Set dot product/multiplication I've been reading How To Prove It Second edition by Daniel J. Velleman, and on page 224, there's an end of sub-chapter question that defines an operation as follows:
Suppose $\mathcal{F}$ and $\mathcal{G}$ are partitions of a set $A$. We define a new family of sets $\mathcal{F} \cdot \mathcal{G}$ as follows:
$$ \mathcal{F} \cdot \mathcal{G} = \lbrace Z \in \mathscr{P}(A) : Z \neq \emptyset \wedge \exists X \in \mathcal{F} \, \exists Y \in \mathcal{G} (Z = X \cap Y) \rbrace$$
Whenever I search for set multiplication all I found is the Cartesian product. Is this "set dot product", so to speak, a one-off thing or is there other sources that uses this notation?
EDIT: The question asks to prove that $\mathcal{F} \cdot \mathcal{G}$ is a partition of $A \times B$, which I have proven. What I'm asking for is whether this notation is common within Set theory or if it's just a convenient notation the author is using within the textbook only.
 A: I've never seen the notation elsewhere, but it's not surprising, given the context.
Partitions on the set $A$ are the same as equivalence relations. If $\mathcal{F}$ is a partition on $A$, denote by $R(\mathcal{F})$ the associated equivalence relation, where
$$
a\mathrel{\rho(\mathcal{F})}b
\quad\text{stands for}\quad
\text{“there exists $X\in\mathcal{F}$ such that $a\in\mathcal{F}$ and $b\in\mathcal{F}\,$”}
$$
If $R$ is an equivalence relation, then the associated partition is the usual quotient set $A/R$.
Prove that $\rho(A/R)=R$ and that $A/\rho(\mathcal{F})=\mathcal{F}$.
If $R$ and $S$ are equivalence relations, then $R\cap S$ is an equivalence relation as well:
$$
a\mathrel{R\cap S}b
\qquad\text{if and only if}\qquad
a\mathrel{R}b\text{ and }a\mathrel{S} b
$$
Then it shouldn't be very difficult to show that
$$
\rho(\mathcal{F}\cdot\mathcal{G})=\rho(\mathcal{F})\cap \rho(\mathcal{G})
$$
Why did the book chose the product symbols? The set of partitions (or equivalently the set of equivalence relations) is a lattice with respect to set inclusion and this dot operation is the infimum of two objects (when using partitions) or the intersection (when using equivalence relations). So it's analogous to using product notation in Boolean algebras.
A: This is just a definition of a new partition made out of the two old partitions.
For example let $$ A=\{1,2,3,4,5,6,7,8,9,10\}$$
$$\mathcal{F}=\{\{1,2,3\},\{4,5,6\},\{7,8,9,10\}\}$$
$$\mathcal{G}=\{\{4,2,3\},\{7,5,6\},\{1,8,9,10\}\}$$
The $$\mathcal{F}\cdot\mathcal{G} = \{\{2,3\},\{5,6\},\{8,9,10\},\{1\},\{4\},\{7\}\}$$
