  We present a systematic development of inductive limits in the categories of
ordered *-vector spaces, Archimedean order unit spaces, matrix ordered spaces,
operator systems and operator C*-systems. We show that the inductive limit
intertwines the operation of passing to the maximal operator system structure
of an Archimedean order unit space, and that the same holds true for the
minimal operator system structure if the connecting maps are complete order
embeddings. We prove that the inductive limit commutes with the operation of
taking the maximal tensor product with another operator system, and establish
analogous results for injective functorial tensor products provided the
connecting maps are complete order embeddings. We identify the inductive limit
of quotient operator systems as a quotient of the inductive limit, in case the
involved kernels are completely biproximinal. We describe the inductive limit
of graph operator systems as operator systems of topological graphs, show that
two such operator systems are completely order isomorphic if and only if their
underlying graphs are isomorphic, identify the C*-envelope of such an operator
system, and prove a version of Glimm's Theorem on the isomorphism of UHF
algebras in the category of operator systems.
