"C-minimal fields"

"C-minimal fields"

Thu, 03/02/2011
17:00 		Francoise Delon (Paris 7) Logic Seminar Add to calendar L3
"C-minimal fields"
A $ C $-relation is the ternary relation induced by an ultrametric distance, in particular a valuation on a field, when we only remember the relation: $ C(x;y,z) $ iff $ d(x,y) < d(y,z) $. A $ C $-structure is a set equipped with a $ C $-relation and possibly additional structure. Following Haskell, Macpherson and Steinhorn, such a structure $ \mathbb M $ is said to be $ C $-minimal if, in any structure $ \mathbb N $ elementarily equivalent to $ \mathbb M $, definable sets in one-space (in one variable) are Boolean combinations of “cones” or “thick cones” (the generalization of “open” and “closed” balls from ultrametric spaces). A $ C $-field is a field equipped with a $ C $-relation compatible with addition and multiplication. It is known that a $ C $-minimal field is valued algebraically closed with $ C $ induced by the valuation. There are obvious analogies between o-minimality and $ C $-minimality... and obvious differences! We investigate more precisely the case of fields.