Partition (abstract algebra)

Definition

Let [ilmath]I[/ilmath] be an arbitrary indexing set, to each element [ilmath]i\in I[/ilmath] we assign a set [ilmath]A_i[/ilmath] which is non-empty. If:

Then the family [ilmath]\{A_i\}_{i\in I} [/ilmath] is called[1] the partition of the set [ilmath]B[/ilmath] into classes [ilmath]A_i[/ilmath] for [ilmath]i\in I[/ilmath]

Equality

• Two partitions are identical (and can be swapped around as needed) if they have the same indexing family and the same set assigned to each element of the indexing family[2]

Subpartition

• We say the partition [ilmath]\{C_j\}_{j\in J} [/ilmath] is finer than [ilmath]\{A_i\}_{i\in I} [/ilmath] (or a subpartition of [ilmath]\{A_i\}_{i\in I} [/ilmath]) if we have: