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
Also see
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