Does a lower bounded set always have an infimum?

Question

Let $A$ be a partially ordered subset of $X$. If $A$ is bounded below, does $\inf(A)$ exist?

Answer 1

Let us assume $A$ is nonempty to avoid pathologies.

The statement does not have to be true even in the nice case that the ambient space $X$ is totally ordered:

Let $\Bbb Q^\times$ be the set of nonzero rational numbers, and let $\Bbb Q_{>0}$ be the set of positive rationals; then $\Bbb Q_{>0}$ has every negative rational as a lower bound, but there is no largest such upper bound.

When we do not assume the space is totally ordered, three elements suffice: put $X = \{a,b,c\}$, $A = \{c\}$ and set $a \preceq c, b\preceq c$ as the only nontrivial comparisons. Then $a$ and $b$ are both lower bounds of $A$, but since they are incomparable, neither is the largest lower bound, $\inf(A)$ does not exist.