Directed set

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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 (i.e., preorder) ≤ having the additional property that every pair of elements has an upper bound; more precisely, for any two elements a and b in A, there exists an element c in A (not necessarily distinct from a,b) with a ≤ c and b ≤ c (directedness).

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[edit] Applications

Directed sets are generalizations of nonempty totally ordered sets. In topology they are used to define nets that generalize sequences and unite the various notions of limit used in analysis. They also give rise to direct limits in abstract algebra and (more generally) category theory.

[edit] Examples

Examples of directed sets include:

  • The set of natural numbers N with the ordinary order ≤ is a directed set (and so is every totally ordered set).
  • The set N \times N of pairs of natural numbers can be made into a directed set by defining (n0 , n1) ≤ (m0, m1) if and only if n0 ≤ m0 and n1 ≤ m1.
  • If x0 is a real number, we can turn the set R − {x0} 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 not a partial order.
  • 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,W: if U ≤ V and V ≤ W, then U ≤ W; since if U contains V and V contains W then U contains W.
    • For every U, V: there exists the set U \cap V such that U ≤ U \cap V and V ≤ U \cap V; since both U and V contain U \cap V.
  • In a poset P, every subset of the form {a| a in P, a ≤x}, where x is a fixed element from P, is directed.

[edit] 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 {1000,0001,1101,1011,1111} ordered bitwise, where {1000,0001} has three upper bounds but no least upper bound.

[edit] Directed subsets

Directed sets need not be antisymmetric and therefore in general are not partial orders. However, the term is also frequently used in the context of posets. In this setting, a subset A of a partially ordered set (P,≤) is called a directed subset iff

  • A is not the empty set,
  • for any two a and b in A, there exists a c in A with a ≤ c and b ≤ c (directedness),

where the order of the elements of A is inherited from P. For this reason, reflexivity and transitivity need not be required explicitly.

Directed subsets are most commonly used in domain theory, where one studies orders for which these sets are required to have a least upper bound. Thus, directed subsets provide a generalization of (converging) sequences in the setting of partial orders as well.

[edit] See also