Finite distributive lattices and finite abelian monoids

Question

A structure of semilattice over T is the same thing than a structure of finite abelian monoid such that $\forall t \in T$, $t² = t$.
Given a semilattice T, we get an abelian monoid by defining $a.b$ = $a \wedge b$ and conversely given an abelian monoid T such that $\forall t \in T$, $t² = t$, we get a semilattice by putting $a \leq b$ iff $a.b = a$ and we get that $a \wedge b = a.b$.
I was wondering if we had a similar result for a finite distributive lattice T. I saw that if T is not just distributive but is a Boolean algebra, we get a correspondence with boolean rings.
However, I found nothing about finite distributive lattice. Someone has an idea ?
Thank you