Join Semilattice is Ordered Structure/Proof 2

Theorem

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

Then $\struct {S, \vee, \preceq}$ is an ordered structure.

That is, $\preceq$ is compatible with $\vee$.

Proof

Let $a, b, c \in S$.

Let $a \preceq b$.

By the definition of join semilattice:

$a \vee b = b$

Thus:

$\paren {a \vee b} \vee c = b \vee c$

Since $\vee$ is associative, commutative, and idempotent:

$\paren {a \vee c} \vee \paren {b \vee c} = b \vee c$

Therefore, $a \vee c \preceq b \vee c$.

From Join is Commutative, we conclude that:

$c \vee a \preceq c \vee b$

$\blacksquare$