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.

$\blacksquare$


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