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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?

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  • $\begingroup$ For an actual poset, the definition of the particular partial order will often make it obvious as to whether or not there is a least element. $\endgroup$
    – quasi
    Feb 11, 2017 at 1:09
  • $\begingroup$ A least element need not exist. $\endgroup$
    – Fimpellizzeri
    Feb 11, 2017 at 1:10

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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.

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