  Polygon equations generalize the prominent pentagon equation in very much the
same way as simplex equations generalize the famous Yang-Baxter equation. In
particular, they appeared as ''cocycle equations'' in Street's category theory
associated with oriented simplices. Whereas the $(N-1)$-simplex equation can be
regarded as a realization of the higher Bruhat order $B(N,N-2)$, the $N$-gon
equation is a realization of the higher Tamari order $T(N,N-2)$. The latter and
its dual $\tilde T(N,N-2)$, associated with which is the dual $N$-gon equation,
have been shown to arise as suborders of $B(N,N-2)$ via a ''three-color
decomposition''. There are two different reductions of $T(N,N-2)$ and $\tilde
T(N,N-2)$, to ${T(N-1,N-3)}$, respectively $\tilde T(N-1,N-3)$. In this work,
we explore the corresponding reductions of (dual) polygon equations, which lead
to relations between solutions of neighboring (dual) polygon equations. We also
elaborate (dual) polygon equations in this respect explicitly up to the octagon
equation.
