{"title":"避免线性秩序中的梅德韦杰夫减少","authors":"Noah Schweber","doi":"10.1002/malq.202200059","DOIUrl":null,"url":null,"abstract":"<p>While every endpointed interval <i>I</i> in a linear order <i>J</i> is, considered as a linear order in its own right, trivially Muchnik-reducible to <i>J</i> itself, this fails for Medvedev-reductions. We construct an extreme example of this: a linear order in which no endpointed interval is Medvedev-reducible to any other, even allowing parameters, except when the two intervals have finite difference. We also construct a scattered linear order which has many endpointed intervals Medvedev-incomparable to itself; the only other known construction of such a linear order yields an ordinal of extremely high complexity, whereas this construction produces a low-level-arithmetic example. Additionally, the constructions here are “coarse” in the sense that they lift to other uniform reducibility notions in place of Medvedev reducibility itself.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Avoiding Medvedev reductions inside a linear order\",\"authors\":\"Noah Schweber\",\"doi\":\"10.1002/malq.202200059\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>While every endpointed interval <i>I</i> in a linear order <i>J</i> is, considered as a linear order in its own right, trivially Muchnik-reducible to <i>J</i> itself, this fails for Medvedev-reductions. We construct an extreme example of this: a linear order in which no endpointed interval is Medvedev-reducible to any other, even allowing parameters, except when the two intervals have finite difference. We also construct a scattered linear order which has many endpointed intervals Medvedev-incomparable to itself; the only other known construction of such a linear order yields an ordinal of extremely high complexity, whereas this construction produces a low-level-arithmetic example. Additionally, the constructions here are “coarse” in the sense that they lift to other uniform reducibility notions in place of Medvedev reducibility itself.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-07-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202200059\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202200059","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Avoiding Medvedev reductions inside a linear order
While every endpointed interval I in a linear order J is, considered as a linear order in its own right, trivially Muchnik-reducible to J itself, this fails for Medvedev-reductions. We construct an extreme example of this: a linear order in which no endpointed interval is Medvedev-reducible to any other, even allowing parameters, except when the two intervals have finite difference. We also construct a scattered linear order which has many endpointed intervals Medvedev-incomparable to itself; the only other known construction of such a linear order yields an ordinal of extremely high complexity, whereas this construction produces a low-level-arithmetic example. Additionally, the constructions here are “coarse” in the sense that they lift to other uniform reducibility notions in place of Medvedev reducibility itself.