# Definition:Directed Ordering

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## Definition

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

Let $\struct {S, \preccurlyeq}$ be such that:

- $\forall x, y \in S: \exists z \in S: x \preccurlyeq z$ and $y \preccurlyeq z$

That is, such that every pair of elements of $S$ has an upper bound in $S$.

Then $\preccurlyeq$ is a **directed ordering**.

This article is complete as far as it goes, but it could do with expansion.In particular: We really need an explanation to the effect that "directed set" can mean both "directed preordering" and "directed ordering"You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also see

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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.28$ - 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 2.$ Topology