore some interesting properties of these relations. Finally, we provide logical characterizations of our relations by a natural extension of linear temporal logic.