General Criterion for a Poset to Admit a Partition Into $\omega$-Chains

Given a poset $P$ with no maximal element the question naturally arises of when $P$ can be partitioned into $\omega$-chains. In this paper we give general criterion for when this can be done. We show that in the case when a poset with no maximal element is countable or a well-founded forest, then it can always be partitioned into $\omega$-chains. We then introduce the lifting property for posets with no maximal element and show that a poset can be partitioned into $\omega$-chains if and only if it has the lifting property.