In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set ${\displaystyle A}$ together with a reflexive and transitive binary relation ${\displaystyle \,\leq \,}$ (that is, a preorder), with the additional property that every pair of elements has an upper bound.^[1] In other words, for any ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle A}$ there must exist ${\displaystyle c}$ in ${\displaystyle A}$ with ${\displaystyle a\leq c}$ and ${\displaystyle b\leq c.}$ 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. 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 ${\displaystyle A}$ with a preorder such that every finite subset of ${\displaystyle A}$ has an upper bound. In this definition, the existence of an upper bound of the empty subset implies that ${\displaystyle A}$ is nonempty.

Examples

The set of natural numbers ${\displaystyle \mathbb {N} }$ with the ordinary order ${\displaystyle \,\leq \,}$ is one of the most important examples of a directed set. Every totally ordered set is a directed set, including ${\displaystyle (\mathbb {N} ,\leq ),}$ ${\displaystyle (\mathbb {N} ,\geq ),}$ ${\displaystyle (\mathbb {R} ,\leq ),}$ and ${\displaystyle (\mathbb {R} ,\geq ).}$

A (trivial) example of a partially ordered set that is not directed is the set ${\displaystyle \{a,b\},}$ in which the only order relations are ${\displaystyle a\leq a}$ and ${\displaystyle b\leq b.}$ A less trivial example is like the following example of the "reals directed towards ${\displaystyle x_{0}}$" but in which the ordering rule only applies to pairs of elements on the same side of ${\displaystyle x_{0}}$ (that is, if one takes an element ${\displaystyle a}$ to the left of ${\displaystyle x_{0},}$ and ${\displaystyle b}$ to its right, then ${\displaystyle a}$ and ${\displaystyle b}$ are not comparable, and the subset ${\displaystyle \{a,b\}}$ has no upper bound).

Product of directed sets

Let ${\displaystyle \mathbb {D} _{1}}$ and ${\displaystyle \mathbb {D} _{2}}$ be directed sets. Then the Cartesian product set ${\displaystyle \mathbb {D} _{1}\times \mathbb {D} _{2}}$ can be made into a directed set by defining ${\displaystyle \left(n_{1},n_{2}\right)\leq \left(m_{1},m_{2}\right)}$ if and only if ${\displaystyle n_{1}\leq m_{1}}$ and ${\displaystyle n_{2}\leq m_{2}.}$ In analogy to the product order this is the product direction on the Cartesian product. For example, the set ${\displaystyle \mathbb {N} \times \mathbb {N} }$ of pairs of natural numbers can be made into a directed set by defining ${\displaystyle \left(n_{0},n_{1}\right)\leq \left(m_{0},m_{1}\right)}$ if and only if ${\displaystyle n_{0}\leq m_{0}}$ and ${\displaystyle n_{1}\leq m_{1}.}$

Directed towards a point

If ${\displaystyle x_{0}}$ is a real number then the set ${\displaystyle I:=\mathbb {R} \backslash \lbrace x_{0}\rbrace }$ can be turned into a directed set by defining ${\displaystyle a\leq _{I}b}$ if ${\displaystyle \left|a-x_{0}\right|\geq \left|b-x_{0}\right|}$Zdroj:https://en.wikipedia.org?pojem=Directed_set
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