ONE DIMENSIONAL T.T.T STRUCTURES
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Publication:2933674
DOI10.1017/JSL.2014.14zbMATH Open1353.03035arXiv1210.5603OpenAlexW2962812419MaRDI QIDQ2933674
Publication date: 5 December 2014
Published in: Journal of Symbolic Logic (Search for Journal in Brave)
Abstract: In this paper we analyze the relationship between o-minimal structures and the notion of omega -saturated one dimensional t.t.t structures. We prove that if removing any point from such a structure splits it into more than one definably connected component then it must be a one dimensional simplex of a finite number of o-minimal structures. In addition, we show that even if removing points doesn't split the structure, additional topological assumptions ensure that the structure is locally o-minimal. As a corollary we obtain the result that if an omega -saturated one dimensional t.t.t structure admits a topological group structure then it is locally o-minimal. We also prove that the number of connected components in a definable family is uniformally bounded which implies that an elementary extension of an omega -saturated one dimensional t.t.t structure is t.t.t as well.
Full work available at URL: https://arxiv.org/abs/1210.5603
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Models with special properties (saturated, rigid, etc.) (03C50) Model theory of ordered structures; o-minimality (03C64)
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