Praeclarum Theorema for Meet Semilattices
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Theorem
Let $(S, \wedge, \preceq)$ be a meet semilattice.
Let $a, b, c, d \in S$.
Let $a \preceq b$.
Let $c \preceq d$.
Then $(a \wedge c) \preceq (b \wedge d)$.
Proof
By Meet Semilattice is Ordered Structure, $\preceq$ is compatible with $\wedge$.
By the definition of ordering, $\preceq$ is transitive.
Thus the theorem holds by Operating on Transitive Relationships Compatible with Operation.
$\blacksquare$
Also See
- Praeclarum Theorema, an analogous result in logic