Maximal WAP and tame quotients of type spaces

TL;DR

This paper investigates the structure of maximal WAP and tame quotients of type spaces in model theory, demonstrating their stability under changes of the monster model and analyzing their associated Ellis groups.

Contribution

It introduces and studies the properties of maximal WAP and tame quotients of type spaces, showing their invariance under model expansions and analyzing their Ellis groups.

Findings

F_{ extrm{WAP}} and F_{ extrm{Tame}} are well-behaved under model enlargements.

The Ellis groups associated with these quotients are independent of the choice of the monster model.

The paper establishes the definability and invariance properties of these quotients in topological dynamics.

Abstract

We study maximal WAP and tame (in the sense of topological dynamics) quotients of $S_X(\mathfrak{C})$, where $\mathfrak{C}$ is a sufficiently saturated (called monster) model of a complete theory $T$, $X$ is a $\emptyset$-type-definable set, and $S_X(\mathfrak{C})$ is the space of complete types over $\mathfrak{C}$ concentrated on $X$. Namely, let $F_{\textrm{WAP}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{WAP}} )$ is WAP, and let $F_{\textrm{Tame}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{Tame}} )$ is tame. We show good behaviour of $F_{\textrm{WAP}}$ and $F_{\textrm{Tame}}$ under changing the monster model $\mathfrak{C}$. Namely, we prove that if $\mathfrak{C}'\succ \mathfrak{C}$ is a bigger monster model, $F'_{\textrm{WAP}}$ and $F'_{\textrm{Tame}}$ are the counterparts of $F_{\textrm{WAP}}$ and $F_{\textrm{Tame}}$ computed for $\mathfrak{C}'$, and $r\colon S_X(\mathfrak{C}')\to S_X(\mathfrak{C})$ is the restriction map, then $r[F'_{\textrm{WAP}}]=F_{\textrm{WAP}}$ and $r[F'_{\textrm{Tame}}]=F_{\textrm{Tame}}$. Using these results, we show that the Ellis (or ideal) groups of $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{WAP}} )$ and $(aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{Tame}})$ do not depend on the choice of the monster model $\mathfrak{C}$.