Directed set

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set together with a reflexive and transitive binary relation (that is, a preorder), with the additional property that every pair of elements has an upper bound.[1] In other words, for any and in there must exist in with and 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. Be aware that 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 definitionEdit

In addition to the definition above, there is an equivalent definition. A directed set is a set   with a preorder such that every finite subset of   has an upper bound. In this definition, the existence of an upper bound of the empty subset implies that   is nonempty.

ExamplesEdit

An element   of a preordered set   is a maximal element if for every  ,   implies  .[5] It is a greatest element if for every  ,  . Some straightforward implications of the definition include:

  • Any preordered set with a greatest element is a directed set with the same preorder.
  • 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.

Examples of directed sets include:

  • The set of natural numbers   with the ordinary order is a directed set (and so is every totally ordered set).
  • 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  
  • If   is a topological space and   is a point in   set of all neighbourhoods of   can be turned into a directed set by writing   if and only if   contains   For every     and   :
    •   since   contains itself.
    • if   and   then   and   which implies   Thus  
    • because   and since both   and   we have   and  
  • If   is a real number then the set   can be turned into a directed set by defining   if   (so "greater" elements are closer to  ). We then say that the reals have been directed towards  . 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   and   equidistant from   where   and   are on opposite sides of   Explicitly, this happens when   for some real   in which case   and   even though   Had this preorder been defined on   instead of   then it would still form a directed set but it would now have a (unique) greatest element, specifically  ; however, it still wouldn't be partially ordered. This example can be generalized to a metric space   by defining on   or   the preorder   if and only if  
  • A (trivial) example of a partially ordered set that is not directed is the set  }, in which the only order relations are   and   A less trivial example is like the previous example of the "reals directed towards  " but in which the ordering rule only applies to pairs of elements on the same side of x0 (ie, if one takes an element   to the left of   and   to its right, then   and   are not comparable, and the subset   has no upper bound).
  • A non-empty family of sets is a directed set with respect to the partial order   (respectively,  ) if and only if the intersection (resp., union) of any two of its members contains as a subset (resp., is contained as a subset of) some third member. In symbols, a family   of sets is directed with respect to   (respectively,  ) if and only if
    for all   there exists some   such that   and   (resp.,   and  )
    or equivalently,
    for all   there exists some   such that   (resp.  ).
    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  
  • 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   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  ).
  • In a poset   every lower closure of an element, i.e. every subset of the form   where   is a fixed element from   is directed.

Contrast with semilatticesEdit

 
Example of a directed set which is not a join-semilattice

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   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 least upper bound, cf. picture. (Also note that without 1111, the set is not directed.)

Directed subsetsEdit

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   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   is inherited from  ; 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.[6] 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.[further explanation needed]

See alsoEdit

NotesEdit

  1. ^ Kelley, p. 65.
  2. ^ Robert S. Borden (1988). A Course in Advanced Calculus. Courier Corporation. p. 20. ISBN 978-0-486-15038-3.
  3. ^ Arlen Brown; Carl Pearcy (1995). An Introduction to Analysis. Springer. p. 13. ISBN 978-1-4612-0787-0.
  4. ^ Siegfried Carl; Seppo Heikkilä (2010). Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory. Springer. p. 77. ISBN 978-1-4419-7585-0.
  5. ^ This implies   if   is a partially ordered set.
  6. ^ Gierz, p. 2.

ReferencesEdit

  • J. L. Kelley (1955), General Topology.
  • Gierz, Hofmann, Keimel, et al. (2003), Continuous Lattices and Domains, Cambridge University Press. ISBN 0-521-80338-1.