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Is there a name for (partially or totally) ordered set $(A,<)$ such that for any $x,y\in A$, $x< y$ there is a $z\in A$ such that $x<z<y$?

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  • $\begingroup$ I don't know if there is an especially snazzy name for this property, but you could rephrase it by saying that all maximal elements are minimal. Less elegant formulations include all maximal elements are incomparable to other elements and all maximal upper bounded chains have at most one element. $\endgroup$
    – Unwisdom
    Mar 6, 2014 at 21:38
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    $\begingroup$ The way you've phrased the question, there's no poset that satisfies this condition (aside from trivial examples like singletons), because it implies in particular that for all $x,y\in A$, $x\ne y$ implies $x<y$. Would you like to rephrase? $\endgroup$
    – Jack Lee
    Mar 6, 2014 at 21:46
  • $\begingroup$ @JackLee: Good point! I misread that myself: I read it as "for any $x,y\in A, x\lneq y$...". This needs to be clarified/confirmed before my previous comment can be used! $\endgroup$
    – Unwisdom
    Mar 6, 2014 at 21:50
  • $\begingroup$ @Unwisdom Fixed the typo. I have no idea why I typed $\ne$ instead of $<$. $\endgroup$
    – Joker_vD
    Mar 6, 2014 at 22:18
  • $\begingroup$ @JackLee Fixed the typos $\endgroup$
    – Joker_vD
    Mar 6, 2014 at 22:19

2 Answers 2

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http://en.wikipedia.org/wiki/Dense_order

From Wikipedia : "In mathematics, a partial order < on a set X is said to be dense if, for all x and y in X for which x < y, there is a z in X such that x < z < y."

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    $\begingroup$ But the property in this case is not density - it postulates the existence of a $z$ greater than $y$, not between $x$ and $y$. $\endgroup$
    – Unwisdom
    Mar 6, 2014 at 21:44
  • $\begingroup$ Yes, "density"! It's just that I've heard this property being called "continuity" by non-math people in arguments so often that I forgot the actual name. $\endgroup$
    – Joker_vD
    Mar 6, 2014 at 22:22
  • $\begingroup$ Well done @user3211995 for answering the question that was intended rather than the one that was actually asked! Is there a badge for being psychic? $\endgroup$
    – Unwisdom
    Mar 6, 2014 at 22:32
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This would be that no chain has a maximal element, the opposite conclusion of Zorn's Lemma.

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  • $\begingroup$ The criterion read $x<y<z$ at time of posting. $\endgroup$ Mar 6, 2014 at 22:22
  • $\begingroup$ Sorry, it was a typo. $\endgroup$
    – Joker_vD
    Mar 6, 2014 at 22:24

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