Inductive Limits in Quasi Ordered Vector Spaces

In this article, we investigate the concept of inductive limit in the category C of quasi ordered $$*$$ -vector spaces with nonnegative order units. We show that C is a category with inductive limits and for any direct system (ᵢ\, \ fi,j≤ j)i,j∈ I in C the state space of inductive limit limXᵢ is homeomorphic to the inverse limit limS(Xᵢ) of state spaces. Then we consider structure of Archimedeanization of inductive limits and we show that the category AQ of Archimedean elements of C is a category with inductive limits.