Partition (abstract algebra)

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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]


  • 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]


  • 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:

See also


  1. The book (Maurin, in the references) uses a strict [ilmath]\subset[/ilmath] however take [1,2,3], [4,5] as a partition of 1-5, then [1],[2,3],[4,5] is a sub-partition, but [4,5][ilmath]\not\subset[/ilmath][4,5] - however [4,5][ilmath]\subseteq[/ilmath][4,5]


  1. Analysis - Part 1: Elements - Krzystof Maurin
  2. Alec's own work - equality is a difficult definition as the partition sets may be associated with different members in the indexing set, this could be important. Example, [1,2,3] and [4,5] partition 1-5, we could associate [ilmath]i\in I[/ilmath] with [1,2,3] and also [ilmath]j\in J[/ilmath] with [1,2,3] but there's no requirement for [ilmath]i=j[/ilmath] -it would be naive to consider these equal if [ilmath]i\ne j[/ilmath]