Partial ordering
 Note to reader: this page defines [ilmath]\sqsubseteq[/ilmath] as the partial ordering under study, this is to try and make the concept distinct from [ilmath]\le[/ilmath], which the reader should have been familiar with from a young age (and thus can taint initial study)
Contents
Definition
Given a relation, [ilmath]\sqsubseteq[/ilmath] in [ilmath]X[/ilmath] (mathematically: [ilmath]\sqsubseteq\subseteq X\times X[/ilmath]^{[Note 1]}) we say [ilmath]\sqsubseteq[/ilmath] is a partial order^{[1]}^{[2]}^{[3]} if:
 The relation [ilmath]\sqsubseteq[/ilmath] is all 3 of the following:
Name  Definition  

1  Reflexive  [ilmath]\forall x\in X[(x,x)\in\sqsubseteq][/ilmath] or equivalently [ilmath]\forall x\in X[x\sqsubseteq x][/ilmath] 
2  Identitive (AKA: antisymmetric)  [ilmath]\forall x,y\in X[((x,y)\in\sqsubseteq\wedge(y,x)\in\sqsubseteq)\implies (x=y)][/ilmath] or equivalently [ilmath]\forall x,y\in X[(x\sqsubseteq y\wedge y\sqsubseteq x)\implies(x=y)][/ilmath] 
3  Transitive  [ilmath]\forall x,y,z\in X[((x,y)\in\sqsubseteq\wedge(y,z)\in\sqsubseteq)\implies(x,z)\in\sqsubseteq][/ilmath] or equivalently [ilmath]\forall x,y,z\in X[(x\sqsubseteq y\wedge y\sqsubseteq z)\implies(x\sqsubseteq z)][/ilmath] 
 Note: [ilmath]\le[/ilmath], [ilmath]\preceq[/ilmath] or [ilmath]\subseteq[/ilmath]^{[Warning 1]} are all commonly used for partial relations too.
 The corresponding strict partial orderings are [ilmath]<[/ilmath], [ilmath]\prec[/ilmath] and [ilmath]\subset[/ilmath]
Alternative definition
Alternatively, a partial order is simply a preorder that is also antisymmetric (AKA Identitive), that is to say^{[4]}:
 Given a preorder in [ilmath]X[/ilmath], so a [ilmath]\preceq[/ilmath] such that [ilmath]\preceq\subseteq X\times X[/ilmath], then [ilmath]\preceq[/ilmath] is also a partial order if:
 [ilmath]\forall x,y\in X[((x,y)\in\preceq\wedge(y,x)\in\preceq)\implies (x=y)][/ilmath] or equivalently
 [ilmath]\forall x,y\in X[(x\preceq y\wedge y\preceq x)\implies(x=y)][/ilmath]
Terminology
A tuple consisting of a set [ilmath]X[/ilmath] and a partial order [ilmath]\sqsubseteq[/ilmath] in [ilmath]X[/ilmath] is called a poset^{[4]}, then we may say that:
 [ilmath](X,\sqsubseteq)[/ilmath] is a poset.
Notation
Be careful, as [ilmath]\preceq[/ilmath], [ilmath]\le[/ilmath] and [ilmath]\sqsubseteq[/ilmath] are all used to denote both partial and preorders, so always be clear which one you mean at the point of definition. That is to say write:
 Let [ilmath](X,\preceq)[/ilmath] be a partial ordering in [ilmath]X[/ilmath]. Or
 Given any [ilmath]\preceq[/ilmath] that is a partial order of [ilmath]X[/ilmath]
So forth
Induced strict partial ordering
Here, let [ilmath]\preceq[/ilmath] be a partial ordering as defined above, then the relation, [ilmath]\prec[/ilmath] defined by:
 [ilmath](x,y)\in\prec\iff[x\ne y\wedge x\preceq y][/ilmath]
 Note: every strict partial ordering induces a partial ordering, given a strict partial ordering, [ilmath]<[/ilmath], we can define a relation [ilmath]\le[/ilmath] as:
 [ilmath]x\le y\iff[x=y\vee x<y][/ilmath] or equivalently (in relational form): [ilmath](x,y)\in\le\iff[x=y\vee (x,y)\in<][/ilmath]
In fact there is a 1:1 correspondence between partial and strict partial orderings, this is why the term "partial ordering" is used so casually, as given a strict you have a partial, given a partial you have a strict.
 See Overview of partial orders for more information
See also
 Poset  the term a tuple consisting of a set equipped with a partial order
 Preorder  like a partial order except it need not be antisymmetric (AKA identitive)
 Preset (is to preorder as poset is to partial order)  a tuple consisting of a set and a preorder on it.
 Strict partial order  which induces and is induced by the same partial order, thus an equivalent statement to a partial order
Notes
 ↑ Here [ilmath]\sqsubseteq[/ilmath] is the name of the relation, so [ilmath](x,y)\in \sqsubseteq[/ilmath] means [ilmath]x\sqsubseteq y[/ilmath]  as usual for relations
Warnings
 ↑ I avoid using [ilmath]\subseteq[/ilmath] for anything other than denoting subsets, the relation and the set it relates on will go together, so you'll already be using [ilmath]\subseteq[/ilmath] to mean subset
References
 ↑ Analysis  Part 1: Elements  Krzysztof Maurin
 ↑ Set Theory  Thomas Jech  Third millennium edition, revised and expanded
 ↑ Real and Abstract Analysis  Edwin Hewitt & Karl Stromberg
 ↑ ^{4.0} ^{4.1} An Introduction to Category Theory  Harold Simmons  1st September 2010 edition
