Algebraic Independence Relations in Randomizations

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.