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The Theory Of Algebraic Semi Rings

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ON BORNOLOGICAL SEMI RINGS
ANWAR N. IMRAN
Abstract. Our main focus in this work is to generalizes the theory of algebraic semi ring from the algebraic setting to the framework of bornological set. More specifically, the concept of new structure bornological semi ring BSR is introduced and the fundamental 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 ring is ensured. Additionally, the description of the category of bornological semi rings is presented and we study the product, coproduct and fiber product. Key words: Bornological Set, Bornological Ring.
1. Introduction
The notion of a bounded subset of (real or complex) topological vector space was introduces by von Neamann. That motivated the definition and more general classes bounded sets and called bornologies. A bornological space is a type of space in functional analysis and it is very useful to solve problem of boundedness for set of elements and functions. That means, the main goal for bornology is to determine the boundedness, location and sitting the boundary for any area. A bornology on a set X is a family β of subsets of X, such that;
• Every singleton set fxg is in β, that means, β covering X ;
• If A ⊂ B, B 2 β then A 2 β;
• β is stable under finite union.
The elements of β are called bounded subsets of X. Let X and Y be two bornological sets, a map f : X ! Y is called bounded if

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