In

mathematics
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, a directed set (or a directed preorder or a filtered set) is a nonempty set $A$ together with a reflexive and transitive binary relation
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$\backslash ,\backslash leq\backslash ,$ (that is, a preorder), with the additional property that every pair of elements has an upper bound. In other words, for any $a$ and $b$ in $A$ there must exist $c$ in $A$ with $a\; \backslash leq\; c$ and $b\; \backslash leq\; c.$ A directed set's preorder is called a .
The notion defined above is sometimes called an . A is defined analogously, meaning that every pair of elements is bounded below.
Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other authors call a set directed if and only if it is directed both upward and downward.
Directed sets are a generalization of nonempty totally ordered set
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s. That is, all totally ordered sets are directed sets (contrast 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
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, directed sets are used to define nets, which generalize sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

s 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

The set ofnatural number
In mathematics, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lang ...

s $\backslash N$ with the ordinary order $\backslash ,\backslash leq\backslash ,$ is one of the most important examples of a directed set (and so is every totally ordered set
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). By definition, a is a function from a directed set and a sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

is a function from the natural numbers $\backslash N.$ Every sequence canonically becomes a net by endowing $\backslash N$ with $\backslash ,\backslash leq.\backslash ,$
A (trivial) example of a partially ordered set that is directed is the set $\backslash ,$ in which the only order relations are $a\; \backslash leq\; a$ and $b\; \backslash leq\; b.$ 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 $x\_0$ (that is, 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 $\backslash $ has no upper bound).
If $x\_0$ is a real number
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then the set $I\; :=\; \backslash R\; \backslash backslash\; \backslash lbrace\; x\_0\; \backslash rbrace$ can be turned into a directed set by defining $a\; \backslash leq\_I\; b$ if $\backslash left,\; a\; -\; x\_0\backslash \; \backslash geq\; \backslash left,\; b\; -\; x\_0\backslash $ (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 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 $\backslash \; =\; \backslash left\backslash $ for some real $r\; \backslash neq\; 0,$ in which case $a\; \backslash leq\_I\; b$ and $b\; \backslash leq\_I\; a$ even though $a\; \backslash neq\; b.$ Had this preorder been defined on $\backslash R$ instead of $\backslash R\; \backslash backslash\; \backslash lbrace\; x\_0\; \backslash rbrace$ 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
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$(X,\; d)$ by defining on $X$ or $X\; \backslash setminus\; \backslash left\backslash $ the preorder $a\; \backslash leq\; b$ if and only if $d\backslash left(a,\; x\_0\backslash right)\; \backslash geq\; d\backslash left(b,\; x\_0\backslash right).$
Maximal and greatest elements

An element $m$ of a preordered set $(I,\; \backslash leq)$ is amaximal element
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if for every $j\; \backslash in\; I$, $m\; \backslash leq\; j$ implies $j\; \backslash leq\; m$.
It is a greatest element if for every $j\; \backslash in\; I,$ $j\; \backslash leq\; m.$
Some straightforward implications of the definition include:
- Any preordered set with a greatest element is a directed set with the same preorder. * For instance, in a poset $P,$ every lower closure of an element; that is, every subset of the form $\backslash $ where $x$ is a fixed element from $P,$ is directed.
- Every maximal element of a directed preordered set is a greatest element. Indeed, a directed preordered set is characterized by equality of the (possibly empty) sets of maximal and of greatest elements.

Product of directed sets

Let $\backslash mathbb\_1$ and $\backslash mathbb\_2$ be directed sets. Then the Cartesian product set $\backslash mathbb\_1\; \backslash times\; \backslash mathbb\_2$ can be made into a directed set by defining $\backslash left(n\_1,\; n\_2\backslash right)\; \backslash leq\; \backslash left(m\_1,\; m\_2\backslash right)$ if and only if $n\_1\; \backslash leq\; m\_1$ and $n\_2\; \backslash leq\; m\_2.$ In analogy to the product order this is the product direction on the Cartesian product. For example, the set $\backslash N\; \backslash times\; \backslash N$ of pairs of natural numbers can be made into a directed set by defining $\backslash left(n\_0,\; n\_1\backslash right)\; \backslash leq\; \backslash left(m\_0,\; m\_1\backslash right)$ if and only if $n\_0\; \backslash leq\; m\_0$ and $n\_1\; \backslash leq\; m\_1.$Subset inclusion

