[Paper Review] The free pseudospace is n-ample, but not (n+1)-ample

Katrin Tent|arXiv (Cornell University)|Sep 29, 2011

Algebraic structures and combinatorial models4 references3 citations

TL;DR

This paper constructs free pseudospaces of dimension $n$ using a uniform inductive method based on $n+1$-colored graphs with specific incidence and cycle axioms. It proves these structures realize $n$-ampleness but not $(n+1)$-ampleness, providing the first known examples of $ heta$-stable theories that are $n$-ample but not $(n+1)$-ample, with prime models corresponding to Tits buildings of certain right-angled Coxeter groups.

ABSTRACT

We give a uniform construction of free pseudospaces of dimension n extending work by Baudisch and Pillay. This yields examples of $ω$-stable theories which are n-ample, but not (n+1)-ample. The prime models of these theories are buildings associated to certain right-angled Coxeter groups.

Motivation & Objective

• To construct a uniform family of free pseudospaces of dimension $n$ for any $n \geq 1$, generalizing earlier work on dimension 2.
• To prove that the theory of the free pseudospace of dimension $n$ is $n$-ample but not $(n+1)$-ample, filling a gap in understanding ampleness in stable theories.
• To establish a connection between these pseudospaces and Tits buildings associated with right-angled Coxeter groups, particularly identifying the prime model as such a building.
• To classify the regular types in the theory and show there are exactly two orthogonality classes, contributing to the model-theoretic understanding of these structures.

Proposed method

• Define an $L_n$-language with $n+1$ vertex types $V_0, \ldots, V_n$ and edge relations $E$ between consecutive types.
• Use inductive axioms $(\Sigma1)_n$ to $(\Sigma4)_n$ to define free pseudospaces: closure under substructures of lower dimension, residue conditions, intersection and generation axioms, and cycle path conditions.
• Construct the theory $T_n$ as the first-order theory of these structures, proving consistency and completeness via an inductive strong extension system.
• Show that models of $T_n$ are buildings of type $A_{\infty,n+1}$ by proving that $E_i$-connectedness and maximal flag containment characterize such buildings.
• Use quantifier elimination in a language with predicates $\delta_w^{i,j}$ for Weyl distances between vertices of types $V_i$ and $V_j$ to analyze type spaces.
• Classify regular types via forking behavior and $U$-rank analysis, distinguishing types based on connectivity to algebraic closures of parameters.

Experimental results

Research questions

• RQ1 Can a uniform construction of free pseudospaces of dimension $n$ be given that generalizes the known dimension-2 case?
• RQ2 Is the theory of the free pseudospace of dimension $n$ $n$-ample but not $(n+1)$-ample?
• RQ3 What is the model-theoretic structure of the prime model of this theory, and how does it relate to buildings?
• RQ4 How many orthogonality classes of regular types exist in the theory of the free pseudospace of dimension $n$?
• RQ5 Can the $U$-rank and Morley rank of regular types be precisely characterized in terms of connectivity to algebraic closures?

Key findings

• The theory $T_n$ of the free pseudospace of dimension $n$ is $n$-ample but not $(n+1)$-ample, providing the first known examples of $ heta$-stable theories with this ampleness property.
• The prime model of $T_n$ is isomorphic to the building $M_n^0$ associated with a right-angled Coxeter group of type $A_{\infty,n+1}$, confirming a structural link to Tits buildings.
• The theory $T_n$ admits quantifier elimination in a language with predicates for Weyl distances $\delta_w^{i,j}$ between vertices of types $V_i$ and $V_j$.
• There are exactly two orthogonality classes of regular types: those non-forking over parameters with no connection to $\operatorname{acl}(A)$, and those connected via flags or residues.
• The Morley rank of a regular type $\operatorname{tp}(a/A)$ is $\omega^n$ if $a$ is not connected to $\operatorname{acl}(A)$, $\omega^{n-j-1}$ or $\omega^{j-1}$ if connected via a residue, and $\omega^{|k-j|-2}$ in other cases.
• Any regular type is non-orthogonal to a type of kind (I) or (IV), where (IV) corresponds to types realized by vertices in dense flags with neighbors in $\operatorname{acl}(A)$.