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?
You cannot prove the existence in general because such an element may not exist. For the simplest example, consider the set where no two elements are comparable.
Given a particular poset, and assuming at least some of the elements are comparable, you still may not have an element which is comparable to every other number. Consider:
$$a\le b\le c$$ $$d\le c.$$ Clearly there is no least element.
Since you are not guaranteed to have a least element, you are unlikely to find much literature which helps in general cases.