{"title":"Simple Models of Randomization and Preservation Theorems","authors":"Karim Khanaki, Massoud Pourmahdian","doi":"arxiv-2408.15014","DOIUrl":null,"url":null,"abstract":"The main purpose of this paper is to present new and more uniform\nmodel-theoretic/combinatorial proofs of the theorems (in [5] and [4]): The\nrandomization $T^{R}$ of a complete first-order theory $T$ with $NIP$/stability\nis a (complete) first-order continuous theory with $NIP$/stability. The proof\nmethod for both theorems is based on the significant use of a particular type\nof models of $T^{R}$, namely simple models, and certain indiscernible arrays.\nUsing simple models of $T^R$ gives the advantage of re-proving these theorems\nin a simpler and quantitative manner. We finally turn our attention to $NSOP$\nin randomization. We show that based on the definition of $NSOP$ given [11],\n$T^R$ is stable if and only if it is $NIP$ and $NSOP$.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.15014","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The main purpose of this paper is to present new and more uniform
model-theoretic/combinatorial proofs of the theorems (in [5] and [4]): The
randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$/stability
is a (complete) first-order continuous theory with $NIP$/stability. The proof
method for both theorems is based on the significant use of a particular type
of models of $T^{R}$, namely simple models, and certain indiscernible arrays.
Using simple models of $T^R$ gives the advantage of re-proving these theorems
in a simpler and quantitative manner. We finally turn our attention to $NSOP$
in randomization. We show that based on the definition of $NSOP$ given [11],
$T^R$ is stable if and only if it is $NIP$ and $NSOP$.