Directed set

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound.[1] In other words, for any a and b in A there must exist c in A with a ≤ c and b ≤ c. A directed set's preorder is called a direction.

The notion defined above is sometimes called an upward directed set. A downward directed set is defined analogously,[2] meaning that every pair of elements is bounded below.[3] Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Be aware that other authors call a set directed if and only if it is directed both upward and downward.[4]

Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast partially ordered sets, which need not be directed). Join semilattices (which are partially ordered sets) are directed sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward.

In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.

Equivalent definition

In addition to the definition above, there is an equivalent definition. A directed set is a set A with a preorder such that every finite subset of A has an upper bound. In this definition, the existence of an upper bound of the empty subset implies that A is nonempty.

Examples

Examples of directed sets include:

The set of natural numbers $ℕ$ with the ordinary order $\leq$ is a directed set (and so is every totally ordered set).
Let $𝔻 1$ and $𝔻 2$ be directed sets. Then the Cartesian product set $𝔻 1 \times 𝔻 2$ can be made into a directed set by defining $(n 1, n 2) \leq (m 1, m 2)$ if and only if $n 1 \leq m 1 and n 2 \leq m 2$ . In analogy to the product order this is the product direction on the Cartesian product.
It follows from previous example that the set $ℕ \times ℕ$ of pairs of natural numbers can be made into a directed set by defining $(n 0, n 1) \leq (m 0, m 1)$ if and only if $n 0 \leq m 0 and n 1 \leq m 1$ .
If T is a topological space and x₀ is a point in T, set of all neighbourhoods of x₀ can be turned into a directed set by writing U ≤ V if and only if U contains V. For every U, V, and W :
- $U \leq U$ since $U$ contains itself.
- if $U \leq V$ and $V \leq W$ , then $U \supseteq V$ and $V \supseteq W$ , which implies $U \supseteq W$ . Thus $U \leq W$ .
- since $x 0 \in U \cap V$ , and since both $U \supseteq U \cap V$ and $V \supseteq U \cap V$ , we have $U \leq U \cap V$ and $V \leq U \cap V$ .
Any preordered set $(I, \leq)$ such that $I$ has a greatest element is a directed set. An element $g \in I$ is called a greatest element of $(I, \leq)$ if $j \leq g$ for every $j \in I$ .
If $(I, \leq)$ is a directed set and if $m \in I$ is a maximal element of $I$ then $I$ necessarily has a unique greatest element, and that is $m$ . An element $m \in I$ is called a maximal element of $(I, \leq)$ if there does not exist any $j \in I$ such that $j \neq m$ and $m \leq j$ .
If $x 0$ is a real number then the set $I := ℝ ∖ {x 0}$ can be turned into a directed set by defining $a \leq I b$ if $| a - x 0 | \geq | b - x 0 |$ (so "greater" elements are closer to $x 0$ ). We then say that the reals have been directed towards $x 0$ . This is an example of a directed set that is neither partially ordered nor totally ordered. This is because antisymmetry breaks down for every pair $a$ and $b$ equidistant from $x 0$ , where $a$ and $b$ are on opposite sides of $x 0$ . Explicitly, this happens when ${a, b} = {x 0 - r, x 0 + r}$ for some real $r \neq 0$ , in which case $a \leq I b$ and $b \leq I a$ even though $a \neq b$ . Had this preorder been defined on $ℝ$ instead of $ℝ ∖ {x 0}$ then it would still form a directed set but it would now have a (unique) greatest element, specifically $x 0$ ; however, it still wouldn't be partially ordered. This example can be generalized to a metric space $(X, d)$ by defining on $X$ or $X ∖ {x 0}$ the preorder $a \leq b$ if and only if $d (a, x 0) \geq d (b, x 0)$ .
A (trivial) example of a partially ordered set that is not directed is the set ${a, b$ }, in which the only order relations are $a \leq a$ and $b \leq b$ . A less trivial example is like the previous example of the "reals directed towards $x 0$ " but in which the ordering rule only applies to pairs of elements on the same side of x₀ (ie, if one takes an element $a$ to the left of $x 0$ , and $b$ to its right, then $a$ and $b$ are not comparable, and the subset ${a, b$ } has no upper bound).
A non-empty family of sets is a directed set with respect to the preorder $\supseteq$ (respectively, $\subseteq$ ) if and only if the intersection (resp., union) of any two of its members contains as a subset (resp., is contained as a subset of) some third member. In symbols, a family $I$ of sets is directed with respect to $\supseteq$ (respectively, $\subseteq$ ) if and only if
for all $A, B \in I$ , there exists some $C \in I$ such that $A \supseteq C$ and $B \supseteq C$ (resp., $A \subseteq C$ and $B \subseteq C$ )

