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. 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 . 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) .
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.
The set of natural numbers with the ordinary order is one of the most important examples of a directed set (and so is every totally ordered set). By definition, a is a function from a directed set and a sequence is a function from the natural numbers Every sequence canonically becomes a net by endowing with
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 partially ordered nor totally ordered.
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Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a
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. 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 . A is defined analogously, meaning that every pair of elements is bounded below.
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection.