Editing Partially ordered set (section)

== Notation ==

Given a set <math>P</math> and a partial order relation, typically the non-strict partial order <math>\leq</math>, we may uniquely extend our notation to define four partial order relations <math>\leq,</math> <math><,</math> <math>\geq,</math> and <math>></math>, where <math>\leq</math> is a non-strict partial order relation on <math>P</math>, <math> < </math> is the associated strict partial order relation on <math>P</math> (the [[irreflexive kernel]] of <math>\leq</math>), <math>\geq</math> is the dual of <math>\leq</math>, and <math> > </math> is the dual of <math> < </math>. Strictly speaking, the term ''partially ordered set'' refers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation, <math>(P,\leq)</math> or <math>(P,<)</math>, or, in rare instances, the non-strict and strict relations together, <math>(P,\leq,<)</math>.<ref>{{cite book |last1=Avigad |first1=Jeremy |last2=Lewis |first2=Robert Y. |last3=van Doorn |first3=Floris |title=Logic and Proof |date=29 March 2021 |edition=Release 3.18.4 |url=https://leanprover.github.io/logic_and_proof/relations.html#more-on-orderings |access-date=24 July 2021 |chapter=13.2. More on Orderings|quote=So we can think of every partial order as really being a pair, consisting of a weak partial order and an associated strict one.}}</ref> 

The term ''ordered set'' is sometimes used as a shorthand for ''partially ordered set'', as long as it is clear from the context that no other kind of order is meant. In particular, [[Total order|totally ordered sets]] can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Some authors use different symbols than <math>\leq</math> such as <math>\sqsubseteq</math><ref>{{cite web |last1=Rounds |first1=William C. |title=Lectures slides |url=http://www.eecs.umich.edu/courses/eecs203-1/203-Mar7.pdf |website=EECS 203: DISCRETE MATHEMATICS |access-date=23 July 2021 |date=7 March 2002}}</ref> or <math>\preceq</math><ref>{{cite book |last1=Kwong |first1=Harris |title=A Spiral Workbook for Discrete Mathematics |date=25 April 2018 |url=https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/07%3A_Relations/7.04%3A_Partial_and_Total_Ordering |access-date=23 July 2021 |language=en |chapter=7.4: Partial and Total Ordering}}</ref> to distinguish partial orders from total orders.

When referring to partial orders, <math>\leq</math> should not be taken as the [[complementary relation|complement]] of <math> > </math>. The relation <math> > </math> is the converse of the irreflexive kernel of <math>\leq</math>, which is always a subset of the complement of <math>\leq</math>, but <math> > </math> is equal to the complement of <math>\leq</math> [[if, and only if]], <math>\leq</math> is a total order.{{efn|A proof can be found [[:File:PartialOrders redundencies.pdf|here]].}}