Ultrafiltres à la façon de Ramsey

Abstract:

Let $\beta {\text {N}}$ be the set of ultrafilters on N; ${\mathcal {U}} \in \beta {\text {N}}$ is “absolu” [6] (Ramsey [4]) if all its free images by continuous maps $\beta {\text {N}} \to \beta {\text {N}}$ are isomorphic. We study here a weaker Ramsey-like property, which implies the existence of fiber products $\mathcal {D} \otimes {_E}\mathcal {D}\left ( { \otimes _\textbf {E}^k\mathcal {D}} \right )$ extending the usual product ${\mathcal {D}} \otimes {\mathcal {D}}\left ( {{ \otimes ^{k}}{\mathcal {D}}} \right )$. This can be translated in the language of model-theory on the one hand as the existence of repeated almagamated sums and on the other hand by some properties of sets of indiscernibles associated with ultrafilters having this property (§5). We show that the class of ultrafilters we study strictly contains the class of Ramsey ultrafilters (§1) and is (§2) strictly (§3) contained in the class of p-point ultrafilters [9] ("$\delta$-stables” [6]) and contains the free images of its elements (§4). In §2 we also give a characterization of p-point ultrafilters in terms of the product ${ \otimes ^k}{\mathcal {D}}$. In §3 we show the link with weakly Ramsey ultrafilters of Blass [3] and more generally we study ultrafilters ${\mathcal {D}}$ on N having only a finite number $i\left ( {\mathcal {D}} \right )$ of free images up to isomorphism and such that $\# \tau {^{ - 1}}\left ( {{\mathcal {D}},{\mathcal {D}}} \right ) = 2i\left ( {\mathcal {D}} \right ) + 1$, where $\# {\tau ^{ - 1}}\left ( {{\mathcal {D}},{\mathcal {D}}} \right )$ is the number of ultrafilters on ${{\text {N}}^2}$ finer than the filter generated by $\left ( {D \times D} \right )$ with $D \in {\mathcal {D}}$.