Homogeneous ultrametric structures

TL;DR

This paper develops a new theory of homogeneous Polish ultrametric structures using Fraisse classes and universal epimorphisms, expanding the hierarchy of Fraisse limits with a strict homogeneity property.

Contribution

It introduces a novel Fraisse limit as an inverse limit with strict homogeneity, advancing the understanding of ultrametric structures in Polish spaces.

Findings

Constructed a new Fraisse limit as an inverse limit.

Established universality with respect to embeddings.

Demonstrated strict homogeneity in the new structures.

Abstract

We develop the theory of homogeneous Polish ultrametric structures. Our starting point is a Fraisse class of finite structures and the crucial tool is the universal homogeneous epimorphism. The new Fraisse limit is an inverse limit, nevertheless its universality is with respect to embeddings and, contrary to the Polish metric Fraisse theory of Ben Yaacov, homogeneity is strict. Our development can be viewed as the third step of building a Borel-like hierarchy of Fraisse limits, where the first step was the classical setting of Fraisse and the second step is the more recent theory, due to Irwin and Solecki, of pro-finite Fraisse limits.