# Definition:Meet (Order Theory)

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*This page is about Meet in the context of Order Theory. For other uses, see Meet.*

## Definition

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

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

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

- $a \wedge b = \inf \set {a, b}$

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

- $(1): \quad c \preceq a$ and $c \preceq b$
- $(2): \quad \forall s \in S: s \preceq a$ and $s \preceq b \implies s \preceq c$

## Also known as

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

## Also see

- Results about
**the meet operation**can be found**here**.

## Sources

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