For P a partially ordered set, let S be a subset of P such that if:
a,c\in S and b\in P and a<=b<=c then b\in S
Is there a name for a subset with this property? The term "dense" subset is already taken and means something else.
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1$\begingroup$ It is probably not worth bumping 5 years old post with such a trivial edit, but for the sake of better readability, the condition is: If $a,c\in S$ and $b\in P$ and $a\le b\le c$ then $b\in S$. $\endgroup$– Martin SleziakCommented Jan 15, 2015 at 13:33
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2 Answers
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A set with this property is called convex.
See e.g. Quasi-uniform spaces, Volume 77 of Lecture notes in pure and applied mathematics, Peter Fletcher, William F. Lindgren, Marcel Dekker, 1982, p.84.
answered Dec 7, 2009 at 8:47
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I remember seeing a definition of interval in a poset as the subset $[a,c] = \{b: a\leq b\leq c\}$. This would seem to be what you're talking about.
Specifically, this is the definition in Stanley: Enumerative Combinatorics vol 1, p 98.
answered Dec 7, 2009 at 22:07
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2$\begingroup$ Every interval is convex, but not every convex set is an interval. For instance, an antichain in a poset is convex, but unless it has only one element it is not an interval. $\endgroup$ Commented Dec 8, 2009 at 0:16
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$\begingroup$ Right, and Stanley requires for the interval that a ≤ b, which isn't stated explicitly in the question. $\endgroup$ Commented Dec 9, 2009 at 1:04