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 ,\leq, (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 \leq c and b \leq c. A directed set's preorder is called a direction. 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 sets. 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, 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

The set of natural numbers \N with the ordinary order ,\leq, is one of the most important examples of a directed set. Every totally ordered set is a directed set, including (\N, \leq), (\N, \geq), and A (trivial****)**** example of**** a partially ordered set that**** is**** **** 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 following example of**** the "reals**** directed**** towards x0"**** but in**** which the ordering**** rule**** only**** applies to**** pairs of**** elements**** on**** the same**** side**** of**** x0 (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**** *{* a,**** b *}* has no**** upper bound).

Product of directed sets

Let and be directed sets. Then the Cartesian product set can be made into a directed set by defining if and only if and In analogy to the product order this is the product direction on the Cartesian product. For example, the set of pairs of natural numbers can be made into a directed set by defining if and only if and

Directed towards a point

If x_0 is a real number then the set can be turned into a directed set by defining a \leq_I b if (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 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 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 the preorder a \leq b if and only if

Maximal and greatest elements

An element m of a preordered set (I, \leq) is a maximal element if for every j \in I, m \leq j implies j \leq m. It is a greatest element if for every j \in I, j \leq m. 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 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.

Subset inclusion

The subset inclusion relation along with its dual 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 (respectively, ) 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 (respectively, ) if and only if or equivalently, Many important examples of directed sets can be defined using these partial orders. For example, by definition, a Filter (set theory) or is a non-empty family of sets that is a directed set with respect to the partial order 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 ). 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 Every λ-system is a directed set with respect to Every filter, topology, and σ-algebra is a directed set with respect to both and

Tails of nets

By definition, a is a function from a directed set and a sequence is a function from the natural numbers \N. Every sequence canonically becomes a net by endowing \N with ,\leq., If is any net from a directed set (I, \leq) then for any index i \in I, the set is called the tail of (I, \leq) starting at i. The family of all tails is a directed set with respect to in fact, it is even a prefilter.

Neighborhoods

If T is a topological space and x_0 is a point in T, the set of all neighbourhoods of x_0 can be turned into a directed set by writing U \leq V if and only if U contains V. For every U, V, and W:

Finite subsets

The set of all finite subsets of a set I is directed with respect to since given any two their union is an upper bound of A and B in This particular directed set is used to define the sum of a generalized series of an I-indexed collection of numbers (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 that is:

Logic

Let 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, S could be a first-order theory (like Zermelo–Fraenkel set theory) or a simpler zeroth-order theory. The preordered set is a directed set because if A, B \in S and if denotes the sentence formed by logical conjunction ,\wedge,, then and where C \in S. If S / \sim is the Lindenbaum–Tarski algebra associated with S then is a partially ordered set that is also a directed set.

Contrast with semilattices

Directed set is a more general concept than (join) semilattice: 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. holds, but does not, since in the last bit 1 > 0), where {1000,0001} 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 always partial orders. However, the term is also used frequently in the context of posets. In this setting, a subset A of a partially ordered set (P, \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, 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. 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.

Footnotes