Definition:Directed Ordering

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.

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Also see

• Definition:Directed Preordering

• 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