Difference between revisions of "Partition (abstract algebra)"

Latest revision as of 16:03, 18 June 2015

Definition

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

• $A_i\cap A_j=\emptyset$ for $i\ne j$ (the $A_i$ are mutually disjoint)
• $B=\bigcup_{i\in I}A_i$

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

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^[2]

Subpartition

• We say the partition $\{C_j\}_{j\in J}$ is finer than $\{A_i\}_{i\in I}$ (or a subpartition of $\{A_i\}_{i\in I}$) if we have:
  • $$\forall j\in J\exists i\in I[C_j\subseteq A_i]$$ (Or in Krzysztof Maurin's notation $$\bigwedge_{j\in J}\bigvee_{i\in I}C_j\subseteq A_i$$ )^{[Note 1]}

See also

• Partition (combinatorics)

Notes

1. Jump up ↑ The book (Maurin, in the references) uses a strict $\subset$ 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]$\not\subset$[4,5] - however [4,5]$\subseteq$[4,5]

References

1. Jump up ↑ Analysis - Part 1: Elements - Krzystof Maurin
2. Jump up ↑ 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 $i\in I$ with [1,2,3] and also $j\in J$ with [1,2,3] but there's no requirement for $i=j$ -it would be naive to consider these equal if $i\ne j$