Join Semilattice is Semilattice

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

• Meet Semilattice is Semilattice

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.22 \ \text {(a)}$
• 1981: Stanley Burris and H.P. Sankappanavar: A Course in Universal Algebra: $\text {II} \ \S 1$ Example $(7)$
• 1982: Peter T. Johnstone: Stone Spaces ... (previous) ... (next): Chapter $\text I$: Preliminaries, Definition $1.3$