# What's the name of this property?

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 $x<z<y$?

• 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. Mar 6, 2014 at 21:38
• 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? Mar 6, 2014 at 21:46
• @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! Mar 6, 2014 at 21:50
• @Unwisdom Fixed the typo. I have no idea why I typed $\ne$ instead of $<$. Mar 6, 2014 at 22:18
• @JackLee Fixed the typos Mar 6, 2014 at 22:19

• But the property in this case is not density - it postulates the existence of a $z$ greater than $y$, not between $x$ and $y$. Mar 6, 2014 at 21:44
• The criterion read $x<y<z$ at time of posting. Mar 6, 2014 at 22:22