In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set ${\displaystyle (X,\leq )}$ is a subset ${\displaystyle S\subseteq X}$ with the following property: if s is in S and if x in X is larger than s (that is, if ${\displaystyle s<x}$), 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.