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Stability of representations of effective partial algebras

✍ Scribed by Jens Blanck; Viggo Stoltenberg-Hansen; John V. Tucker

Publisher
John Wiley and Sons
Year
2011
Tongue
English
Weight
171 KB
Volume
57
Category
Article
ISSN
0044-3050

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✦ Synopsis

An algebra is effective if its operations are computable under some numbering. When are two numberings of an effective partial algebra equivalent? For example, the computable real numbers form an effective field and two effective numberings of the field of computable reals are equivalent if the limit operator is assumed to be computable in the numberings (theorems of Moschovakis and Hertling). To answer the question for effective algebras in general, we give a general method based on an algebraic analysis of approximations by elements of a finitely generated subalgebra. Commonly, the computable elements of a topological partial algebra are derived from such a finitely generated algebra and form a countable effective partial algebra. We apply the general results about partial algebras to the recursive reals, ultrametric algebras constructed by inverse limits, and to metric algebras in general.

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