Algebraic Independence Relations in Randomizations

TL;DR

This paper investigates algebraic independence in the randomizations of complete first-order theories, revealing how properties like extension and local character behave under different conditions.

Contribution

It characterizes algebraic and pointwise algebraic independence in randomizations, showing how these properties depend on the original theory T.

Findings

Algebraic independence in T^R satisfies extension and local character when algebraic and definable closures coincide in T.

Pointwise algebraic independence in T^R satisfies extension for countable sets and has finite character.

Certain properties like base monotonicity depend on whether algebraic independence in T holds.

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.