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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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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 $b\in S$. – Martin Sleziak Jan 15 '15 at 13:33
up vote 11 down vote accepted

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.

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Thank you very much! – Adam Dec 8 '09 at 9:17

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.

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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. – David Eppstein Dec 8 '09 at 0:16
Right, and Stanley requires for the interval that a ≤ b, which isn't stated explicitly in the question. – Mikael Vejdemo-Johansson Dec 9 '09 at 1:04

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