# Logic seminar: Ricardo Palomino

Dates: | 15 May 2024 |
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Times: | 15:15 - 16:30 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |

Title: Local real closed SV-rings of finite rank and their model theory

Abstract: A commutative and unital ring A is a survaluation ring, or SV-ring for short, if A/ is a valuation ring for all prime ideals of A. SV-rings were first introduced in the early 1990s within the context of rings of continuous functions, and since then, they have mostly gained the attention of the mathematical community which works on lattice-ordered groups and rings. The class of all SV-rings turns out to not be elementary in the language of rings, but some subclasses are; in particular, the class of local real closed SV-rings of rank at most m ? ? is elementary, and this talk will be focused on the algebra and model theory of SV-rings in this latter class. It turns out that a ring is a local real closed SV-ring of rank at most m ? ? if and only if it is isomorphic to a finite iterated fibre product of at most m non-trivial real closed valuation rings (i.e., proper convex subrings of real closed fields) along surjective ring homomorphisms; I will sketch the proof of this structure theorem, and then use it, together with some constructions involving those valuation rings corresponding to the canonical valuation on real closed Hahn fields, to show that much of the good model theory of real closed valuation rings is preserved to various elementary subclasses of local real closed SV-rings of finite rank.