Meet Semilattice is Ordered Structure
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Theorem
Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.
Then $\struct {S, \wedge, \preceq}$ is an ordered structure.
Proof
For $\struct {S, \wedge, \preceq}$ to be an ordered structure is equivalent to, for all $a,b,c \in S$:
- $a \preceq b \implies a \wedge c \preceq b \wedge c$
- $a \preceq b \implies c \wedge a \preceq c \wedge b$
Since Meet is Commutative, it suffices to prove the first of these implications.
By definition of meet:
- $b \wedge c = \inf \set {b, c}$
where $\inf$ denotes infimum.
- $a \wedge c \preceq a$
- $a \wedge c \preceq c$
Now also $a \preceq b$, and by transitivity of $\preceq$ we find that:
- $a \wedge c \preceq b$
Thus $a \wedge c$ is a lower bound for $\set {b, c}$.
Hence:
- $a \wedge c \preceq b \wedge c$
by definition of infimum.
$\blacksquare$