| Résume | Reduced products are the natural generalisation of ultraproducts when one uses any filter instead of ultrafilters. It is a classical result, due to Keisler and Galvin, that a first-order formula is preserved under reduced products if and only if it is equivalent to a Horn formula. However, another fragment is of interest when studying reduced products: the one consisting of Palyutin formulas (also called h-formulas). First, Palyutin formulas satisfy some kind of Łoś's Theorem for reduced products, contrary to Horn formulas in general. Moreover, they are used to obtain a nice characterisation of complete theories that are preserved under reduced products. This allows one, for instance, to prove an analogue of Keisler-Shelah's Theorem for reduced products, or to study stability of these structures.
After reviewing these results in the classical setting, we will see that very similar tools can be developed to study reduced products of metric structures, as defined by Lopes. |
