# Uniform Interpolation and Compact Congruences

## TL;DR

This paper explores uniform interpolation in algebraic varieties, linking it to compact congruences and demonstrating how these properties can ensure the existence of a model completion for the variety's first-order theory.

## Contribution

It establishes a connection between uniform interpolation, compact congruences, and model completions, extending previous results to a broader algebraic context.

## Key findings

- Uniform interpolation relates to properties of compact congruences.
- These properties can guarantee the existence of a model completion.
- The results extend prior work by Ghilardi and Zawadowski.

## Abstract

Uniform interpolation properties are defined for equational consequence in a variety of algebras and related to properties of compact congruences on first the free and then the finitely presented algebras of the variety. It is also shown, following related results of Ghilardi and Zawadowski, that a combination of these properties provides a sufficient condition for the first-order theory of the variety to admit a model completion.