Simplex and Polygon Equations

It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a \mixed order". We describe simplex equations (including the Yang{Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of \polygon equations" realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-ske- letons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N-simplex equation to the (N + 1)-gon equation, its dual, and a compatibility equation.