Algebraic Independence Relations in Randomizations

2017/03/31 by Uri Andrews, Andrews, Uri, Isaac Goldbring +3
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #Constraint Satisfaction and Optimization #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.1703.10913

openalex publication_date 2017/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Independence relations in randomizations
• Independence in randomizations
• Definable closure in randomizations
• Explanation of Independence
• Axiomatic Theory of Independence Relations in Model Theory