# Definition:Meet (Order Theory)

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## Contents

## Definition

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

Let $a, b \in S$, and suppose that their infimum $\inf \left\{{a, b}\right\}$ exists in $S$.

Then $a \wedge b$, the **meet of $a$ and $b$**, is defined as:

- $a \wedge b = \inf \left\{{a, b}\right\}$

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

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

## Also known as

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

## Also see

## Sources

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