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Quasi-Ordered Rings : a Uniform Study of Orderings and Valuations

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Quasi-Ordered Rings : a Uniform Study of Orderings and Valuations Quasi-Ordered Rings: a uniform Study of Orderings and Valuations Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften vorgelegt von Müller, Simon an der Mathematisch-naturwissenschaftliche Sektion Fachbereich Mathematik & Statistik Konstanz, 2020 Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-2-22s99ripn6un3 Tag der mündlichen Prüfung: 10.07.2020 1. Referentin: Prof. Dr. Salma Kuhlmann 2. Referent: Prof. Dr. Tobias Kaiser Abstract The subject of this thesis is a systematic investigation of quasi-ordered rings. A quasi-ordering ; sometimes also called preordering, is usually understood to be a binary, reflexive and transitive relation on a set. In his note Quasi-Ordered Fields ([19]), S. M. Fakhruddin introduced totally quasi-ordered fields by imposing axioms for the compatibility of with the field addition and multiplication. His main result states that any quasi-ordered field (K; ) is either already an ordered field or there exists a valuation v on K such that x y , v(y) ≤ v(x) holds for all x; y 2 K: Hence, quasi-ordered fields provide a uniform approach to the classes of ordered and valued fields. At first we generalise quasi-orderings and the result by Fakhruddin that we just mentioned to possibly non-commutative rings with unity. We then make use of it by stating and proving mathematical theorems simultaneously for ordered and valued rings. Key results are: (1) We develop a notion of compatibility between quasi-orderings and valua- tions. Given a quasi-ordered ring (R; ); this yields, among other things, a characterisation of all valuations v on R such that canonically induces a quasi-ordering on the residue class domain Rv: Moreover, it leads to a uniform definition of the rank of (R; ): (2) Given a valued ring (R; v); we characterise all v-compatible quasi-orderings on R in terms of the quasi-orderings on Rv by establishing a Baer-Krull theorem for quasi-ordered rings. From this, we also derive such a theorem for ordered rings, respectively valued rings. (3) We equip the set of all quasi-orderings on a ring R with a coarsening relation ≤; which is defined as follows: 1 ≤ 2 :, 8x; y 2 R: (0 1 x 1 y ) x 2 y) That way, we unify three different notions at once, namely coarsenings of valuations, inclusions of orderings, and compatibility of orderings and valuations. We prove that the set of all quasi-orderings on R with fixed support forms a rooted tree with respect to ≤; i.e. a partially ordered set with a unique maximal element such that for each such quasi-ordering all its coarsenings are linearly ordered. As an application of this tree structure theorem we deduce that the set of all quasi-orderings on R; partially ordered by a slight modification of the coarsening relation ≤; is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring. (4) We develop a notion of the quasi-real spectrum of a ring, in analogy to the real spectrum as known from real algebraic geometry. For that purpose, we generalise both the Harrison and the Tychonoff topology to the space of all quasi-orderings on a given ring. We establish the results that this space is a spectral space with respect to the Harrison topology and a Boolean space with respect to the Tychonoff topology. Furthermore, we introduce partially quasi-ordered rings and show that they provide a uniform approach to strict partial valuations and division closed partial orderings. Last but not least, we give a model-theoretic application of quasi-orderings. We subsume the results that the theories of algebraically closed valued fields and real closed fields adjoined with a convex valuation admit quantifier elimination. Deutsche Zusammenfassung (German Summary) Gegenstand dieser Thesis ist das Studium quasiangeordneter Ringe. Gemeinhin versteht man unter einer Quasianordnung ; auch Pr¨aordnung genannt, eine bin¨are, reflexive und transitive Relation auf einer Menge. In seinem Paper Quasi-Ordered Fields ([19]) f¨uhrtS. M. Fakhruddin total quasiangeordnete K¨orper ein, indem er Axiome f¨urdie Vertr¨aglichkeit von mit Addition und Multiplikation formuliert. Sein Hauptresultat besagt, dass jeder quasiangeordnete K¨orper (K; ) entweder schon ein angeordneter K¨orper ist, oder eine Bewertung v auf K existiert, so dass x y , v(y) ≤ v(x) f¨uralle x; y 2 K gilt. Somit bieten quasiangeordnete K¨orper einen einheitlichen Zugang zu angeordneten und bewerteten K¨orpern. In dieser Arbeit verallgemeinern wir Quasianordnungen und das eben beschriebene Resultat zun¨achst auf m¨oglicherweise nicht-kommutative Ringe mit Eins. Dies machen wir uns zu Nutze, um mathematische S¨atze simultan f¨urangeordnete und bewertete Ringe zu beweisen. Zentrale Ergebnisse dieser Arbeit sind: (1) Wir entwickeln einen Begriff von Kompatibilit¨atzwischen Quasianordnun- gen und Bewertungen. Ist (R; ) ein quasiangeordneter Ring, so f¨uhrtuns dies unter anderem zu einer Charakterisierung