  We examine the lattice of all order congruences of a finite poset from the
viewpoint of combinatorial algebraic topology. We will prove that the order
complex of the lattice of all nontrivial order congruences (or order-preserving
partitions) of a finite $n$-element poset $P$ with $n\geq 3$ is homotopy
equivalent to a wedge of spheres of dimension $n-3$. If $P$ is connected, then
the number of spheres is equal to the number of linear extensions of $P$. In
general, the number of spheres is equal to the number of cyclic extensions of
$P$.
