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Projective Fraïssé limits of trees

Włodzimierz J. Charatonik, Robert P Roe

Abstract

The following paper has been withdrawn from consideration for publication because there are mistakes. In particular, Theorem 3.9 does not hold. Examples were found of finite trees with monotone epimorphisms which do not amalgamate. Further, finite rooted trees with monotone epimorphisms do not amalgamate. A revision, with additional co-authors A. Kwiatkowska and S. Yang, is posted on arXiv ({\it Projective Fraïssé limits of trees with confluent epimorphisms} 2312.16915). In that article, it is shown that the family of finite trees having ramification vertices of order at most 3 with monotone epimorphisms does form a projective Fraïssé family and the topological realization of its Fraïssé limit is the Wa\. zewski dendrite $D_3$. Further, two families of finite rooted trees with restrictions on what confluent epimorphisms are allowed are also shown to form projective Fraïssé families. The topological realization of the Fraïssé limit of one of these families is shown to be the Mohler-Nikiel universal dendroid. We continue study of projective Fraïssé limit developed by Irvin, Panagiotopoulos and Solecki. We modify the ideas of monotone, confluent, and light mappings from continuum theory as well as several properties of continua so as to apply to topological graphs. As the topological realizations of the projective Fraïssé limits we obtain the dendrite $D_3$ as well as quite new, interesting continua for which we do not yet have topological characterizations.

Projective Fraïssé limits of trees

Abstract

The following paper has been withdrawn from consideration for publication because there are mistakes. In particular, Theorem 3.9 does not hold. Examples were found of finite trees with monotone epimorphisms which do not amalgamate. Further, finite rooted trees with monotone epimorphisms do not amalgamate. A revision, with additional co-authors A. Kwiatkowska and S. Yang, is posted on arXiv ({\it Projective Fraïssé limits of trees with confluent epimorphisms} 2312.16915). In that article, it is shown that the family of finite trees having ramification vertices of order at most 3 with monotone epimorphisms does form a projective Fraïssé family and the topological realization of its Fraïssé limit is the Wa\. zewski dendrite . Further, two families of finite rooted trees with restrictions on what confluent epimorphisms are allowed are also shown to form projective Fraïssé families. The topological realization of the Fraïssé limit of one of these families is shown to be the Mohler-Nikiel universal dendroid. We continue study of projective Fraïssé limit developed by Irvin, Panagiotopoulos and Solecki. We modify the ideas of monotone, confluent, and light mappings from continuum theory as well as several properties of continua so as to apply to topological graphs. As the topological realizations of the projective Fraïssé limits we obtain the dendrite as well as quite new, interesting continua for which we do not yet have topological characterizations.
Paper Structure (10 sections, 52 theorems, 2 equations)

This paper contains 10 sections, 52 theorems, 2 equations.

Table of Contents

  1. Introduction and Definitions
  2. Connectedness properties of topological graphs
  3. Monotone epimorphisms
  4. Confluent epimorphisms
  5. Rooted trees
  6. Kelley topological graphs
  7. End vertices and ramification vertices in topological graphs
  8. Rooted trees and epimorphism preserving end vertices
  9. Confluent epimorphisms and rooted trees
  10. Confluent epimorphisms preserving end vertices

Key Result

Theorem 1.4

Let $\mathcal{F}$ be a projective Fraı̈ssé class with a fixed family of epimorphisms among the structures of $\mathcal{F}$. There exists a unique topological graph $\mathbb{F}$ such that This topological graph $\mathbb F$ is called the projective Fraı̈ssé limit of $\mathcal{F}$.

Theorems & Definitions (141)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • proof
  • Definition 1.5
  • Theorem 1.6
  • proof
  • Definition 2.1
  • Definition 2.2
  • ...and 131 more