# Definition:Join (Order Theory)

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## Contents

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $a, b \in S$.

Let their supremum $\sup \left\{{a, b}\right\}$ exist in $S$.

Then the **join of $a$ and $b$** is defined as:

- $a \vee b = \sup \left\{{a, b}\right\}$

Expanding the definition of supremum, one sees that $c = a \vee b$ if and only if:

- $a \preceq c$ and $b \preceq c$ and $\forall s \in S: a \preceq s \land b \preceq s \implies c \preceq s$

## Also known as

Some sources refer to this as the **union** of $a$ and $b$.

## Also see

## Sources

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$