Editing Inequality (mathematics) (section)

== Formal definitions and generalizations ==

A (non-strict) '''partial order''' is a [[binary relation]] ≤ over a [[Set (mathematics)|set]] ''P'' which is [[Reflexive relation|reflexive]], [[Antisymmetric relation|antisymmetric]], and [[Transitive relation|transitive]].<ref>{{cite book|title=Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics|author1=Simovici, Dan A.|author2=Djeraba, Chabane|publisher=Springer|year=2008|isbn=9781848002012|chapter=Partially Ordered Sets|chapter-url=https://books.google.com/books?id=6i-F3ZNcub4C&pg=PA127|name-list-style=amp}}</ref> That is, for all ''a'', ''b'', and ''c'' in ''P'', it must satisfy the three following clauses:

* ''a'' ≤ ''a'' ([[Reflexive relation|reflexivity]])
* if ''a'' ≤ ''b'' and ''b'' ≤ ''a'', then ''a'' = ''b'' ([[Antisymmetric relation|antisymmetry]])
* if ''a'' ≤ ''b'' and ''b'' ≤ ''c'', then ''a'' ≤ ''c'' ([[Transitive relation|transitivity]])

A set with a partial order is called a '''[[partially ordered set]]'''.<ref>{{Cite web|url=http://mathworld.wolfram.com/PartiallyOrderedSet.html|title=Partially Ordered Set|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref> Those are the very basic axioms that every kind of order has to satisfy. 

A strict partial order is a relation < that satisfies 
* ''a'' ≮ ''a'' ([[Reflexive relation#irreflexive|irreflexivity]]),
* if ''a'' < ''b'', then ''b'' ≮ ''a'' ([[Asymmetric relation|asymmetry]]),
* if ''a'' < ''b'' and ''b'' < ''c'', then ''a'' < ''c'' ([[Transitive relation|transitivity]]),
where {{char|≮}} means that {{char|<}} does not hold.

Some types of partial orders are specified by adding further axioms, such as:

* [[Total order]]: For every ''a'' and ''b'' in ''P'', ''a'' ≤ ''b'' or ''b'' ≤ ''a'' .
* [[Dense order]]: For all ''a'' and ''b'' in ''P'' for which ''a'' < ''b'', there is a ''c'' in ''P'' such that ''a'' < ''c'' < ''b''.
* [[Least-upper-bound property]]: Every non-empty [[Set (mathematics)|subset]] of ''P'' with an [[upper bound]] has a [[Least upper bound|''least'' upper bound]] (supremum) in ''P''.

=== Ordered fields ===
{{Main|Ordered field}}
If (''F'', +, ×) is a [[Field (mathematics)|field]] and ≤ is a [[total order]] on ''F'', then (''F'', +, ×, ≤) is called an '''[[ordered field]]''' if and only if:
* ''a'' ≤ ''b'' implies ''a'' + ''c'' ≤ ''b'' + ''c'';
* 0 ≤ ''a'' and 0 ≤ ''b'' implies 0 ≤ ''a'' × ''b''.

Both {{tmath|(\mathbb Q, +, \times, \leq)}} and {{tmath|(\mathbb R, +, \times, \leq)}} are [[ordered field]]s, but {{math|≤}} cannot be defined in order to make {{tmath|(\mathbb C, +, \times, \leq)}} an [[ordered field]],<ref>{{Cite web|url=http://www.math.ubc.ca/~feldman/m320/fields.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.math.ubc.ca/~feldman/m320/fields.pdf |archive-date=2022-10-09 |url-status=live|title=Fields|last=Feldman|first=Joel|date=2014|website=math.ubc.ca|access-date=2019-12-03}}</ref> because −1 is the square of ''i'' and would therefore be positive.

Besides being an ordered field, '''R''' also has the [[Least-upper-bound property]]. In fact, '''R''' can be defined as the only ordered field with that quality.<ref>{{cite book |last1=Stewart |first1=Ian |title=Why Beauty Is Truth: The History of Symmetry |date=2007 |publisher=Hachette UK |isbn=978-0-4650-0875-9 |page=106 |url=https://books.google.com/books?id=1ek3DgAAQBAJ&pg=PT106}}</ref>