Definition. Let $P$ denote a poset. Then $P$ is amazing iff every $A \subseteq P$ has the property that if $A$ has a unique minimal element (call it $m$), then $A$ has a minimum element (namely $m.$)
Question. Does there exist an amazing poset that is neither totally-ordered nor well-founded?
Discussion. Both total-orderedness and well-foundedness imply amazing. An example of a poset that isn't amazing is $\mathbb{Z} \sqcup \mathbb{Z}$. I haven't been able to think of any amazing posets that aren't either well-founded, or totally ordered.
How about $\mathbb Z$ with the $0$ split into two different elements? The zeroes are not comparable, but both are smaller than every positive number and larger than every negative number.
More generally if $A$ is a totally ordered but ill-founded set and $B$ is any set, with $\ge 2$ elements, then $A\times B$ where $(a,b)<(a',b')$ iff $a<a'$.