View source for Partition (abstract algebra)

==Definition==
Let {{M|I}} be an arbitrary [[Indexing set|indexing set]], to each element {{M|i\in I}} we assign a set {{M|A_i}} which is non-empty. If:
* {{M|1=A_i\cap A_j=\emptyset}} for {{M|i\ne j}} ([[Mutually disjoint sets|the {{M|A_i}} are mutually disjoint]])
* {{M|1=B=\bigcup_{i\in I}A_i}}
Then the ''family'' {{M|\{A_i\}_{i\in I} }} is called<ref name="Analysis">Analysis - Part 1: Elements - Krzystof Maurin</ref> ''the partition of the set {{M|B}} into classes {{M|A_i}} for {{M|i\in I}}''
==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<ref>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 {{M|i\in I}} with [1,2,3] and also {{M|j\in J}} with [1,2,3] but there's no requirement for {{M|1=i=j}} -it would be naive to consider these equal if {{M|i\ne j}}</ref>
==Subpartition==
* We say the partition {{M|\{C_j\}_{j\in J} }} is ''finer'' than {{M|\{A_i\}_{i\in I} }} (or a ''subpartition'' of {{M|\{A_i\}_{i\in I} }}) if we have:
** <math>\forall j\in J\exists i\in I[C_j\subseteq A_i]</math> (Or in [[Krzysztof Maurin's notation]] <mm>\bigwedge_{j\in J}\bigvee_{i\in I}C_j\subseteq A_i</mm>)<ref group="Note">The book uses a strict {{M|\subset}} 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]{{M|\not\subset}}[4,5] - however [4,5]{{M|\subseteq}}[4,5]</ref>
==See also==
* [[Partition (combinatorics)]]

==Notes==
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==References==
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{{Definition|Set Theory|Abstract Algebra}}