2025/05/31 by Krapp, Lothar Sebastian, Kuhlmann, Salma, Vogel, Lasse
#03C40 (Primary) 12J15 #03C64 #06A05 (Secondary) #06F20 #12J10 #12L12 #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2506.00443
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 to that of its value set with respect to the natural valuation. We fully describe the definable rank on the group level. We also give a detailed comparison of field- and group-level, in particular for ordered fields with henselian natural valuation. We investigate definability of final segments in ordered sets and develop a tool for further study.