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 ${\displaystyle \left|a-x_{0}\right|\geq \left|b-x_{0}\right|}$ (so "greater" elements are closer to ${\displaystyle x_{0}}$). We then say that the reals have been directed towards ${\displaystyle x_{0}.}$ 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 ${\displaystyle a}$ and ${\displaystyle b}$ equidistant from ${\displaystyle x_{0},}$ where ${\displaystyle a}$ and ${\displaystyle b}$ are on opposite sides of ${\displaystyle x_{0}.}$ Explicitly, this happens when ${\displaystyle \{a,b\}=\left\{x_{0}-r,x_{0}+r\right\}}$ for some real ${\displaystyle r\neq 0,}$ in which case ${\displaystyle a\leq _{I}b}$ and ${\displaystyle b\leq _{I}a}$ even though ${\displaystyle a\neq b.}$ Had this preorder been defined on ${\displaystyle \mathbb {R} }$ instead of ${\displaystyle \mathbb {R} \backslash \lbrace x_{0}\rbrace }$ then it would still form a directed set but it would now have a (unique) greatest element, specifically ${\displaystyle x_{0}}$; however, it still wouldn't be partially ordered. This example can be generalized to a metric space ${\displaystyle (X,d)}$ by defining on ${\displaystyle X}$ or ${\displaystyle X\setminus \left\{x_{0}\right\}}$ the preorder ${\displaystyle a\leq b}$ if and only if ${\displaystyle d\left(a,x_{0}\right)\geq d\left(b,x_{0}\right).}$

Maximal and greatest elements

An element ${\displaystyle m}$ of a preordered set ${\displaystyle (I,\leq )}$ is a maximal element if for every ${\displaystyle j\in I,}$ ${\displaystyle m\leq j}$ implies ${\displaystyle j\leq m.}$^[5] It is a greatest element if for every ${\displaystyle j\in I,}$ ${\displaystyle j\leq m.}$

Any preordered set with a greatest element is a directed set with the same preorder. For instance, in a poset ${\displaystyle P,}$ every lower closure of an element; that is, every subset of the form ${\displaystyle \{a\in P:a\leq x\}}$ where ${\displaystyle x}$ is a fixed element from ${\displaystyle P,}$ 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.

Subset inclusion

The subset inclusion relation ${\displaystyle \,\subseteq ,\,}$ along with its dual ${\displaystyle \,\supseteq ,\,}$ define partial orders on any given family of sets. A non-empty family of sets is a directed set with respect to the partial order ${\displaystyle \,\supseteq \,}$ (respectively, ${\displaystyle \,\subseteq \,}$) 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 ${\displaystyle I}$ of sets is directed with respect to ${\displaystyle \,\supseteq \,}$ (respectively, ${\displaystyle \,\subseteq \,}$) if and only if

for all ${\displaystyle A,B\in I,}$ there exists some ${\displaystyle C\in I}$ such that ${\displaystyle A\supseteq C}$ and ${\displaystyle B\supseteq C}$ (respectively, ${\displaystyle A\subseteq C}$ and ${\displaystyle B\subseteq C}$)

or equivalently,

for all ${\displaystyle A,B\in I,}$ there exists some ${\displaystyle C\in I}$ such that ${\displaystyle A\cap B\supseteq C}$ (respectively, ${\displaystyle A\cup B\subseteq C}$).

Many important examples of directed sets can be defined using these partial orders. For example, 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 ${\displaystyle \,\supseteq \,}$ 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 ${\displaystyle \,\supseteq \,}$). 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 ${\displaystyle \,\supseteq \,.}$ Every λ-system is a directed set with respect to ${\displaystyle \,\subseteq \,.}$ Every filter, topology, and σ-algebra is a directed set with respect to both ${\displaystyle \,\supseteq \,}$ and ${\displaystyle \,\subseteq \,.}$

Finite subsets

The set ${\displaystyle \operatorname {Finite} (I)}$ of all finite subsets of a set ${\displaystyle I}$ is directed with respect to ${\displaystyle \,\subseteq \,}$ since given any two ${\displaystyle A,B\in \operatorname {Finite} (I),}$ their union ${\displaystyle A\cup B\in \operatorname {Finite} (I)}$ is an upper bound of ${\displaystyle A}$ and ${\displaystyle B}$ in ${\displaystyle \operatorname {Finite} (I).}$ This particular directed set is used to define the sum ${\displaystyle {\textstyle \sum \limits _{i\in I}}r_{i}}$ of a generalized series of an ${\displaystyle I}$-indexed collection of numbers ${\displaystyle \left(r_{i}\right)_{i\in I}}$ (or more generally, the sum of elements in an abelian topological group, such as vectors in a topological vector space) as the limit of the net of partial sums ${\displaystyle F\in \operatorname {Finite} (I)\mapsto {\textstyle \sum \limits _{i\in F}}r_{i};}$ that is:

$${\displaystyle \sum _{i\in I}r_{i}~:=~\lim _{F\in \operatorname {Finite} (I)}\ \sum _{i\in F}r_{i}~=~\lim \left\{\sum _{i\in F}r_{i}\,:F\subseteq I,F{\text{ finite }}\right\}.}$$

Logic

Let ${\displaystyle S}$ be a formal theory, which is a set of sentences with certain properties (details of which can be found in the article on the subject). For instance, ${\displaystyle S}$ could be a first-order theory (like Zermelo–Fraenkel set theory) or a simpler zeroth-order theory. The preordered set ${\displaystyle (S,\Leftarrow )}$ is a directed set because if ${\displaystyle A,B\in S}$ and if ${\displaystyle C:=A\wedge B}$ denotes the sentence formed by logical conjunction ${\displaystyle \,\wedge ,\,}$ then ${\displaystyle A\Leftarrow C}$ and ${\displaystyle B\Leftarrow C}$ where ${\displaystyle C\in S.}$ If ${\displaystyle S/\sim }$ is the Lindenbaum–Tarski algebra associated with ${\displaystyle S}$ then ${\displaystyle \left(S/\sim ,\Leftarrow \right)}$ is a partially ordered set that is also a directed set.