  We introduce a family of toric algebras defined by maximal chains of a finite
distributive lattice. Applying results on stable set polytopes we conclude that
every such algebra is normal and Cohen-Macaulay, and give an interpretation of
its Krull dimension in terms of the combinatorics of the underlying lattice.
When the lattice is planar, we show that the corresponding chain algebra is
generated by a sortable set of monomials and is isomorphic to a Hibi ring of
another finite distributive lattice. As a consequence it has a defining toric
ideal with a quadratic Gr\"obner basis, and its $h$-vector counts ascents in
certain standard Young tableaux. If instead the lattice has dimension $n>2$, we
show that the defining ideal has minimal generators of degree at least $n$.
