An extension of stone's partitioning theorem for ordered sets

Extract

In [1], A. H. Stone proved that for a cardinal number k ≥ 1 a set with a transitive relation can be partitioned into k cofinal subsets provided each element of the set has at least k successors. Using methods quite different from those of Stone, we show that for k ≥ ℵ₀ the same condition on successors guarantees that a set on which there are defined not more than k transitive relations can be partitioned into k sets each of which is cofinal with respect to each of the relations. We also show that such a partition exists even if some of the relations are not transitive as long as the non-transitive relations have no more than k elements.