In mathematics, an upper set of a partially ordered set {\displaystyle } 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, 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 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.