The subset inclusion relation $\backslash ,\backslash subseteq,\backslash ,$ along with its dual $\backslash ,\backslash supseteq,\backslash ,$ definepartial order
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s on any given family of sets.
A non-empty family of sets is a directed set with respect to the partial order $\backslash ,\backslash supseteq\backslash ,$ (respectively, $\backslash ,\backslash subseteq\backslash ,$) if and only if the intersection (respectively, union) of any two of its members contains as a subset (respectively, is contained as a subset of) some third member.
In symbols, a family $I$ of sets is directed with respect to $\backslash ,\backslash supseteq\backslash ,$ (respectively, $\backslash ,\backslash subseteq\backslash ,$) if and only if
:for all $A,\; B\; \backslash in\; I,$ there exists some $C\; \backslash in\; I$ such that $A\; \backslash supseteq\; C$ and $B\; \backslash supseteq\; C$ (respectively, $A\; \backslash subseteq\; C$ and $B\; \backslash subseteq\; C$)
or equivalently,
:for all $A,\; B\; \backslash in\; I,$ there exists some $C\; \backslash in\; I$ such that $A\; \backslash cap\; B\; \backslash supseteq\; C$ (respectively, $A\; \backslash cap\; B\; \backslash subseteq\; C$).
Many important examples of directed sets can be defined using these partial orders.
For example, by definition, a or is a non-empty family of sets that is a directed set with respect to the partial order
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$\backslash ,\backslash supseteq\backslash ,$ and that also does not contain the empty set (this condition prevents triviality because otherwise, the empty set would then be a greatest element with respect to $\backslash ,\backslash supseteq\backslash ,$).
Every -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 $\backslash ,\backslash supseteq\backslash ,.$ Every λ-system is a directed set with respect to $\backslash ,\backslash subseteq\backslash ,.$ Every filter, topology
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, and σ-algebra is a directed set with respect to both $\backslash ,\backslash supseteq\backslash ,$ and $\backslash ,\backslash subseteq\backslash ,.$
If $x\_\; =\; \backslash left(x\_i\backslash right)\_$ is any net from a directed set $(I,\; \backslash leq)$ then for any index $i\; \backslash in\; I,$ the set $x\_\; :=\; \backslash left\backslash $ is called the tail of $(I,\; \backslash leq)$ starting at $i.$ The family $\backslash operatorname\backslash left(x\_\backslash right)\; :=\; \backslash left\backslash $ of all tails is a directed set with respect to $\backslash ,\backslash supseteq;\backslash ,$ in fact, it is even a prefilter.
If $T$ is a topological space and $x\_0$ is a point in $T,$ set of all neighbourhoods of $x\_0$ can be turned into a directed set by writing $U\; \backslash leq\; V$ if and only if $U$ contains $V.$ For every $U,$ $V,$ and $W$:
* $U\; \backslash leq\; U$ since $U$ contains itself.
* if $U\; \backslash leq\; V$ and $V\; \backslash leq\; W,$ then $U\; \backslash supseteq\; V$ and $V\; \backslash supseteq\; W,$ which implies $U\; \backslash supseteq\; W.$ Thus $U\; \backslash leq\; W.$
* because $x\_0\; \backslash in\; U\; \backslash cap\; V,$ and since both $U\; \backslash supseteq\; U\; \backslash cap\; V$ and $V\; \backslash supseteq\; U\; \backslash cap\; V,$ we have $U\; \backslash leq\; U\; \backslash cap\; V$ and $V\; \backslash leq\; U\; \backslash cap\; V.$
Let $\backslash operatorname(X)$ denote the set of all finite subsets of $X.$ Then $\backslash operatorname(X)$ is directed with respect to $\backslash ,\backslash subseteq\backslash ,$ since given any two $A,\; B\; \backslash in\; \backslash operatorname(X),$ the union $A\; \backslash cup\; B\; \backslash in\; \backslash operatorname(X)$ is an upper bound of $A$ and $B$ in $\backslash operatorname(X).$ This particular directed set is used to define the sum $\backslash sum\_\; a\_x$ of a generalized series of an $X$-indexed collection of numbers $\backslash left(a\_x\backslash right)\_$ (or elements in an abelian topological group, such as vectors in a topological vector space) as the limit of the net of partial sums $A\; \backslash in\; \backslash operatorname(X)\; \backslash mapsto\; \backslash sum\_\; a\_x;$ that is:
$$\backslash sum\_\; a\_x\; :=\; \backslash lim\_\; \backslash \; \backslash sum\_\; a\_x\; =\; \backslash lim\; \backslash left\backslash .$$
Contrast with semilattices

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 ordered bitwise (e.g. $1000\; \backslash leq\; 1011$ holds, but $0001\; \backslash leq\; 1000$ does not, since in the last bit 1 > 0), where has three upper bounds but no 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 alwayspartial order
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s. However, the term is also used frequently in the context of posets. In this setting, a subset $A$ of a partially ordered set $(P,\; \backslash leq)$ 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
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, 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.Gierz, p. 2. 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.
See also

* * * * *Notes

References

* J. L. Kelley (1955), ''General Topology''. * Gierz, Hofmann, Keimel, ''et al.'' (2003), ''Continuous Lattices and Domains'', Cambridge University Press. . {{Order theory Binary relations General topology Order theory