Editing Order theory (section)

== Basic definitions ==
This section introduces ordered sets by building upon the concepts of [[set theory]], [[arithmetic]], and [[binary relation]]s.

=== Partially ordered sets ===<!-- This section is linked from [[Indifference curve]] -->
Orders are special binary relations. Suppose that ''P'' is a set and that ≤ is a relation on ''P'' ('relation ''on'' a set' is taken to mean 'relation ''amongst'' its inhabitants', i.e. ≤ is a subset of the cartesian product ''P'' × ''P''). Then ≤ is a '''partial order''' if it is [[reflexive relation|reflexive]], [[antisymmetric relation|antisymmetric]], and [[transitive relation|transitive]], that is, if for all ''a'', ''b'' and ''c'' in ''P'', we have that:

: ''a'' ≤ ''a'' (reflexivity)
: if ''a'' ≤ ''b'' and ''b'' ≤ ''a'' then ''a'' = ''b'' (antisymmetry)
: if ''a'' ≤ ''b'' and ''b'' ≤ ''c'' then ''a'' ≤ ''c'' (transitivity).

A set with a [[partially ordered set|partial order]] on it is called a '''partially ordered set''', '''poset''', or just '''ordered set''' if the intended meaning is clear. By checking these properties, one immediately sees that the well-known orders on [[natural number]]s, [[integer]]s, [[rational number]]s and [[real number|reals]] are all orders in the above sense. However, these examples have the additional property that any two elements are comparable, that is, for all ''a'' and ''b'' in ''P'', we have that:

: ''a'' ≤ ''b'' or ''b'' ≤ ''a''.

A partial order with this property is called a [[total order]]. These orders can also be called '''linear orders''' or '''chains'''. While many familiar orders are linear, the [[subset]] order on sets provides an example where this is not the case. Another example is given by the divisibility (or "is-a-[[divisor|factor]]-of") relation |. For two natural numbers ''n'' and ''m'', we write ''n''|''m'' if ''n'' [[division (mathematics)|divides]] ''m'' without remainder. One easily sees that this yields a partial order. For example neither 3 divides 13 nor 13 divides 3, so 3 and 13 are not comparable elements of the divisibility relation on the set of integers.
The identity relation = on any set is also a partial order in which every two distinct elements are incomparable. It is also the only relation that is both a partial order and an [[equivalence relation]] because it satisfies both the antisymmetry property of partial orders and the [[Symmetric relation|symmetry]] property of equivalence relations. Many advanced properties of posets are interesting mainly for non-linear orders.

=== Visualizing a poset ===
[[File:Lattice of the divisibility of 60.svg|thumb|right|250px|[[Hasse diagram]] of the set of all divisors of 60, partially ordered by divisibility]]
[[Hasse diagram]]s can visually represent the elements and relations of a partial ordering. These are [[graph drawing]]s where the [[vertex (graph theory)|vertices]] are the elements of the poset and the ordering relation is indicated by both the [[graph theory|edges]] and the relative positioning of the vertices. Orders are drawn bottom-up: if an element ''x'' is smaller than (precedes) ''y'' then there exists a path from ''x'' to ''y'' that is directed upwards. It is often necessary for the edges connecting elements to cross each other, but elements must never be located within an edge. An instructive exercise is to draw the Hasse diagram for the set of natural numbers that are smaller than or equal to 13, ordered by | (the ''[[Divisor|divides]]'' relation).

Even some infinite sets can be diagrammed by superimposing an [[ellipsis]] (...) on a finite sub-order. This works well for the natural numbers, but it fails for the reals, where there is no immediate successor above 0; however, quite often one can obtain an intuition related to diagrams of a similar kind{{vague|date=January 2017}}.

=== Special elements within an order ===
In a partially ordered set there may be some elements that play a special role. The most basic example is given by the '''least element''' of a [[poset]]. For example, 1 is the [[least element]] of the positive integers and the [[empty set]] is the least set under the subset order. Formally, an element ''m'' is a least element if:

:  ''m'' ≤ ''a'', for all elements ''a'' of the order.

The notation 0 is frequently found for the least element, even when no numbers are concerned. However, in orders on sets of numbers, this notation might be inappropriate or ambiguous, since the number 0 is not always least. An example is given by the above divisibility order |, where 1 is the least element since it divides all other numbers. In contrast, 0 is the number that is divided by all other numbers. Hence it is the '''greatest element''' of the order.  Other frequent terms for the least and greatest elements is '''bottom''' and '''top''' or '''zero''' and '''unit'''.

