Exact sequences of direct systems of hypergroups

This paper investigates the categorical properties of direct limits in the context of canonical hypergroups.Fundamental results are established concerning the construction and the preservation of exactness for direct limits of directed systems of hypergroups.The main contributions are as follows: (1) a rigorous proof of the universal property of direct limits in the category of canonical hypergroups, demonstrating their existence via an explicit quotient construction; (2) a theorem showing that exact sequences of directed systems induce exact sequences at the limit level; and (3) the development of essential commutative diagram techniques for hypergroup homomorphisms in directed systems.These results extend classical algebraic constructions to the hypergroup setting, where the multivalued nature of operations requires a careful treatment of equivalence classes and compatibility conditions.