Chain algebras of finite distributive lattices

We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. When the lattice is planar, the corresponding chain algebra is isomorphic to a Hibi ring. 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 will show that the defining ideal has minimal generators of degree at least $n$. We will also give a combinatorial interpretation of the Krull dimension of a chain algebra