Definition:Meet (Order Theory)

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

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

• Results about the meet operation can be found here.

Sources

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