Given a poset $(P,\leq)$, how could you prove the existence of a least element? Or rather, which posets admit a least element?
An element $x \in P$ is a least element if $x \leq p$ for all $p \in P$. I know that if the poset is a complete lattice (the least upper bound and greatest lower bound exist for any subset of $P$) then there is a category theoretic argument for the exist of a least object, but i was wondering if there was any literature on proving the existence in general?