Bornological structures on some algebraic systems

This work concerns the notion of determining the boundedness of some algebraic structures such as groups and rings. Firstly, a new structure bornological semigroup is considered to determine the boundedness of algebraic structure semigroups. Then, some properties are investigated. Some of these properties are shared with bornological groups, but some properties are not. Further properties of bornological groups are studied to give sufficient condition of bornology to bornologize every group. In particular, we show that a left (right) translation in bornological groups is a bornological isomorphism and therefore the bornological groups structures are homogeneous. Next, bornological group actions (BGA) are constructed to prove some basic results which hold true just for bornological actions. In particular, we show that a bornological group action can be deduced from its boundedness at the identity and a bornological group acts on a bornological set by a bornological isomorphism. The effect of bornological action is to partition bornological sets into orbital bornological sets. Furthermore, the morphisms between G-bornological sets to be bounded maps are introduced. This motivated us to construct the category of G-bornological sets. For this purpose, we construct chorology theory for bornological groups based on bounded cochains and study some of its basic properties. We show that the cohomology theory of bounded cochains and the cohomology theory of homogenous cochains are isomorphic. Furthermore, the equivalent classes of bornological group in terms of a semi-bounded sets and s-bounded maps are presented to restrict the condition of boundedness for bornological group. Lastly, the concept of bornological semi rings is introduced to determine the boundedness of rings and semi rings, and the fundamental constructions in the class of bornological semi rings are discussed. The general results in this chapter concerning projective limits and inductive limits as well as an isomorphism theorem are established.

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