Upper set

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> 
 
 
 
 (
 X
 ,
 ≤
 )
 
 
 {\displaystyle (X,\leq )}
 
 is a subset 
 
 
 
 S
 ⊆
 X
 
 
 {\displaystyle S\subseteq X}
 
 with the following property: if s is in S and if x in X is larger than s (that is, if 
 
 
 
 s
 <
 x
 
 
 {\displaystyle s<x}
 
), then x is in S. In other words, this means that any x element of X that is 
 
 
 
 
 ≥
 
 
 
 {\displaystyle \,\geq \,}
 
 to some element of S is necessarily also an element of S. 
The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset S of X with the property that any element x of X that is 
 
 
 
 
 ≤
 
 
 
 {\displaystyle \,\leq \,}
 
 to some element of S is necessarily also an element of S.

Upper set open-in-new

Upper set