A Theory of Partitions . Part I

For simplicity, we use the following convention: Y is a non empty set, P1, P2 are partitions of Y , A, B are subsets of Y , i is a natural number, x, y, x1, x2, z0 are sets, and X, V , d, t, S1, S2 are sets. The following proposition is true (1) If X ∈ P1 and V ∈ P1 and X ⊆ V, then X = V. Let us consider S1, S2. We introduce S1 ⋐ S2 and S2 ⋑ S1 as synonyms of S1 is finer than S2. We now state several propositions: (2) For every partition P1 of Y holds P1 ⋑ P1. (3) ⋃ (S1 \ {∅}) = ⋃ S1. (4) For all partitions P1, P2 of Y such that P1 ⋑ P2 and P2 ⋑ P1 holds P2 ⊆ P1. (5) For all partitions P1, P2 of Y such that P1 ⋑ P2 and P2 ⋑ P1 holds P1 = P2.