Upper set

In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set $$(X, \leq)$$ is a subset $$S \subseteq X$$ with the following property: if s is in S and if x in X is larger than s (that is, if $$s < x$$), then x is in S. In other words, this means that any x element of X that is $$\,\geq\,$$ 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 $$\,\leq\,$$ to some element of S is necessarily also an element of S.

Definition
Let $$(X, \leq)$$ be a preordered set. An ' in $$X$$ (also called an ', an , or an  set) is a subset $$U \subseteq X$$ that is "closed under going up", in the sense that
 * for all $$u \in U$$ and all $$x \in X,$$ if $$u \leq x$$ then $$x \in U.$$

The dual notion is a ' (also called a ', ', ', ', or '), which is a subset $$L \subseteq X$$ that is "closed under going down", in the sense that
 * for all $$l \in L$$ and all $$x \in X,$$ if $$x \leq l$$ then $$x \in L.$$

The terms ' or ' are sometimes used as synonyms for lower set. This choice of terminology fails to reflect the notion of an ideal of a lattice because a lower set of a lattice is not necessarily a sublattice.

Properties

 * Every partially ordered set is an upper set of itself.
 * The intersection and the union of any family of upper sets is again an upper set.
 * The complement of any upper set is a lower set, and vice versa.
 * Given a partially ordered set $$(X, \leq),$$ the family of upper sets of $$X$$ ordered with the inclusion relation is a complete lattice, the upper set lattice.
 * Given an arbitrary subset $$Y$$ of a partially ordered set $$X,$$ the smallest upper set containing $$Y$$ is denoted using an up arrow as $$\uparrow Y$$ (see upper closure and lower closure).
 * Dually, the smallest lower set containing $$Y$$ is denoted using a down arrow as $$\downarrow Y.$$
 * A lower set is called principal if it is of the form $$\downarrow\{x\}$$ where $$x$$ is an element of $$X.$$
 * Every lower set $$Y$$ of a finite partially ordered set $$X$$ is equal to the smallest lower set containing all maximal elements of $$Y$$
 * $$\downarrow Y = \downarrow \operatorname{Max}(Y)$$ where $$\operatorname{Max}(Y)$$ denotes the set containing the maximal elements of $$Y.$$
 * A directed lower set is called an order ideal.
 * For partial orders satisfying the descending chain condition, antichains and upper sets are in one-to-one correspondence via the following bijections: map each antichain to its upper closure (see below); conversely, map each upper set to the set of its minimal elements. This correspondence does not hold for more general partial orders; for example the sets of real numbers $$\{ x \in \R: x > 0 \}$$ and $$\{ x \in \R: x > 1 \}$$ are both mapped to the empty antichain.

Upper closure and lower closure
Given an element $$x$$ of a partially ordered set $$(X, \leq),$$ the upper closure or upward closure of $$x,$$ denoted by $$x^{\uparrow X},$$ $$x^{\uparrow},$$ or $$\uparrow\! x,$$ is defined by $$x^{\uparrow X} =\; \uparrow\! x = \{ u \in X : x \leq u\}$$ while the lower closure or downward closure of $$x$$, denoted by $$x^{\downarrow X},$$ $$x^{\downarrow},$$ or $$\downarrow\! x,$$ is defined by $$x^{\downarrow X} =\; \downarrow\! x = \{l \in X : l \leq x\}.$$

The sets $$\uparrow\! x$$ and $$\downarrow\! x$$ are, respectively, the smallest upper and lower sets containing $$x$$ as an element. More generally, given a subset $$A \subseteq X,$$ define the upper/upward closure and the lower/downward closure of $$A,$$ denoted by $$A^{\uparrow X}$$ and $$A^{\downarrow X}$$ respectively, as $$A^{\uparrow X} = A^{\uparrow} = \bigcup_{a \in A} \uparrow\!a$$ and $$A^{\downarrow X} = A^{\downarrow} = \bigcup_{a \in A} \downarrow\!a.$$

In this way, $$\uparrow x = \uparrow\{x\}$$ and $$\downarrow x = \downarrow\{x\},$$ where upper sets and lower sets of this form are called principal. The upper closure and lower closure of a set are, respectively, the smallest upper set and lower set containing it.

The upper and lower closures, when viewed as functions from the power set of $$X$$ to itself, are examples of closure operators since they satisfy all of the Kuratowski closure axioms. As a result, the upper closure of a set is equal to the intersection of all upper sets containing it, and similarly for lower sets. (Indeed, this is a general phenomenon of closure operators. For example, the topological closure of a set is the intersection of all closed sets containing it; the span of a set of vectors is the intersection of all subspaces containing it; the subgroup generated by a subset of a group is the intersection of all subgroups containing it; the ideal generated by a subset of a ring is the intersection of all ideals containing it; and so on.)

Ordinal numbers
An ordinal number is usually identified with the set of all smaller ordinal numbers. Thus each ordinal number forms a lower set in the class of all ordinal numbers, which are totally ordered by set inclusion.