Least and [[greatest element]]s may fail to exist, as the example of the real numbers shows. But if they exist, they are always unique. In contrast, consider the divisibility relation | on the set {2,3,4,5,6}. Although this set has neither top nor bottom, the elements 2, 3, and 5 have no elements below them, while 4, 5 and 6 have none above. Such elements are called '''minimal''' and '''maximal''', respectively. Formally, an element ''m'' is [[minimal element|minimal]] if:

: ''a'' ≤ ''m'' implies ''a'' = ''m'', for all elements ''a'' of the order.

Exchanging ≤ with ≥ yields the definition of [[maximal element|maximality]]. As the example shows, there can be many maximal elements and some elements may be both maximal and minimal (e.g. 5 above). However, if there is a least element, then it is the only minimal element of the order. Again, in infinite posets maximal elements do not always exist - the set of all ''finite'' subsets of a given infinite set, ordered by subset inclusion, provides one of many counterexamples. An important tool to ensure the existence of maximal elements under certain conditions is '''[[Zorn's Lemma]]'''.

Subsets of partially ordered sets inherit the order. We already applied this by considering the subset {2,3,4,5,6} of the natural numbers with the induced divisibility ordering. Now there are also elements of a poset that are special with respect to some subset of the order. This leads to the definition of '''[[upper bound]]s'''. Given a subset ''S'' of some poset ''P'', an upper bound of ''S'' is an element ''b'' of ''P'' that is above all elements of ''S''. Formally, this means that

: ''s'' ≤ ''b'', for all ''s'' in ''S''.

Lower bounds again are defined by inverting the order. For example, −5 is a lower bound of the natural numbers as a subset of the integers. Given a set of sets, an upper bound for these sets under the subset ordering is given by their [[union (set theory)|union]]. In fact, this upper bound is quite special: it is the smallest set that contains all of the sets. Hence, we have found the '''[[least upper bound]]''' of a set of sets. This concept is also called '''supremum''' or '''join''', and for a set ''S'' one writes sup(''S'') or <math>\bigvee S</math> for its least upper bound. Conversely, the '''[[greatest lower bound]]''' is known as '''[[infimum]]''' or '''meet''' and denoted inf(''S'') or <math>\bigwedge S</math>. These concepts play an important role in many applications of order theory. For two elements ''x'' and ''y'', one also writes <math>x\vee y</math> and <math>x\wedge y</math> for sup({''x'',''y''}) and inf({''x'',''y''}), respectively.

For example, 1 is the infimum of the positive integers as a subset of integers.

For another example, consider again the relation | on natural numbers. The least upper bound of two numbers is the smallest number that is divided by both of them, i.e. the [[least common multiple]] of the numbers. Greatest lower bounds in turn are given by the [[greatest common divisor]].

=== Duality ===
In the previous definitions, we often noted that a concept can be defined by just inverting the ordering in a former definition. This is the case for "least" and "greatest", for "minimal" and "maximal", for "upper bound" and "lower bound", and so on. This is a general situation in order theory: A given order can be inverted by just exchanging its direction, pictorially flipping the Hasse diagram top-down. This yields the so-called '''dual''', '''inverse''', or '''opposite order'''.

Every order theoretic definition has its dual: it is the notion one obtains by applying the definition to the inverse order. Since all concepts are symmetric, this operation preserves the theorems of partial orders. For a given mathematical result, one can just invert the order and replace all definitions by their duals and one obtains another valid theorem. This is important and useful, since one obtains two theorems for the price of one. Some more details and examples can be found in the article on [[duality (order theory)|duality in order theory]].

=== Constructing new orders ===
There are many ways to construct orders out of given orders. The dual order is one example. Another important construction is the [[cartesian product]] of two partially ordered sets, taken together with the [[product order]] on pairs of elements. The ordering is defined by (''a'', ''x'') ≤ (''b'', ''y'') if (and only if) ''a'' ≤ ''b'' and ''x'' ≤ ''y''. (Notice carefully that there are three distinct meanings for the relation symbol ≤ in this definition.) The [[disjoint union]] of two posets is another typical example of order construction, where the order is just the (disjoint) union of the original orders.

Every partial order ≤ gives rise to a so-called [[strict order]] <, by defining ''a'' < ''b'' if ''a'' ≤ ''b'' and not ''b'' ≤ ''a''. This transformation can be inverted by setting ''a'' ≤ ''b'' if ''a'' < ''b'' or ''a'' = ''b''. The two concepts are equivalent although in some circumstances one can be more convenient to work with than the other.