Upper set

A Hasse diagram of the power set of the set {1,2,3,4} with the upper set ↑{1} colored green. The white sets form the lower set ↓{2,3,4}.

In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X)^[1] of a partially ordered set (X, ≤) is a subset S ⊆ X with the following property: if s is in S and if x in X is larger than s (i.e. if s ≤ x), then x is in S. In words, this means that any x element of X that is ≥ 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 ≤ to some element of S is necessarily also an element of S.

Definition[]

Let ${\displaystyle (X,\leq )}$ be a preordered set. An upper set in ${\displaystyle X}$ (also called an upward closed set, an upset, or an isotone set)^[1] is a subset ${\displaystyle U\subseteq X}$ such that if ${\displaystyle u\in U}$ and if ${\displaystyle x\in X}$ satisfies ${\displaystyle u\leq x,}$ then ${\displaystyle x\in U.}$ That is, ${\displaystyle U}$ satisfies:

for all ${\displaystyle u\in U}$ and all ${\displaystyle x\in X,}$ if ${\displaystyle u\leq x}$ then ${\displaystyle x\in U.}$

The dual notion is a lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal), which is a subset ${\displaystyle L\subseteq X}$ such that that if ${\displaystyle l\in L}$ and if ${\displaystyle x\in X}$ satisfies ${\displaystyle x\leq l,}$ then ${\displaystyle x\in L.}$ That is, ${\displaystyle L}$ satisfies:

for all ${\displaystyle l\in L}$ and all ${\displaystyle x\in X,}$ if ${\displaystyle x\leq l}$ then ${\displaystyle x\in L.}$

The terms order ideal or ideal are sometimes used as synonyms for lower set.^[2][3][4] 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.^[2]

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, ≤), 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 ↑Y (see upper closure and lower closure).
  • Dually, the smallest lower set containing Y is denoted using a down arrow as ↓Y.
• A lower set is called principal if it is of the form ↓{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: Y = ↓Max(Y) where 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 1-1 correspondence. For the first direction, take the upper closure (see below) of the antichain. For the second direction, take the minimal elements of the upper set. This correspondence does not hold for more general partial orders.

Upper closure and lower closure[]

Given an element ${\displaystyle x}$ of a partially ordered set ${\displaystyle (X,\leq ),}$ the upper closure or upward closure of ${\displaystyle x,}$ denoted by ${\displaystyle x^{\uparrow X},}$ ${\displaystyle x^{\uparrow },}$ or ${\displaystyle \uparrow x,}$ is defined by:

${\displaystyle x^{\uparrow X}=\uparrow x=\left\{u\in X:x\leq u\right\}}$

while the lower closure or downward closure of x, denoted by ${\displaystyle x^{\downarrow X},}$ ${\displaystyle x^{\downarrow },}$ or ${\displaystyle \downarrow x,}$ is defined by:

${\displaystyle x^{\downarrow X}=\downarrow x=\left\{l\in X:l\leq x\right\}.}$

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

${\displaystyle A^{\uparrow X}=A^{\uparrow }=\bigcup _{a\in A}\uparrow \!a}$      and      ${\displaystyle A^{\downarrow X}=A^{\downarrow }=\bigcup _{a\in A}\downarrow \!a.}$

In this way, ↑x = ↑{x} and ↓x = ↓{x}, where upper sets and lower sets of this form are called principal. The upper closures and lower closures of a set are, respectively, the smallest upper set and lower set containing it.

The upper and lower closures, when viewed as function 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.

One can also speak of the strict upper closure of an element ${\displaystyle x\in X}$ defined as {y∈X : x<y}, and more generally, the strict upper closure of a subset ${\displaystyle A\subseteq X,}$ which is defined as the union of the strict upper closures of its elements, and we can make analogous definitions for strict lower closures. However note that these 'closures' are not actually closure operators, since for example the strict upper closure of a singleton set {x} does not contain {x}.

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.

See also[]

• Cofinal set – a subset U of a partially ordered set (X, ≤) that contains for every element ${\displaystyle x\in X,}$ some element y such that ${\displaystyle x\leq y.}$
• Abstract simplicial complex (also called: Independence system) - a set-family that is downwards-closed with respect to the containment relation.

References[]

• Blanck, J. (2000). "Domain representations of topological spaces" (PDF). Theoretical Computer Science. 247 (1–2): 229–255. doi:10.1016/s0304-3975(99)00045-6.
• ; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
• Hoffman, K. H. (2001), The low separation axioms (T₀) and (T₁)