Meet Semilattice is Ordered Structure

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.

By Meet Precedes Operands:


 * $a \wedge c \preceq a$
 * $a \wedge c \preceq c$

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.

Also see

 * Join Semilattice is Ordered Structure