Chain algebras of finite distributive lattices

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