# Definition:Partial Ordering

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

Then the ordering $\preceq$ is a **partial ordering** on $S$ if and only if $\preceq$ is not connected.

That is, if and only if $\struct {S, \preceq}$ has at least one pair which is non-comparable:

- $\exists x, y \in S: x \npreceq y \land y \npreceq x$

## Also defined as

Some sources define a **partial ordering** to be the structure known on $\mathsf{Pr} \infty \mathsf{fWiki}$ as an ordering, that is, whose nature (**partial** or total) is unspecified.

## Also known as

A **partial ordering** as defined here is sometimes referred to as a **weak partial ordering**, to distinguish it from a strict partial ordering

## Examples

### Arbitrary Example

Let $X = \set {x, y, z}$.

Let $\RR = \set {\tuple {x, x}, \tuple {x, y}, \tuple {x, z}, \tuple {y, y}, \tuple {z, z} }$.

Then $\RR$ is a partial ordering on $X$.

The strict partial ordering on $X$ corresponding to $\RR$ is its reflexive reduction:

- $\RR^{\ne} = \set {\tuple {x, y}, \tuple {x, z} }$

## Also see

- Results about
**partial orderings**can be found here.

## Sources

- 1963: George F. Simmons:
*Introduction to Topology and Modern Analysis*... (previous) ... (next): $\S 1$: Sets and Set Inclusion - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 3.3$: Ordered sets. Order types - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets