Projective Fra\"{\i}ss\'{e} limits of trees

TL;DR

This paper discusses the development of projective Fra"{sse9 limits of trees, correcting previous mistakes, and explores their topological realizations, including the Wa
7zewski dendrite and the Mohler-Nikiel universal dendroid.

Contribution

It revises the theory of projective Fra"{sse9 limits for trees, establishing new families that form such limits and identifying their topological realizations.

Findings

The family of finite trees with ramification vertices of order at most 3 forms a projective Fra"{sse9 family.

The topological realization of this family is the Wa
7zewski dendrite D_3.

Additional families lead to the Mohler-Nikiel universal dendroid.

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The following paper has been withdrawn from consideration for publication because there are mistakes.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 Fra\"{\i}ss\'{e} 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 Fra\"{\i}ss\'e family and the topological realization of its Fra\"{\i}ss\'e 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 Fra\"{\i}ss\'e families. The topological realization of the Fra\"{\i}ss\'e limit of one of these families is shown to be the Mohler-Nikiel universal dendroid. We continue study of projective Fra\"{\i}ss\'e 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 Fra\"{\i}ss\'e limits we obtain the dendrite $D_3$ as well as quite new, interesting continua for which we do not yet have topological characterizations.