View source for Overview of partial orders

: '''Note:''' this page contains material intended to help the study of partial orderings and solve some common misconceptions, it is advised that someone trying to learn about (rather than reference or check something) about partial orders read this.

==Definition==
A ''partial order'' is a [[relation]] on a set {{M|X}}, which we shall call {{M|\mathcal{R}\subseteq X\times X}} that is{{rAPIKM}}{{rRAAAHS}}{{rSTTJ}}:
{| class="wikitable" border="1"
|-
!
! Name
! Definition
|-
! 1
| [[Relation#Types_of_relation|Reflexive]]
| {{M|1=\forall x\in X[(x,x)\in\mathcal{R}]}} or equivalently<br/>{{M|1=\forall x\in X[x\mathcal{R} x]}}
|-
! 2
| [[Relation#Types_of_relation|Identitive]] ({{AKA}}: [[Relation#Types_of_relation|antisymmetric]])
| {{M|1=\forall x,y\in X[((x,y)\in\mathcal{R}\wedge(y,x)\in\mathcal{R})\implies (x=y)]}} or equivalently<br/>
{{M|1=\forall x,y\in X[(x\mathcal{R} y\wedge y\mathcal{R} x)\implies(x=y)]}}
|-
! 3
| [[Relation#Types_of_relation|Transitive]]
| {{M|1=\forall x,y,z\in X[((x,y)\in\mathcal{R}\wedge(y,z)\in\mathcal{R})\implies(x,z)\in\mathcal{R}]}} or equivalently<br/>
{{M|1=\forall x,y,z\in X[(x\mathcal{R} y\wedge y\mathcal{R} z)\implies(x\mathcal{R} z)]}}
|}
* '''Note that:''' there is no requirement for {{M|(x,y)\in\mathcal{R}\iff x\mathcal{R}y}} (if {{M|x\ne y}}) this is probably why it is called a partial ordering (rather than a [[total ordering]] which is a partial ordering where all elements are [[comparable]])
* For example:
*# The [[divisor relation]] is a partial ordering (see [[Divisor partial ordering]]) and this is clearly not a total ordering
*# The [[subset relation]] is a partial ordering and is clearly not a total ordering
{{Todo|References}}

==Various views==
Let us now write {{M|\mathcal{R} }} as {{M|\preceq}} and rather than {{M|x\mathcal{R}y}} or {{M|(x,y)\in\mathcal{R}}} write {{M|x\preceq y}} accordingly. 

There are 2 things we must cover:
# What does {{M|\succeq}} mean?
# What does {{M|\prec}} (and {{M|\succ}}) mean?
===Dealing with {{M|x\preceq y\iff y\succeq x}}===
There are 2 views about what {{M|x\prec y}} means. We'll do the strongest (most formal) first
====Dual relation====
Consider the relation {{M|\mathcal{R}^{-1} }}
====Defining {{M|x\preceq y\iff y\succeq x}}====
This is a bit of a cop-out but we can simply say that {{M|y\succeq x}} is another way of writing {{M|x\preceq y}}
===Dealing with {{M|\prec}} and {{M|\succ}}===
Define the relation:
{{M|x \prec y\iff x\preceq y \wedge x\ne y}}
{{Todo|Finish off page, find references}}
==References==
<references/>
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[[Category:Order Theory]]