Definition:Subset Product

Definition
Let $\left({S, \circ}\right)$ be an algebraic structure.

We can define an operation on the power set $\mathcal P \left({S}\right)$ as follows:


 * $\forall A, B \in \mathcal P \left({S}\right): A \circ_\mathcal P B = \left\{{a \circ b: a \in A, b \in B}\right\}$

This is called the operation induced on $\mathcal P \left({S}\right)$ by $\circ$, and $A \circ_\mathcal P B$ is called the subset product of $A$ and $B$.

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

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

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:


 * $a \circ S$ means the same as $\left \{{a}\right\} \circ S$;


 * $S \circ a$ means the same as $S \circ \left \{{a}\right\}$.

Setwise Addition
One of the most common examples of this construct is 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.

Alternative Terms
Also known as a complex.