Editing Order theory (section)

== Background and motivation ==
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.g. "2 is less than 3", "10 is [[greater than]] 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of [[number]]s, such as the [[integer]]s and the [[real number|reals]]. The idea of being greater than or less than another number is one of the basic intuitions of [[number systems]] in general (although one usually is also interested in the actual [[Subtraction|difference]] of two numbers, which is not given by the order). Other familiar examples of orderings are the [[alphabetical order]] of words in a dictionary and the [[genealogy|genealogical]] property of [[lineal descent]] within a group of people.

The notion of order is very general, extending beyond contexts that have an immediate, intuitive feel of sequence or relative quantity. In other contexts orders may capture notions of containment or specialization. Abstractly, this type of order amounts to the [[subset|subset relation]], e.g., "[[Pediatricians]] are [[physicians]]," and "[[Circles]] are merely special-case [[ellipse]]s."

Some orders, like "less-than" on the natural numbers and alphabetical order on words, have a special property: each element can be ''compared'' to any other element, i.e. it is smaller (earlier) than, larger (later) than, or identical to. However, many other orders do not. Consider for example the subset order on a collection of [[Set (mathematics)|sets]]: though the set of birds and the set of dogs are both subsets of the set of animals, neither the birds nor the dogs constitutes a subset of the other. Those orders like the "subset-of" relation for which there exist ''incomparable'' elements are called ''[[partial order]]s''; orders for which every pair of elements is comparable are ''[[total order]]s''.

Order theory captures the intuition of orders that arises from such examples in a general setting. This is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makes much sense, because one can derive numerous theorems in the general setting, without focusing on the details of any particular order. These insights can then be readily transferred to many less abstract applications.

Driven by the wide practical usage of orders, numerous special kinds of ordered sets have been defined, some of which have grown into mathematical fields of their own. In addition, order theory does not restrict itself to the various classes of ordering relations, but also considers appropriate [[function (mathematics)|functions]] between them. A simple example of an order theoretic property for functions comes from [[Functional analysis|analysis]] where [[Monotonic function|monotone]] functions are frequently found.