Amalgamating poset extensions and generating free lattices

Abstract

Given a poset P , a meet-extension e X : P → X and a joinextension e Y : P → Y , we define the canonical amalgamation of e X and e Y as an abstraction of a certain natural poset structure defined on X ∪ Y , and equipped with embeddings of X and Y corresponding to inclusions. This is essentially the extension of P obtained by 'freely' combining e X and e Y . By making particular choices for e X and e Y , the canonical amalgamation is the 'intermediate structure' that appears in the construction of canonical extensions. Our characterization is based on a kind of universal property. We find sufficient conditions for an order-preserving map f : P → Q to admit a lift to the canonical amalgamation that preserves some amount of the meet and join structure parametrized by a pair of cardinals. Using this, we show that the free lattice generated by P while preserving certain meets and joins can be obtained by constructing a chain of canonical amalgamations then taking the colimit. The objects in this chain can be thought of as approximations to the free lattice, and have their own universal properties.