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.

$$

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.