Join Semilattice is Ordered Structure/Proof 1
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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
For $\struct {S, \vee, \preceq}$ 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:
- $a \vee c = \sup \set {a, c}$
where $\sup$ denotes supremum.
- $b \preceq b \vee c$
- $c \preceq b \vee c$
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 $\set {a, c}$.
Hence:
- $a \vee c \preceq b \vee c$
by definition of supremum.
$\blacksquare$