Real Algebraic and Analytic Geometry |

Bounded super real closed rings.

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homepage: http://personalpages.manchester.ac.uk/staff/Marcus.Tressl/index.php

Submission: 2008, July 28.

*Abstract:
This note is a complement to the paper "M. Tressl, Super real closed rings",
where super real closed rings are introduced and studied.
A bounded super real closed ring A is a commutative unital ring
A together with an operation F_{A}:A^{n} -> A
for every bounded continuous map F:ℝ^{n} -> ℝ,
so that all term equalities between the F's
remain valid for the F_{A}'s. We show that
bounded super real closed rings are precisely the convex
subrings of super real closed rings:
for every bounded super real closed ring there is a largest
super real closed subring B contained in A and a
smallest super real closed ring C containing A.
Moreover B is convex in C.
The assignment A↦C is
an idempotent mono-reflector
from bounded to arbitrary super real closed rings which allows to
transfer many of the algebraic
results from the unbounded to the bounded situation.*

Mathematics Subject Classification (2000): 03C60, 46E25.

Keywords and Phrases: real closed rings, rings of continuous functions, model theory.

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