Definition:Meet (Order Theory)

From ProofWiki
Jump to navigation Jump to search

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:

$c \preceq a$ 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

  • Results about meet (and join) can be found here.


Sources