name for "solid" subset of a partially ordered set? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T05:07:53Z http://mathoverflow.net/feeds/question/8088 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8088/name-for-solid-subset-of-a-partially-ordered-set name for "solid" subset of a partially ordered set? Adam 2009-12-07T08:37:19Z 2009-12-07T22:07:26Z <p>For P a partially ordered set, let S be a subset of P such that if:</p> <p>a,c\in S and b\in P and a&lt;=b&lt;=c then b\in S</p> <p>Is there a name for a subset with this property? The term "dense" subset is already taken and means something else.</p> http://mathoverflow.net/questions/8088/name-for-solid-subset-of-a-partially-ordered-set/8089#8089 Answer by David Eppstein for name for "solid" subset of a partially ordered set? David Eppstein 2009-12-07T08:47:11Z 2009-12-07T08:47:11Z <p>A set with this property is called <i>convex</i>.</p> <p>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.</p> http://mathoverflow.net/questions/8088/name-for-solid-subset-of-a-partially-ordered-set/8136#8136 Answer by Mikael Vejdemo-Johansson for name for "solid" subset of a partially ordered set? Mikael Vejdemo-Johansson 2009-12-07T22:07:26Z 2009-12-07T22:07:26Z <p>I remember seeing a definition of <em>interval</em> in a poset as the subset $[a,c] = \{b: a\leq b\leq c\}$. This would seem to be what you're talking about.</p> <p>Specifically, this is the definition in Stanley: Enumerative Combinatorics vol 1, p 98.</p>