Definition:Subset Product

Definition
Let $\struct {S, \circ}$ be an algebraic structure.

We can define an operation on the power set $\powerset S$ as follows:


 * $\forall A, B \in \powerset S: A \circ_\PP B = \set {a \circ b: a \in A, b \in B}$

This is called the operation induced on $\powerset S$ by $\circ$, and $A \circ_\PP B$ is called the subset product of $A$ and $B$.

It is usual to write $A \circ B$ for $A \circ_\PP B$.

If $A = \O$ or $B = \O$, then $A \circ B = \O$.

Subset Product with Singleton
When one of the subsets in a subset product is a singleton, we can (and often do) dispose of the set braces. Thus:

Also defined as
When the operation $\circ$ is in fact addition ($+$), the induced operation $+$ is then also called setwise addition.

When used, it is best to state explicitly that $+$ means setwise addition.

This is because some sources use $A + B$ also to denote set union and disjoint union.

Also known as
Also known as a complex.

As there are a number of conflicting definitions for the word complex in the context of group theory, it is highly recommended that the word not be used. on {{ProofWiki} in this context.

Also see

 * Definition:Direct Image Mapping
 * Definition:Minkowski Sum: for when $S$ is a vector space and the operation is vector addition