I'm reading an article and it says that, if $l$ has a $\bar{l}$ complement in a distributive lattice $L$, then $\bar{l}$ is a pseudocomplement of $l$. Can someone point me a hint, I'm new with lattice theory.
$\bar{l}$ is complement of $l$ if $l\vee\bar{l}=\top$ and $l\wedge\bar{l}=\bot$.
A pseudocomplement of $l\in L$ is an element $l^{\star}$ such that $m\leq l^{\star}$ if and only if $m\wedge l=\bot$.
If $m \le \bar{l}$ then $m \land l \le \bar{l} \land l = \bot$ is immediate (in any lattice $x \le y$ implies $x \land z \le y \land z$ for any $x,y,z$). And always $\bot \le m \land l$ so $m \land l = \bot$.
OTOH, if $m \land l = \bot$, then $\bar{l}= \bar{l} \lor \bot = \bar{l} \lor (m \land l)$ and we apply distributivity to get $\bar{l}= (\bar{l} \lor m) \land (\bar{l} \lor l) = (\bar{l} \lor m) \land \top = \bar{l} \lor m$ so in conclusion
$$\bar{l} = \bar{l} \lor m \implies m \le \bar{l}$$ as required.
So a complement in a distributive lattice is a pseudocomplement.