Join Semilattice is Semilattice

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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:

Join is Commutative
Join is Associative
Join is Idempotent

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

$\blacksquare$


Also see