<p>The study of the properties and optimality of critical state vectors for vector sets with state attributes is an important research direction in vector optimization. In this paper, we consider a special type of vector sets in which both the state variables and the state vectors have normal or fault state attributes. We first provide the definitions for the upper and lower sets of a vector and a subset in the vector set by using the partial order relation induced by the non-negative quadrant cone, and then obtain some corresponding properties. Furthermore, we propose the definition of the critical state vector in the vector set and get some related basic properties of the critical state vector. Meanwhile, we establish the equivalent characterization of the vector set with state attributes, obtain the Pareto optimality of the critical state vectors in the vector set with state attributes, and then we show that the set of the critical state vectors is consistent with a Pareto optimal solution of the bi-objective optimization problem established in this paper. Moreover, we also use the state space of the power system as an example to explain the critical state vector set.</p>