Meet Semilattice is Semilattice

Theorem

Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.

Then $\struct {S, \wedge}$ is a semilattice.

Proof

By definition of meet semilattice, $\wedge$ is closed.

The other three defining properties for a semilattice follow respectively from:

Meet is Commutative

Meet is Associative

Meet is Idempotent

Hence $\struct {S, \wedge}$ is a semilattice.

$\blacksquare$

Also see

• Join Semilattice is Semilattice