Uncountable structures are not classifiable up to bi-embeddability

IF 0.9 1区数学 Q1 LOGIC

Journal of Mathematical Logic Pub Date : 2019-03-18 DOI:10.1142/S0219061320500014

F. Calderoni, H. Mildenberger, L. Ros

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引用次数: 4

Abstract

Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic 164(12) (2013) 1454–1492], we show that whenever [Formula: see text] is a cardinal satisfying [Formula: see text], then the embeddability relation between [Formula: see text]-sized structures is strongly invariantly universal, and hence complete for ([Formula: see text]-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357(12) (2005) 4839–4866; S.-D. Friedman and L. Motto Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic 76(1) (2011) 243–266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic 79(3) (2014) 928–954; F. Calderoni and L. Motto Ros, Universality of group embeddability, Proc. Amer. Math. Soc. 146 (2018) 1765–1780].

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不可数结构在双嵌入性范围内是不可分类的

从[L.]莫图·罗，大尺寸模型上可嵌入关系的描述集理论复杂性，安。纯粹的达成。逻辑164(12)(2013)1454-1492]，我们表明，只要[公式:见文]是一个基数满足[公式:见文]，那么[公式:见文]大小的结构之间的嵌入关系是强不变普遍的，因此对于([公式:见文]-)解析拟序是完备的。我们还证明，在上述结果中，我们可以进一步将我们的注意力限制在各种自然类型的结构上，包括(广义)树、图或群。这就把[A]的主要结果充分推广到不可数情况。罗森达，完全解析等价关系，译。阿米尔。数学。Soc. 357(12) (2005) 4839-4866;南达科他州。傅利民和L. Motto Ros，解析等价关系和双嵌入性，J.符号逻辑76(1)(2011)243-266;J. Williams，泛可数Borel准序，J.符号逻辑79(3)(2014)928-954;王志强，群体可嵌入性的普遍性研究，中国科学院学报。数学。Soc. 146 (2018) 1765-1780 [j]。

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来源期刊

Journal of Mathematical Logic MATHEMATICS-LOGIC

CiteScore

1.60

自引率

11.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Mathematical Logic (JML) provides an important forum for the communication of original contributions in all areas of mathematical logic and its applications. It aims at publishing papers at the highest level of mathematical creativity and sophistication. JML intends to represent the most important and innovative developments in the subject.