A Brief Note On The Semi Rings 5

2472 Words Sep 18th, 2016 10 Pages
ON BORNOLOGICAL SEMI RINGS 5
Theorem 3.8. Let (A; β) be a bornological semi ring and f : A ! C be an epimorphism of semi rings. Then βf = fB ⊂ C : f −1(B) 2 βg is a semi ring bornology on C:
Proof. Let B1; B2 2 βf: Then there exist B1 A; B2 A 2 β such that f(B1 A) = B1 and f(B2 A) = B2 so f(B1 A + B2 A) = f(B1 A) + f(B2 A) = B1 + B2 and f(B1 A · B2 A) = f(B1 A) · f(B2 A) = B1 · B2, therefore βf is a semi ring bornology on C. 
Proposition 3.9. Let (A; βi)i2I be a collection of a bornological semi ring, then i2Iβi is a semi ring bornology on A.
Proof. It is known that β = Ti2I βiis a bornology, we must show that it also is a semi ring bornology. Let B1; B2 2 β, then B1; B2 2 βi for any i 2 I, hence B1 + B2 2 βi and
B1 · B2 2 βi since βi are semi rings bornology. Therefore B1 + B2 2 β and B1 · B2 2 β,
i.e., the additive and multiplication operation are bounded. 
A morphism of bornological semi rings : (A; βA) ! (C; βC) is defined to be a bounded semi ring homomorphism if the image under of every bounded subsemi ring of A is bounded. It is clear that the composition of two bounded semi rings homomorphisms is bounded, so we can define the category of bornological semi rings denote
Born(SRings), with objects bornological semi rings and arrows bounded semi ring homomorphisms. The category of bornological semi ring admits limits, colimits with the following description. Let (Ai; βi)i2I be a family of bornological semi rings, then the direct product
Qi2I(Ai; βi)i2I is defined to be…
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