  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-skeletons 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.
