In these cases:

$$r(x,y) = \bigvee \\{a \in A : \; \Delta(a) \leq_{A\times A} (x,y) \\}$$

$$l(x,y) = \bigwedge \\{a \in A : \; (x,y) \leq_{A\times A} \Delta(a) \\}$$

You're assuming that you can just take infima \$$\bigwedge\$$ and suprema \$$\bigvee\$$. In a simple lattice \$$(L, \wedge, \vee)\$$ you don't have those operations available.