Submission date: 9 May 2024

Abstract:

We propose the notions of uniform local weak o-minimality and *-local weak o-minimality. Local monotonicity theorems hold in definably complete locally o-minimal structures and uniformly locally o-minimal structures of the second kind. In this paper, we demonstrate new local monotonicity theorems for uniformly locally weakly o-minimal structures of the second kind and for locally o-minimal structures under the assumption called the univariate *-continuity property. We also prove that several formulas for dimension of definable sets which hold in definably complete locally o-minimal structures also hold in *-locally weakly o-minimal structures possessing the univariate *-continuity property.

Mathematics Subject Classification: 03C64

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Full text arXiv 2405.06140: pdf, ps.