Definition:Meet (Order Theory)

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 $c = a \wedge b$ :


 * $(1): \quad c \preceq a$ 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

 * Definition:Join (Order Theory)
 * Definition:Meet Semilattice
 * Definition:Lattice