Definable ranks of almost real closed fields (2506.00443v1)

Abstract: We introduce the notion of the definable rank of an ordered field, ordered abelian group and ordered set, respectively. We study the relation between the definable rank of an ordered field and the definable rank of the value group of its natural valuation. Similarly we compare the definable rank of an ordered abelian group and the value set of its natural valuation. We consider in particular the case of an almost real closed field and give conditions under which the definable ranks on the level of field, of value group and of value set coincide.