Difference between revisions of "Partial ordering"

Latest revision as of 10:11, 20 February 2016

Note to reader: this page defines $\sqsubseteq$ as the partial ordering under study, this is to try and make the concept distinct from $\le$, which the reader should have been familiar with from a young age (and thus can taint initial study)

Definition

Given a relation, $\sqsubseteq$ in $X$ (mathematically: $\sqsubseteq\subseteq X\times X$^{[Note 1]}) we say $\sqsubseteq$ is a partial order^[1][2][3] if:

• The relation $\sqsubseteq$ is all 3 of the following:

• Note: $\le$, $\preceq$ or $\subseteq$^{[Warning 1]} are all commonly used for partial relations too.
  • The corresponding strict partial orderings are $<$, $\prec$ and $\subset$

Alternative definition

Alternatively, a partial order is simply a preorder that is also anti-symmetric (AKA Identitive), that is to say^[4]:

• Given a preorder in $X$, so a $\preceq$ such that $\preceq\subseteq X\times X$, then $\preceq$ is also a partial order if:
• $\forall x,y\in X[((x,y)\in\preceq\wedge(y,x)\in\preceq)\implies (x=y)]$ or equivalently
  • $\forall x,y\in X[(x\preceq y\wedge y\preceq x)\implies(x=y)]$

Terminology

A tuple consisting of a set $X$ and a partial order $\sqsubseteq$ in $X$ is called a poset^[4], then we may say that:

• $(X,\sqsubseteq)$ is a poset.

Notation

Be careful, as $\preceq$, $\le$ and $\sqsubseteq$ 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 $(X,\preceq)$ be a partial ordering in $X$. Or
• Given any $\preceq$ that is a partial order of $X$

So forth

Induced strict partial ordering

Here, let $\preceq$ be a partial ordering as defined above, then the relation, $\prec$ defined by:

• $(x,y)\in\prec\iff[x\ne y\wedge x\preceq y]$

is a strict partial ordering

• Note: every strict partial ordering induces a partial ordering, given a strict partial ordering, $<$, we can define a relation $\le$ as:
  • $x\le y\iff[x=y\vee x<y]$ or equivalently (in relational form): $(x,y)\in\le\iff[x=y\vee (x,y)\in<]$

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 anti-symmetric (AKA identitive)
  • Preset (is to preorder as poset is to partial order) - a tuple consisting of a set and a pre-order on it.
• Strict partial order - which induces and is induced by the same partial order, thus an equivalent statement to a partial order

Notes

1. Jump up ↑ Here $\sqsubseteq$ is the name of the relation, so $(x,y)\in \sqsubseteq$ means $x\sqsubseteq y$ - as usual for relations

Warnings

1. Jump up ↑ I avoid using $\subseteq$ for anything other than denoting subsets, the relation and the set it relates on will go together, so you'll already be using $\subseteq$ to mean subset

References

1. Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin
2. Jump up ↑ Set Theory - Thomas Jech - Third millennium edition, revised and expanded
3. Jump up ↑ Real and Abstract Analysis - Edwin Hewitt & Karl Stromberg
4. ↑ ^{Jump up to: 4.0 4.1} An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition