In mathematics, a directed set is a nonempty setA together with a reflexive and transitivebinary relation ≤, 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 ≤ c and b ≤ c. The notion defined above is sometimes called an upward directed set. A downward directed set is defined analogously, meaning that every pair of elements is bounded below. Some authors 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. Directed sets are a generalization of nonempty totally ordered sets. That is, all totallyordered sets are directed sets. Join semilattices 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 generalizesequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and 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 numbersN with the ordinary order ≤ is a directed set.
Let D1 and D2 be directed sets. Then the Cartesian product set D1 D2 can be made into a directed set by defining ≤ if and only if n1 ≤ m1 and n2 ≤ m2. In analogy to the product order this is the product direction on the Cartesian product.
It follows from previous example that the setNN of pairs of natural numbers can be made into a directed set by defining ≤ if and only if n0 ≤ m0 and n1 ≤ m1.
If x0 is a real number, we can turn the set R − into a directed set by writing a ≤ b if and only if
|a − x0| ≥ |b − x0|. We then say that the reals have been directed towards x0. This is an example of a directed set that is not ordered.
A example of a partially ordered set that is not directed is the set, in which the only order relations are a ≤ a and b ≤ b. A less trivial example is like the previous 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.
If T is a topological space and x0 is a point in T, we turn the set of all neighbourhoods of x0 into a directed set by writing U ≤ V if and only if U contains V.
* For every U: U ≤ U; since U contains itself.
* For every U, V, and W: if U ≤ V and V ≤ W, then we have UV and VW, which implies UW. Thus U ≤ W.
* For every U and V: since x0UV, and since both UU V and VU V, we have U ≤ U V and V ≤ U V.
In a posetP, every lower closure of an element, i.e. every subset of the form where x is a fixed element from P, is directed.
Contrast with semilattices
Directed sets are a more general concept than 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, where has three upper bounds but no least upper bound, cf. picture.
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 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, 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.
