Independence in randomizations

TL;DR

This paper explores the properties of independence relations in the randomization of a complete first order theory, showing that under certain conditions, the randomized theory exhibits real rosiness and a strict independence relation.

Contribution

It demonstrates that if the original theory has the exchange property and algebraic closure equals definable closure, then its randomization has a strict independence relation and is real rosy.

Findings

Randomization preserves certain independence properties.

If T has exchange and acl=dcl, then T^R has a strict independence relation.

For o-minimal T, T^R is real rosy.

Abstract

The randomization of a complete first order theory $T$ is the complete continuous theory $T^R$ with two sorts, a sort for random elements of models of $T$, and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of $T$. We show that if $T$ has the exchange property and $\operatorname{acl}=\operatorname{dcl}$, then $T^R$ has a strict independence relation in the home sort, and hence is real rosy. In particular, if $T$ is o-minimal, then $T^R$ is real rosy.