# On Extensions of Partial Isomorphisms

## TL;DR

This paper explores extensions of finite structures in relational languages, characterizes minimal HL-extensions, and introduces ultraextensive structures, revealing properties of their automorphism groups.

## Contribution

It provides a complete description of finite minimal HL-extensions and introduces ultraextensive structures, expanding understanding of automorphism groups in model theory.

## Key findings

- Finite minimal HL-extensions are fully characterized.
- The group-theoretic property is closed under free products.
- Every countable structure can be extended to a countable ultraextensive structure.

## Abstract

In this paper we study a notion of HL-extension (HL standing for Herwig--Lascar) for a structure in a finite relational language $\mathcal{L}$. We give a description of all finite minimal HL-extensions of a given finite $\mathcal{L}$-structure. In addition, we study a group-theoretic property considered by Herwig--Lascar and show that it is closed under taking free products. We also introduce notions of coherent extensions and ultraextensive $\mathcal{L}$-structures and show that every countable $\mathcal{L}$-structure can be extended to a countable ultraextensive structure. Finally, it follows from our results that the automorphism group of any countable ultraextensive $\mathcal{L}$-structure has a dense locally finite subgroup.