Join Semilattice is Ordered Structure

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

Then $\left({S, \vee, \preceq}\right)$ is an ordered structure.

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

Proof 1
For $\left({S, \vee, \preceq}\right)$ to be an ordered structure is equivalent to, for all $a,b,c \in S$:


 * $a \preceq b \implies a \vee c \preceq b \vee c$
 * $a \preceq b \implies c \vee a \preceq c \vee b$

Since Join is Commutative, it suffices to prove the first of these implications.

By definition of join:


 * $a \vee c = \sup \left\{{a, c}\right\}$

where $\sup$ denotes supremum.

By Join Succeeds Operands:


 * $b \preceq b \vee c$
 * $c \preceq b \vee c$

Now also $a \preceq b$, and by transitivity of $\preceq$ we find that:


 * $a \preceq b \vee c$

Thus $b \vee c$ is an upper bound for $\left\{{a, c}\right\}$.

Hence:


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

by definition of supremum.

Proof 2
Let $a, b, c \in S$.

Let $a \preceq b$.

By the definition of join semilattice:
 * $a \vee b = b$

Thus:
 * $\left({a \vee b}\right) \vee c = b \vee c$

Since $\vee$ is associative, commutative, and idempotent:
 * $\left({a \vee c}\right) \vee \left({b \vee c}\right) = b \vee c$

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

From Join is Commutative, we conclude that:
 * $c \vee a \preceq c \vee b$

Also see

 * Meet Semilattice is Ordered Structure