Extending partial orders to dense linear orders Original Research Article

J. Łoś raised the following question: Under what conditions can a countable partially ordered set be extended to a dense linear order merely by adding instances of comparability (without adding new points)? We show that having such an extension is a Σ1l-complete property and so there is no Borel answer to Łośʹs question. Additionally, we show that there is a natural Π1l-norm on the partial orders which cannot be so extended and calculate some natural ranks in that norm.