Rad HAZU, Matematičke znanosti, Vol. 27 (2023), 1-9.

THE SURJECTIVITY AND THE CONTINUITY OF DEFINABLE FUNCTIONS IN SOME DEFINABLY COMPLETE LOCALLY O-MINIMAL EXPANSIONS AND THE GROTHENDIECK RING OF ALMOST O-MINIMAL STRUCTURES

Mourad Berraho

Abstract.   In this paper, we first show that in a definably complete locally o-minimal expansion of an ordered abelian group (M, <,+, 0, ...) and for a definable subset X ⊆ Mⁿ which is closed and bounded in the last coordinate such that the set π_n−1(X) is open, the mapping π_n−1 is surjective from X to M^n-1, where π_n−1 denotes the coordinate projection onto the first n−1 coordinates. Afterwards, we state some of its consequences. Also we show that the Grothendieck ring of an almost o-minimal expansion of an ordered divisible abelian group which is not o-minimal is null. Finally, we study the continuity of the derivative of a given definable function in some ordered structures.

2020 Mathematics Subject Classification.   03C64.

Key words and phrases.   Coordinate projection, Grothendieck rings, definably complete locally o-minimal expansion of a densely linearly ordered abelian group.

DOI: https://doi.org/10.21857/y26keclz69

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