Linear extensions and order-preserving poset partitions

We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We 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 ⩾ 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 classes of linear extensions of P.