Definition:Subset Product

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:

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

This is called this 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 $$S \circ \left \{{a}\right\}$$.