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:
- $\RR^{\ne} = \set {\tuple {x, y}, \tuple {x, z} }$
Parallel Lines
Let $S$ denote the set of all infinite straight lines embedded in a cartesian plane.
Let $\LL$ denote the relation on $S$ defined as:
- $a \mathrel \LL b$ if and only if:
- $a$ is parallel $b$
- if $a$ is not parallel to the $y$-axis, then coincides with or lies below $b$
- but if $b$ is parallel to the $y$-axis, then $a$ coincides with or lies to the right of $b$
Its dual $\LL^{-1}$ is defined as:
- $a \mathrel {\LL^{-1} } b$ if and only if:
- $a$ is parallel $b$
- if $a$ is not parallel to the $y$-axis, then coincides with or lies above $b$
- but if $b$ is parallel to the $y$-axis, then $a$ coincides with or lies to the left of $b$.
Then $\LL$ and $\LL^{-1}$ are partial orderings on $S$.
Ancestry
Let $P$ denote the set of all people who have ever lived.
Let $\DD$ denote the relation on $P$ defined as:
- $a \mathrel \DD b$ if and only if $a$ is a descendant of or the same person as $b$.
Its dual $\DD^{-1}$ is defined as:
- $a \mathrel {\DD^{-1} } b$ if and only if $a$ is an ancestor of or the same person as $b$.
Then $\DD$ and $\DD^{-1}$ are partial orderings on $P$.
Also see
- Results about partial orderings can be found here.
Internationalization
Partial Ordering is translated:
In German : | Halbordnung |
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