Join Semilattice is Semilattice

From ProofWiki
Jump to navigation Jump to search


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

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


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

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

Join is Commutative
Join is Associative
Join is Idempotent

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


Also see