This paper investigates direct limits and quotients in the category of canonical hypergroups.Consider a direct system consisting of canonical hypergroups Gi and homomorphisms ij between them.Also assume a compatible family of subhypergroups Hi, one for each index i.Let G denote the direct limit of the system Gi, and let H denote the direct limit of the system Hi.Our main result is a natural isomorphism between the direct limit of the quotient hypergroups Gi/H * i and the quotient G/ H.A key step in the proof is showing that equivalence in the direct limit comes from equivalence at some finite stage of the system.This extends classical direct limit theorems from groups and modules to hypergroups, offering new tools for hypergroup structure theory.
Ostadhadi-Dehkordi et al. (Mon,) studied this question.