Real Algebraic and Analytic Geometry

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245. Marcus Tressl:
Bounded super real closed rings.


Submission: 2008, July 28.

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 FA:An -> A for every bounded continuous map F:ℝn -> ℝ, so that all term equalities between the F's remain valid for the FA'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.

Full text, 16p.: dvi 95k, ps.gz 148k, pdf 202k.

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