  A simple but elegant result of Rival states that every sublattice $L$ of a
finite distributive lattice $\mathcal{P}$ can be constructed from $\mathcal{P}$
by removing a particular family $\mathcal{I}_L$ of its irreducible intervals.
  Applying this in the case that $\mathcal{P}$ is a product of a finite set
$\mathcal{C}$ of chains, we get a one-to-one correspondence $L \mapsto
\mathcal{D}_\mathcal{P}(L)$ between the sublattices of $\mathcal{P}$ and the
preorders spanned by a canonical sublattice $\mathcal{C}^\infty$ of
$\mathcal{P}$.
  We then show that $L$ is a tight sublattice of the product of chains
$\mathcal{P}$ if and only if $\mathcal{D}_\mathcal{P}(L)$ is asymmetric. This
yields a one-to-one correspondence between the tight sublattices of
$\mathcal{P}$ and the posets spanned by its poset $J(\mathcal{P})$ of non-zero
join-irreducible elements.
  With this we recover and extend, among other classical results, the
correspondence derived from results of Birkhoff and Dilworth, between the tight
embeddings of a finite distributive lattice $L$ into products of chains, and
the chain decompositions of its poset $J(L)$ of non-zero join-irreducible
elements.
