СДУ хабаршысы - 2008

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Remarks on weak o-minimality
(Suleyman Demirel University, 2008) Kudaibergenov K. Zh.
This paper investigates several necessary and sufficient conditions for weak o-minimality in expansions of linearly ordered structures. Building on earlier results, we provide a simplified proof of the main theorem of Kulpeshov, which characterizes weak o-minimality in terms of convexity of realizations of types. Additionally, two further equivalent conditions for weak o-minimality are established, involving types containing cuts and convexity properties of definable sets. A new, more conceptual proof of Pillay and Steinhorn’s characterization of full o-minimality is also presented. Furthermore, we examine weakly o-minimal ordered rings and show that every weakly o-minimal Archimedean ordered ring is necessarily a real closed field. This result follows from structural properties of definable subgroups in weakly o-minimal groups and known criteria for ordered fields. Overall, the paper clarifies foundational connections between weak o-minimality, type spaces, and definable order structures.