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Our main focus in this work is to generalize the theory of algebraic semi rings from the algebraic setting to the framework of bornological sets. More specifically, the concept of a new structure bornological semi ring (BSR) is introduced and some constructions in the class of bornological semi rings are discussed. In particular, the existence of arbitrary projective limits and arbitrary inductive limits of bornological semi rings is ensured. Additionally, the description of the category of bornological semi rings is presented. We also discuss the concept of product, coproduct and fibre product in the category of bornological semi ring. In the context under consideration, general results concerning projective limits and inductive limits as well as an isomorphism theorem are established.

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