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Monotone theories

Banff International Research Station for Mathematical Innovation and Discovery

By a result of Simon it is known that a theory of a coloured linear order has quantifier elimination after naming all unary $L$-definable sets and all $L$-definable binary monotone relations. Motivated by this result we define the notion of monotone theories, theories of linear orders in which binary definable sets have previous description. More precisely, an $\aleph_0$-saturated structure $M$ is said to be monotone if there exists an $L$-definable linear order $<$ such that every $A$-definable subset of $M^2$ is a finite Boolean combination of unary $A$-definable sets and $A$-definable $<$-monotone relations, in which case we also say that $M$ is monotone with respect to $<$. A theory is said to be monotone if it has a monotone $\aleph_0$-saturated model. We prove that the class of monotone theories coincides with the class of weakly quasi-o-minimal theories introduced by Kudaibergenov. Moreover, we describe definable linear orders in monotone theories and show that monotonicity of a theory does not depend on the choice of the linear order. Joint work with Predrag Tanovi\'c.

Mathematics; Mathematical logic and foundations; Combinatorics

video/mp4

Author affiliation: University of Wroclaw

2019-04-14

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69677

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/