A Theory of Partitions . Part I

Let us consider S1, S2. We introduce S1 b S2 and S2 c S1 as synonyms of S1 is finer than S2. The following four propositions are true: (3)1 ⋃ (S1 \{ / 0}) = ⋃ S1. (4) For all partitions P1, P2 of Y such that P1 c P2 and P2 c P1 holds P2 ⊆ P1. (5) For all partitions P1, P2 of Y such that P1 c P2 and P2 c P1 holds P1 = P2. (7)2 For all partitions P1, P2 of Y such that P1 c P2 holds P1 is coarser than P2.