# Join Semilattice is Semilattice

Jump to navigation
Jump to search

## Theorem

Let $\left({S, \vee, \preceq}\right)$ be a join semilattice.

Then $\left({S, \vee}\right)$ is a semilattice.

## Proof

By definition of join semilattice, $\vee$ is closed.

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

Hence $\left({S, \vee}\right)$ is a semilattice.

$\blacksquare$