Submission date: 31 March 2017

Abstract:

We study the properties of algebraic independence and pointwise algebraic independence in a class of continuous theories, the randomizations T^R of complete first order theories T. If algebraic and definable closure coincide in T, then algebraic independence in T^R satisfies extension and has local character with the smallest possible bound, but has neither finite character nor base monotonicity. For arbitrary T, pointwise algebraic independence in T^R satisfies extension for countable sets, has finite character, has local character with the smallest possible bound, and satisfies base monotonicity if and only if algebraic independence in T does.

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Full text arXiv 1703.10913: pdf, ps.