or equivalently,

for all $A, B \in I$ , there exists some $C \in I$ such that $A \cap B \supseteq C$ (resp. $A \cup B \subseteq C$ ).
Every $π$ -system, which is a non-empty family of sets that is closed under the intersection of any two of its members, is a directed set with respect to $\supseteq$ . Every λ-system is a directed set with respect to $\subseteq$ . Every filter, topology, and σ-algebra is a directed set with respect to both $\supseteq$ and $\subseteq$ .
By definition a prefilter or filter base is a non-empty family of sets that is a directed set with respect to the partial order $\supseteq$ and that also doesn't contain the empty set (this condition prevents triviality because otherwise, the empty set would then be a greatest element with respect to $\supseteq$ ).
In a poset $P$ , every lower closure of an element, i.e. every subset of the form ${a : a \in P, a \leq x}$ where $x$ is a fixed element from $P$ , is directed.

Contrast with semilattices

Example of a directed set which is not a join-semilattice

Directed sets are a more general concept than (join) semilattices: every join semilattice is a directed set, as the join or least upper bound of two elements is the desired c. The converse does not hold however, witness the directed set {1000,0001,1101,1011,1111} ordered bitwise (e.g. 1000 ≤ 1011 holds, but 0001 ≤ 1000 does not, since in the last bit 1 > 0), where {1000,0001} has three upper bounds but no least upper bound, cf. picture. (Also note that without 1111, the set is not directed.)

Directed subsets

The order relation in a directed set is not required to be antisymmetric, and therefore directed sets are not always partial orders. However, the term directed set is also used frequently in the context of posets. In this setting, a subset A of a partially ordered set (P,≤) is called a directed subset if it is a directed set according to the same partial order: in other words, it is not the empty set, and every pair of elements has an upper bound. Here the order relation on the elements of A is inherited from P; for this reason, reflexivity and transitivity need not be required explicitly.

A directed subset of a poset is not required to be downward closed; a subset of a poset is directed if and only if its downward closure is an ideal. While the definition of a directed set is for an "upward-directed" set (every pair of elements has an upper bound), it is also possible to define a downward-directed set in which every pair of elements has a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a filter.

Directed subsets are used in domain theory, which studies directed-complete partial orders.[5] These are posets in which every upward-directed set is required to have a least upper bound. In this context, directed subsets again provide a generalization of convergent sequences.

Notes

Kelley, p. 65.
Robert S. Borden (1988). A Course in Advanced Calculus. Courier Corporation. p. 20. ISBN 978-0-486-15038-3.
Arlen Brown; Carl Pearcy (1995). An Introduction to Analysis. Springer. p. 13. ISBN 978-1-4612-0787-0.
Siegfried Carl; Seppo Heikkilä (2010). Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory. Springer. p. 77. ISBN 978-1-4419-7585-0.
Gierz, p. 2.

References

J. L. Kelley (1955), General Topology.
Gierz, Hofmann, Keimel, et al. (2003), Continuous Lattices and Domains, Cambridge University Press. ISBN 0-521-80338-1.

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[1] Kelley, p. 65.

[2] Robert S. Borden (1988). A Course in Advanced Calculus. Courier Corporation. p. 20. ISBN 978-0-486-15038-3.

[Brown-Pearcy-3] Arlen Brown; Carl Pearcy (1995). An Introduction to Analysis. Springer. p. 13. ISBN 978-1-4612-0787-0.

[CarlHeikkilä2010-4] Siegfried Carl; Seppo Heikkilä (2010). Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory. Springer. p. 77. ISBN 978-1-4419-7585-0.

[5] Gierz, p. 2.