Join Semilattice is Semilattice

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Theorem

Let $\struct {S, \vee, \preceq}$ be a join semilattice.


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


Proof

Recall the definition of join semilattice:

Let $\struct {S, \preceq}$ be an ordered set.

Suppose that for all $a, b \in S$:

$a \vee b \in S$

where $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$.


Then the ordered structure $\struct {S, \vee, \preceq}$ is called a join 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 $\struct {S, \vee}$ is a semilattice.

$\blacksquare$


Also see


Sources