Meet is Increasing

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Theorem

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

Let $f: S \times S \to S$ be a mapping such that

$\forall s, t \in S: f\left({s, t}\right) = s \wedge t$

Then:

$f$ is increasing as a mapping from Cartesian product $\left({S\times S, \precsim}\right)$ of $\left({S, \preceq}\right)$ and $\left({S, \preceq}\right)$ into $\left({S, \preceq}\right)$.

Proof

Let $\left({x, y}\right)$, $\left({z, t}\right) \in S \times S$ such that

$\left({x, y}\right) \precsim \left({z, t}\right)$

By definition of Cartesian product of ordered sets:

$x \preceq z$ and $y \preceq t$

By Meet Semilattice is Ordered Structure:

$x \wedge y \preceq z \wedge t$

By definition of $f$:

$f\left({x, y}\right) \preceq f\left({z, t}\right)$

Thus by definition

$f$ is increasing mapping.

$\blacksquare$

Sources