all derjenigen Bewertungen v auf R; f¨urdie kanonisch eine Quasianordnung auf dem Restklassenring Rv induziert. Ferner leiten wir hieraus eine einheitliche Definition f¨urden Rang eines quasiangeordneten Rings (R; ) ab. (2) F¨ureinen bewerteten Ring (R; v) charakterisieren wir die v-kompatiblen Quasianordnungen auf R durch die Quasianordnungen auf Rv mittels eines Baer-Krull Theorems. Hieraus leiten wir dann entsprechende Theoreme f¨ur angeordnete Ringe, beziehungsweise bewertete Ringe, ab. (3) Wir versehen den Raum aller Quasianordnungen auf einem Ring R mit einer Vergr¨oberungs-Relation ≤; welche wie folgt definiert ist: 1 ≤ 2 :, 8x; y 2 R: (0 1 x 1 y ) x 2 y) Auf diese Weise vereinheitlichen wir drei unterschiedliche Sachverhalte auf einmal, n¨amlich Vergr¨oberungen von Bewertungen, Inklusionen von Anord- nungen sowie Kompatibilit¨atzwischen Anordnungen und Bewertungen. Wir beweisen, dass die Menge aller Quasianordnungen auf R mit festem Support eine Baumstruktur bez¨uglich ≤ besitzt, d.h. eine partiell ange- ordnete Menge mit einem Maximum bildet, so dass zu jedem Element die Menge all seiner Vergr¨oberungen linear angeordnet ist. Als eine Anwendung dieses Baumstruktur-Theorems erhalten wir, dass die Menge aller Quasianordnungen auf R; versehen mit einer leichten Modi- fizierung der Vergr¨oberungs-Relation ≤; eine spektrale Menge bildet, d.h. ordnungsisomorph zum Spektrum eines kommutativen Rings ist. (4) Wir entwickeln den Begriff des quasi-reellen Spektrums eines Rings, in Analogie zu dem aus der reellen algebraischen Geometrie bekannten reellen Spektrum. Hierzu verallgemeinern wir die Harrison und die Tychonoff Topologie auf den Raum aller Quasianordnungen eines Rings. Wir be- weisen, dass dies bez¨uglich der Harrison Topologie ein spektraler Raum und bez¨uglich der Tychonoff Topologie ein boolscher Raum ist. Zudem f¨uhrenwir partielle Quasianordnungen auf Ringen ein und zeigen, dass diese strikte partielle Bewertungen und divisionsabgeschlossene partielle Anordnungen vereinheitlichen. Schließlich fassen wir mittels Quasianordnungen die Resultate zusammen, dass die Theorien der algebraisch abgeschlossenen bewerteten K¨orper und der reell abgeschlossenen K¨orper mit konvexer Bewertung Quantorenelimina- tion besitzen. Contents Introduction. 9 1. The Notion of Quasi-Ordered Rings . 15 1.1. Introduction of Quasi-Ordered Rings . 20 1.2. The Dichotomy of Quasi-Ordered Rings - Commutative Case . 23 1.3. The Dichotomy of Quasi-Ordered Rings - General Case . 26 1.4. Special Quasi-Orderings . 29 1.5. Manis Quasi-Orderings and the Inverse Property . 31 2. On the Interplay of Quasi-Orderings and Valuations . 35 2.1. Compatibility of Quasi-Orderings and Valuations . 36 2.2. The Rank of a Quasi-Ordered Ring. 44 3. Baer-Krull Theorems for Quasi-Ordered Fields and Rings. 47 3.1. A Baer-Krull Theorem for Quasi-Ordered Fields . 48 3.2. A Baer-Krull Theorem for Quasi-Ordered Rings . 54 3.3. A Baer-Krull Theorem for Ordered Rings. 59 3.4. A Baer-Krull Theorem for Valued Rings . 61 4. The Tree Structure of Quasi-Orderings . 63 4.1. A Coarsening Relation for Quasi-Orderings . 63 4.2. The Tree Structure Theorem for Quasi-Ordered Rings . 65 4.3. The Tree Structure of Special and Manis Quasi-Orderings . 71 4.4. Applications of the Tree Structure Theorem . 72 4.4.1. On Spectral Sets of Quasi-Orderings. 72 4.4.2. Dependency of Quasi-Orderings . 74 4.4.3. Digression on the Topology Induced by a Quasi-Ordering . 75 5. The Quasi-Real Spectrum of a Ring . 77 5.1. The Quasi-Real Spectrum as a Set . 79 5.2. The Quasi-Real Spectrum as a Topological Space . 80 6. Partial Quasi-Orderings on Rings. 87 6.1. Partial Valuations and Partial Orderings . 87 6.2. Partially Quasi-Ordered Rings. 89 6.3. The Dichotomy of Partially Quasi-Ordered Rings . 92 6.4. Extending Partial Quasi-Orderings . 94 7. Quasi-Real Closed Fields and Quantifier Elimination . 97 Acknowledgements . 100 References . 101 List of Symbols . 104 9 Introduction Goals. The purpose of the present thesis is to provide a systematic and uniform study of the classes of ordered and valued rings. At this point we cannot put enough emphasis on the word uniformity, since we will treat orderings and valuations on rings with multiplicative identity simultaneously throughout the entire thesis - not only when formulating our results, but also when proving them. This is achieved by the notion of quasi-orderings, which provides an axiomatic subsumption of orderings and valuations. Quasi-ordered fields were introduced in 1987 by Fakhruddin in his note [19]. Let K be a field and a binary, reflexive, transitive and total relation on K: Write x ∼ y if x y x; and otherwise x y: Then (K; ) is called a quasi-ordered field if the following axioms are satisfied for all x; y; z 2 K : (Q1) If x ∼ 0; then x = 0: (Q2) If x y and 0 z; then xz yz: (Q3) If x y and z y; then x + z y + z: Fakhruddin's main theorem states that any quasi-ordered field (K; ) is either an ordered field or else there is a valuation v on K such that x y , v(y) ≤ v(x) for all x; y 2 K ([19, Theorem 2.1]).
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