Multiplicative Auxiliary Relation iff Congruent

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below lattice.

Let $\mathcal R$ be an auxiliary relation on $S$.

Then $\mathcal R$ is multiplicative :


 * $\forall a, b, x, y \in S: \tuple {a, x}, \tuple {b, y} \in \RR \implies \tuple {a \wedge b, x \wedge y} \in \RR$

That is $\RR$ is a congruence relation for $\wedge$.

Sufficient Condition
Let $\RR$ be multiplicative.

Let $a, b, x, y \in S$ such that
 * $\tuple {a, x}, \tuple {b, y} \in \RR$

By Meet Precedes Operands:
 * $a \wedge b \preceq a$ and $a \wedge b \preceq b$

By definition of reflexivity:
 * $x \preceq x$ and $y \preceq y$

By definition of auxiliary relation:
 * $\tuple {a \wedge b, x}, \tuple {a \wedge b, y} \in \RR$

Thus by definition of multiplicative relation:
 * $\tuple {a \wedge b, x \wedge y} \in \RR$

Necessary Condition
Suppose that
 * $\forall a, b, x, y \in S: \tuple {a, x}, \tuple {b, y} \in \RR \implies \tuple {a \wedge b, x \wedge y} \in \RR$

Let $a, x, y \in S$ such that
 * $\tuple {a, x}, \tuple {a, y} \in \RR$

By Meet is Idempotent:
 * $a \wedge a = a$

Thus by assumption:
 * $\tuple {a, x \wedge y} \in \RR$