OS Math. Logik, Mengenlehre und Modelltheorie: On the Tree Structure of Orderings and Valuations on arbitrary unital Rings

Montag, 10. Dezember 2018
15:15 – 16:45 Uhr

F 426

C. Antos-Kuby, S. Kuhlmann

Simon Müller

Diese Veranstaltung ist Teil der Veranstaltungsreihe „Oberseminar Mathematische Logik, Mengenlehre und Modelltheorie“.

We first introduce the notion of quasi-orderings on arbitrary unital rings, which axiomatically subsumes the classes of all orderings and valuations. This enables us to uniformly define a coarser relation $\geq$ on the set of all quasi-orderings, generalizing the respective notion known from valuation theory. Our main result states that, given a prime ideal $\mathfrak{q}$ of a unital ring $R,$ the set of all quasi-orderings on $R$ with support $\mathfrak{q}$ is a tree w.r.t. $\geq$, i.e. a partially ordered set admitting a maximum, such that for any such quasi-ordering $\preceq$ the set of its coarsenings is linearly ordered. We conclude the talk by discussing to what extend the tree structure theorem can be expanded to quasi-orderings with different supports.