Partition (abstract algebra)

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 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

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 $$\bigwedge_{j\in J}\bigvee_{i\in I}C_j\subseteq A_i$$ )

See also

• Partition (combinatorics)