The natural numbers $\mathbb{N}$ carry two (order-theoretic) lattice structures: One, say $L_1$, is the division lattice (where the join is the least common multiple and the meet is the greatest common divisor). Another, say $L_2$, is the one induced from the total order on $\mathbb{N}$ (i.e. join=maximum and meet=minimum).
The natural numbers $\mathbb{N}$ are not simply a set but they come with two monoid structures: addition and multiplication. The lattice $L_2$ becomes a residuated lattice with addition, while $L_1$ does not become one with multiplication. Nevertheless, in both $L_1$ and $L_2$ we have the following identity
$$(a \land b)\ast(a\lor b)=a \ast b$$ where the symbol $\ast$ denotes multiplication in the case of $L_1$ and addition in the case of $L_2$.
Do lattices (which are additionally monoids) satisfying the above property have a name? What are other lattices (appearing in nature) with that property? Is my observation at all interesting or not particularly fruitful?