Editing Order theory (section)

== Related mathematical areas ==
Although most mathematical areas ''use'' orders in one or the other way, there are also a few theories that have relationships which go far beyond mere application. Together with their major points of contact with order theory, some of these are to be presented below.

=== Universal algebra ===
As already mentioned, the methods and formalisms of [[universal algebra]] are an important tool for many order theoretic considerations. Beside formalizing orders in terms of [[algebraic structure]]s that satisfy certain identities, one can also establish other connections to algebra. An example is given by the correspondence between [[Boolean algebra (structure)|Boolean algebra]]s and [[Boolean ring]]s. Other issues are concerned with the existence of [[Free object|free constructions]], such as [[free lattice]]s based on a given set of generators. Furthermore, closure operators are important in the study of universal algebra.

=== Topology ===
In [[topology]], orders play a very prominent role. In fact, the collection of [[open set]]s provides a classical example of a complete lattice, more precisely a complete [[Heyting algebra]] (or "'''frame'''" or "'''locale'''"). [[filter (mathematics)|Filters]] and [[net (mathematics)|nets]] are notions closely related to order theory and the [[closed set|closure operator of sets]] can be used to define a topology. Beyond these relations, topology can be looked at solely in terms of the open set lattices, which leads to the study of [[pointless topology]]. Furthermore, a natural preorder of elements of the underlying set of a topology is given by the so-called [[specialization order]], that is actually a partial order if the topology is [[T0 space|T<sub>0</sub>]].

Conversely, in order theory, one often makes use of topological results. There are various ways to define subsets of an order which can be considered as open sets of a topology. Considering topologies on a poset (''X'', ≤) that in turn induce ≤ as their specialization order, the [[Comparison of topologies|finest]] such topology is the [[Alexandrov topology]], given by taking all upper sets as opens. Conversely, the [[Comparison of topologies|coarsest]] topology that induces the specialization order is the [[upper topology]], having the complements of [[ideal (order theory)|principal ideals]] (i.e. sets of the form {''y'' in ''X'' | ''y'' ≤ ''x''} for some ''x'') as a [[subbase]]. Additionally, a topology with specialization order ≤ may be [[Specialization (pre)order#Important properties|order consistent]], meaning that their open sets are "inaccessible by directed suprema" (with respect to ≤). The finest order consistent topology is the [[Scott topology]], which is coarser than the Alexandrov topology. A third important topology in this spirit is the [[Lawson topology]]. There are close connections between these topologies and the concepts of order theory. For example, a function preserves directed suprema if and only if it is [[continuous function (topology)|continuous]] with respect to the Scott topology (for this reason this order theoretic property is also called [[Scott-continuous|Scott-continuity]]).

=== Category theory ===
The visualization of orders with [[Hasse diagram]]s has a straightforward generalization: instead of displaying lesser elements ''below'' greater ones, the direction of the order can also be depicted by giving directions to the edges of a graph. In this way, each order is seen to be equivalent to a [[directed acyclic graph]], where the nodes are the elements of the poset and there is a directed path from ''a'' to ''b'' if and only if ''a'' ≤ ''b''. Dropping the requirement of being acyclic, one can also obtain all preorders.

When equipped with all transitive edges, these graphs in turn are just special [[category theory|categories]], where elements are objects and each set of morphisms between two elements is at most singleton. Functions between orders become functors between categories. Many ideas of order theory are just concepts of category theory in small. For example, an infimum is just a [[product (category theory)|categorical product]]. More generally, one can capture infima and suprema under the abstract notion of a [[limit (category theory)|categorical limit]] (or ''colimit'', respectively). Another place where categorical ideas occur is the concept of a (monotone) [[Galois connection]], which is just the same as a pair of [[adjoint functor]]s.

But category theory also has its impact on order theory on a larger scale. Classes of posets with appropriate functions as discussed above form interesting categories. Often one can also state constructions of orders, like the [[product order]], in terms of categories. Further insights result when categories of orders are found [[equivalence of categories|categorically equivalent]] to other categories, for example of topological spaces. This line of research leads to various ''[[representation theorem]]s'', often collected under the label of [[Stone duality]].