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

• Definition:Strict Partial Ordering
• Definition:Total Ordering
• Definition:Well-Ordering

• 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