☆ 3.9 Article

A dichotomy for T-convex fields with a monomial group

MATHEMATICAL LOGIC QUARTERLY (2023)

On the antichain tree property

JinHoo Ahn et al.

Summary: In this paper, the authors investigate a new type of tree property in model theory called the antichain tree property (ATP). They develop combinatorial techniques for ATP and show that ATP can always be witnessed by a formula with a single free variable. They also prove the equivalence of ATP and k-ATP and provide a criterion for theories to not have ATP (being NATP). The authors present algebraic examples of ATP and NATP, including pure groups, pure fields, and valued fields. They conclude by constructing an antichain tree in Skolem arithmetic and atomless Boolean algebras.

JOURNAL OF MATHEMATICAL LOGIC (2023)

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Article Mathematics

Liouville closed HT-fields

Elliot Kaplan

Summary: This article studies the complete, model complete o-minimal theory T and its extensions of HT-fields. An HT-field is a model K equipped with a T-derivation partial differential such that the underlying ordered differential field of (K, partial differential) is an H-field. We investigate HT-fields and their extensions. Our main result is that if T is power bounded, then every HT-field K has either exactly one or exactly two minimal Liouville closed HT-field extensions up to K-isomorphism. The assumption of power boundedness can be relaxed to allow for certain exponential cases, such as T = Th(Ran,exp). (c) 2023 Elsevier Inc. All rights reserved.

JOURNAL OF ALGEBRA (2023)

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Article Mathematics