# Canonical equivalence relations on fronts on $FIN_k$

## TL;DR

This paper proves that for any equivalence relation on a barrier in a certain infinite combinatorial space, there exists a subset where the relation simplifies to a canonical form, aiding in classification.

## Contribution

It establishes a canonical form result for equivalence relations on barriers in the space of finite functions, extending previous structural theorems.

## Key findings

- Existence of a subset with canonical equivalence relation form
- Generalization to all k in the space FIN_k
- Framework for classifying relations on barriers

## Abstract

We prove that for every equivalence relation on a barrier on the space $\langle FIN^{[\infty]}_k,\leq,r\rangle$, for any $k$, there exists $Y\in FIN_k^{[\infty]}$ so that the restriction of the coloring on $\langle Y\rangle$ is canonical.