Upper and lower sets

A Hasse diagram of the divisors of $210$, ordered by the relation is divisor of, with the upper set $\uparrow 2$ colored green. The white sets form the lower set $\downarrow 105.$

In mathematics, an upper set $S$ of a partially ordered set $X$ is a subset such that if s is in S and if x in X is larger than s, then x is in S. A lower set is defined similarly as being a subset S of X with the property that any element x of X that precedes an element of S is necessarily also an element of S.

Upper sets and lower sets are also known by many other names. An upper set may also be called an upward closed set, an up-set, an isotone set, or an order filter, while a lower set may also be called a downward closed set, down-set, decreasing set, semi-ideal, or order ideal. However, the terms "order ideal" and "order filter" are also used for a more restrictive notion.