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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.

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    $\begingroup$ Is the introduction of random terminology ("amazing") really necessary? Anyway, you probably know by now that a significant number of people dislike PSQ and consider them bad questions... $\endgroup$ – Najib Idrissi Nov 4 '14 at 11:00
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    $\begingroup$ @MphLee Consider the poset $\{x\} \cup \mathbb{Z}$ where $x$ is unrelated to $\mathbb{Z}$; then it has a unique minimal element, namely $x$ (because no element is less than $x$ and it's the only one like that), but it's not a minimum, because for example you don't have $x < 0$. $\endgroup$ – Najib Idrissi Nov 4 '14 at 11:03
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    $\begingroup$ @NajibIdrissi, I think its totally necessary. Discussing the problem would be very clumsy without some terminology. If you're critiquing the particular terminology I chose, then yes, I agree its amazingly cringeworthy (pun intended). But I'd rather not waste time trying to find good terminology. You can spend half an hour on thesaurus.com this way. Better to spend it doing actual math. $\endgroup$ – goblin Nov 4 '14 at 12:57
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    $\begingroup$ @NajibIdrissi, sure, but you want to actually discuss the property. I don't mind the idea of calling it (*), but it still kind of forces you to alter your natural way of talking (to accommodate the fact that you haven't got an adjective). Anyway, I don't understand. You seem to be opposed to one-off terminology. If so, why? If not, what is bothering you? $\endgroup$ – goblin Nov 4 '14 at 13:05
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    $\begingroup$ Then just say "the property" if you want... There really is no need to invent another adjective when math is already bloated with completely random terminology. This Wikipedia list is already depressing enough. Who thought "amicable number" or "narcissistic number" were good terminology...? $\endgroup$ – Najib Idrissi Nov 4 '14 at 13:08
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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 $a<a'$.

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    $\begingroup$ @CarlMummert: A two-element poset is well-founded. (The positive elements in my example are not needed, but the negative ones are there specifically to make it ill-founded). $\endgroup$ – Henning Makholm Nov 4 '14 at 12:12
  • $\begingroup$ Nice example. $\;$ $\endgroup$ – goblin Nov 4 '14 at 12:55
  • $\begingroup$ Henning, do you know of a more intuitive characterization of the property of interest? More generally, I'd be interested in more information, if you have it. The question that's bugging me mostly, I think, is "how do you actually build these things?" Any insight would be appreciated. $\endgroup$ – goblin Nov 4 '14 at 13:49
  • $\begingroup$ @goblin: Do you mean amazingness? I'm not sure I have any good coherent intuition about it. "Minimal elements are always minimums" would feel like a sensible property, but "unique minimal elements are minimums" feels a bit off to me. Where do you have this property from? $\endgroup$ – Henning Makholm Nov 4 '14 at 14:24
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    $\begingroup$ @goblin: Okay -- it's so sensible that it coincides with an already-known concept. There can be too much of a good thing, I suppose :-) $\endgroup$ – Henning Makholm Nov 4 '14 at 23:56

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