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.
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. 


A set with this property is called convex. See e.g. Quasiuniform spaces, Volume 77 of Lecture notes in pure and applied mathematics, Peter Fletcher, William F. Lindgren, Marcel Dekker, 1982, p.84. 


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. 

