Meet Semilattice is Ordered Structure

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Theorem

Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice.


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


Proof

For $\left({S, \wedge, \preceq}\right)$ 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:

$b \wedge c = \inf \left\{{b, c}\right\}$

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 $\left\{{b, c}\right\}$.

Hence:

$a \wedge c \preceq b \wedge c$

by definition of infimum.

$\blacksquare$


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