Embedding partial orders in complete lattices

Given a partially ordered set we can always embed that partial order within a complete lattice through Dedekind completion so that partial order becomes a suborder of a complete lattice. For finite unique extrema partial orders this simply means adding bounds to the partial order to make it into a lattice. Otherwise there may need to be elements placed between others such as in the case of the [2 2] weak order.

If we represent partial orders as set systems then the process of embedding the partial order within a complete lattice is equivalent to the process of embedding the set system within a Moore family. We can always acquire such a set system by taking the Moore completion of the family of sets.

The general applicability of lattice theory to problems of order theory through the process of embedding partial orders in complete lattices makes me think that lattice theory should play a fundamental role in our understanding of order theory. With an understanding of complete lattices we can better understand partial orders in general.