On partitioning ordered sets into cofinal subsets

Extract

Let (X, ≺) denote a non-empty (partially) ordered set, or more generally a non-empty set X with an arbitrary transitive relation ≺ on it. The relation ≺ will be fixed throughout what follows, so to simplify the notation we often write (X, ≺) as X. A successor of x ∈ X is an element y ∈ X such that x ≺ y; thus x may or may not be a successor of itself. As usual, a subset A ⊂ X is cofinal if each x ∈ X has a successor in A. A partition of X is a family of (pairwise) disjoint nonempty subsets of X whose union is X.