# Amalgamating poset extensions and generating free lattices

## TL;DR

This paper explores the relationship between free lattices generated by posets and their canonical extensions, providing a construction method via colimits of intermediate structures.

## Contribution

It introduces a novel approach to construct free lattices from posets using colimits, linking them to canonical extensions.

## Key findings

- Free lattices can be constructed as colimits of intermediate structures.
- A new connection between free lattices and canonical extensions is established.
- The method provides explicit construction techniques for free lattices.

## Abstract

We investigate connections between the free lattice generated by a poset while preserving certain bounds and the canonical extension of a poset. Explicitly, we describe how the free lattice generated by a poset while preserving certain bounds can be constructed as a colimit of `intermediate structures' as they occur in the construction of a canonical extension of a poset.