Definition:Partial Ordering

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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

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