On Direct Product Decomposition of Partially Ordered Sets

Now we want to show that the proposition holds also when m, n are infinite. A partially ordered set is called connected if it cannot be decomposed into the sum of any two sets, and irreducible if it cannot be decomposed into the product of any two sets which have two or more elements. For instance directed sets are connected. LEMMA 1. If a partially ordered set S is connected, there exist for any two elements xo , x e S some finite number of xi, xi e S such that