On Wikipedia I can see examples of distributive lattices. But I don't get how they are distributive. To my eyes, they are not even modular.
E.g. how does it fulfill the modular property of
$$(x \wedge y <: x) \wedge (x \wedge y <: y) \Leftrightarrow (x <: x \vee y) \wedge (y <: x \vee y)$$?
E.g. if I take $b, d$ in the left lattice, where $c$ is their meet, but they are not covered by their join (it's $a$). Or $b,d$ in the right lattice, where again $c$ is their meet, they are not covered by their join either (because it's $d$)